Research ArticleA New Approach in Regression Analysis forModeling Adsorption Isotherms
Dana D MarkoviT1 Branislava M LekiT2 Vladana N RajakoviT-OgnjanoviT2
Antonije E Onjia3 and Ljubinka V RajakoviT1
1 Faculty of Technology and Metallurgy University of Belgrade Karnegijeva 4 11000 Belgrade Serbia2 Faculty of Civil Engineering University of Belgrade Bulevar Kralja Aleksandra 73 11000 Belgrade Serbia3 Vinca Institute of Nuclear Sciences University of Belgrade PO Box 522 11001 Belgrade Serbia
Correspondence should be addressed to Dana D Markovic dmijovicgmailcom
Received 30 August 2013 Accepted 4 November 2013 Published 30 January 2014
Academic Editors G Ding C Kordulis and C Waterlot
Copyright copy 2014 Dana D Markovic et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Numerous regression approaches to isotherm parameters estimation appear in the literature The real insight into the propermodeling pattern can be achieved only by testing methods on a very big number of cases Experimentally it cannot be done ina reasonable time so the Monte Carlo simulation method was applied The objective of this paper is to introduce and comparenumerical approaches that involve different levels of knowledge about the noise structure of the analytical method used for initialand equilibrium concentration determination Six levels of homoscedastic noise and five types of heteroscedastic noise precisionmodels were considered Performance of the methods was statistically evaluated based on median percentage error and meanabsolute relative error in parameter estimates The present study showed a clear distinction between two cases When equilibriumexperiments are performed only once for the homoscedastic case the winning error function is ordinary least squares while forthe case of heteroscedastic noise the use of orthogonal distance regression or Margartrsquos percent standard deviation is suggested Itwas found that in case when experiments are repeated three times the simple method of weighted least squares performed as wellas more complicated orthogonal distance regression method
1 Introduction
Adsorption is a mass transfer process that plays a centralrole in potable water purification wastewater treatmentboth analytical and preparative chromatograph and differenttypes of chemical analyses as a technique for sample pre-concentration and speciation of analytes The predominantscientific basis for sorbent selection and design of an adsorp-tion system is the knowledge about equilibrium partitioningbetween two phases often expressed in the formof adsorptionisotherm Based on the isotherms the following importantfactors can be estimated capacity of the sorbent the methodof sorbent regeneration and the product purities [1] Addi-tionally transport behavior of environmentally significantreactive species is controlled by the sorption behavior of these
solutes to soil surfaces Adsorption isotherms are incorpo-rated into geochemical modeling software (such asMINEQLVisualMINTEQ and ChemEQL) in order to understand andpredict the mobility of the sorbed substances Determinationof the optimum isotherm equation and accurate estimates ofthe isotherm parameters are apparently important for all thementioned purposes
A frequently applied method for determining sin-gle solute adsorption isotherms is the conventional batchmethod based on mixing known amounts of adsorbent withsolutions of various initial concentrations (119862
119900) and measur-
ing the equilibrium concentrations (119862119890exp) Solving the mass
balance corresponding equilibrium loadings (119902119890exp) can be
simply calculated [2] Once pairs (119862119890exp 119902119890exp) are obtained
they are plotted and subjected to the fitting procedure
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 930879 17 pageshttpdxdoiorg1011552014930879
2 The Scientific World Journal
Candidate theoretical models are subsequently fitted to theexperimental data (parameters of themodel functions 119902
119890exp =119891(119862119890exp) are determined) and finally the best fitting model is
chosen to represent the experimental system Two differentsteps of the described procedure can be noticed firstly themethod used for obtaining parameter values and secondlythe method used for the isotherm selection
Themost commonly used empirical adsorption isothermmodels are the Langmuir and Freundlich isotherms [3]In the past decades the equations of these two parameterfunctions were routinely linearized and the parameters weredirectly obtained by linear regression The preferred oneamong the linearized equations would have been chosen bythe coefficient of determination (1198772) closer to one Nonlinearregression being an iterative procedure gained popularityin the era of microcomputers Parameter estimates in thismethod are obtained through minimization of the quadraticerror between experimental data 119902
119890exp and model outputs119902119890calc for all sample points Literature survey summarized ina review paper of Foo and Hameed [4] showed that besidesordinary least squares (OLS) researchers use many othererror functions namely hybrid fractional error function(HYBRD) Marquardtrsquos percent standard deviation (MPSD)average relative error (ARE) and sum of absolute errors(EABS) The coefficient of determination (1198772) root meansquared error (RMSE) all of the mentioned error functionsand sometimes Akaike information criterion [5] are calcu-lated to measure the goodness of fit and as a criterion for theselection of optimum isotherm
However El-Khaiary noticed that both dependent andindependent variables used for construction of isothermequations are affected by experimental errors and first usedthe method known as orthogonal distance regression (ODR)for the isotherm parameter estimation [6]
Having so many options open the researcher has todecide which one to apply The paper of El-Khaiary andMalash [7] contains the insightful analysis of the probablymost common error misuse of linearization Studies com-paring the accuracy of different error functions in predictingthe isotherm parameters and the optimum isotherm arepresented in the literature [8ndash10] An important limitationto these earlier studies is that they have been conductedprimarily on experimental data However there are a fewdrawbacks of such approach the true underlying isothermfunction is not known and the final conclusion about whichof the applied criteria has properly discovered it cannotbe drawn Also the values of the true parameters are notknown and it is not possible to decide which of the modelingapproaches achieved the most accurate parameter estimatesYet another problem is that even proving the validity ofsome method in just one particular case one cannot easilygeneralize the conclusion and suggest the use of the methodwithout sound background theory
Valuable information can be obtained when laboratoryexperiments are simulated through extensive Monte Carlocalculations This technique allows for both complete spec-ification and absolute control of all relevant parameters acondition that real experiments never approximate well An
advantage of Monte Carlo simulations is that they can berepeated thousands of times in a reasonable time and at verylow cost
This study was performed with the aim to answer thequestion which modeling approach should be applied inparticular case A few aspects of the problem were addressedDo the isotherm equation type and number of parametersmake the difference How do the properties of the analyticalmethod for the initial and equilibrium concentrations deter-mination affect the parameter estimation procedure Whatis the preferred method if one has some information aboutthe measurement error structure And what is the winningmethod in the case when the only available information isthe isotherm dataset that consists of 5ndash10 points with noreplication
The Monte Carlo technique was used as a tool to testthe differences between nonlinear and orthogonal distanceregressionmethods Tendencies withinmodeling approacheswere revealed on a large number of generated datasetsallowing the precision and accuracy of parameter estimatesto be determined by comparison with true parameter valuesFive isotherm models in the presence of five noise precisionmodels (NPMs)were analyzed by eightmodeling approachesThree levels of reality were distinguishedmdashtheoretical levelat the one side when the noise structure is exactly knownand the two experimental levels at the other side one in theabsence of data about noise structure and the second whenthe estimates of standard deviations could be obtained
As a result of this investigation a clear strategy for datareduction in the field of adsorption is presented
2 Theoretical Background
21 Adsorption Isotherms Over the years a wide varietyof equilibrium isotherm models have been formulated Ingeneral an adsorption isotherm is the relationship betweenquantity of the component retained on a solid phase (119902
119890)
and the remaining sorbate concentration in the fluid phase(119862119890) mathematically expressed as 119902
119890= 119891(119862
119890) The main
drawback of the isotherms is that the isotherm does notprovide automatically any information about the reactionsinvolved and mechanistic interpretations must be carefullyverified [11] Additionally they cannot take into account theeffect of ionic strength pH of the solution composition of themedia and temperature Despite these limitations isothermsare largely employed to describe sorption phenomena Gileset al [12] classified isotherms as ldquoCrdquo ldquoLrdquo ldquoHrdquo and ldquoSrdquo basedon the 4 main shapes of isotherms commonly observedAccording to this classification ldquoCrdquo isotherm is a line ofzero-origin and ldquoLrdquo and ldquoHrdquo are concave curves supportedby the fact that the ratio between the concentration of thecompound remaining in solution and adsorbed on the solidincreases when the solute concentration increases The ldquoHrdquotype isotherm is only a particular case of the ldquoLrdquo isothermwhere the initial slope is very high Progressive saturationof the solid is supported by these concave isotherms andtwo possibilities are distinguished the curve reaches a strictasymptotic plateau (the solid has a limited sorption capacity)
The Scientific World Journal 3
and the curve does not reach any plateau (the solid doesnot show clearly a limited sorption capacity) ldquoSrdquo type ofadsorption isotherm is sigmoidal-shaped and thus has got apoint of inflection It is always a result of at least two oppositemechanisms Compared to the ldquoLrdquo and ldquoHrdquo isotherms the ldquoSrdquoclass occurs less frequently [13] and it will not be addressedin this paper
From a mathematical point of view isotherm equationscan be grouped into rational power and transcendentalfunctions [3] Important for the convergence propertiesand computational difficulty is the number of parametersMost of the isotherms used for liquid-phase adsorptiondescription are two or three parameter isotherms while forthe adsorption of gases hybrid isotherms with significantlyhigher number of parameters are also present in the literature[14] The equations of five adsorption isotherms addressed inthis paper are listed in Table 1
They were chosen to be widely used and to repre-sent different types of mathematical functions (LangmuirRedlich-Peterson and Sips isotherms are rational functionsFreundlich isotherm is a power function and Jovanovicisotherm is a transcendental function) and different numberof parameters (Langmuir Freundlich and Jovanovic are two-parameter isotherms and Redlich-Peterson and Sips arethree-parameter isotherms) To avoid unnecessary repeti-tions detailed characteristics of the isotherms are not pre-sented Additional information can be found in the literature
22 Method of Least Squares Let independent data pairs(119909119894 119910119894) 119894 = 1 119899 be observed from the underlying
true values (119883119894119884119894) and accept the assumption that only
dependent variable 119910119894is affected by measurement error
119909119894= 119883119894
119910119894= 119884119894+ 120576119894
(1)
where 120576119894is additive zero mean white Gaussian noise
The noise is assumed to be homoscedastic with constantpopulation standard deviation 120590
120576 written in short notation
120576119894sim 119873(0 120590
120576) Although (1) is not absolutely satisfied in
practice and it is often the case that 119909119894have errors these
errors can be safely ignored if they are much smaller than thecorresponding errors in 119910
119894[6]
Assume the smooth function119884 = 119891(119883 120579) is a truemodelwhere 120579 isin 119877119901 is a vector of true parameters With more datapoints than parameters (119899 gt 119901) it is not possible to solvethe model and calculate the values of the true parametersInstead the question is how to obtain the best compromiseso that the model predictions (119910
119894) are on the whole as close as
possible to the observed data values Closeness for any singleobservation may be measured by the vertical distance (120595
119894)
from the data point to the fitted curve
119910119894= 119910119894+ 120595119894 (2)
Closeness averaged over the entire data set is often measuredby the sum of the squares of the individual distances Any
point = (1205791 1205792 120579
119901) in Θ = 120579 = (120579
1 1205792 120579
119901) isin R119901
120579119894gt 0 119894 = 1 119901 which minimizes the functional
ErrFun () =119899
sum
119894=1
1199082
119894(119891 (119909
119894 ) minus 119910
119894)2
(3)
where 119908119894are data weights and equation for the fitted curve
reads 119910119894= 119891(119909
119894 ) is called the least squares estimate of the
unknown parameters if it exists [20] Condition 120579119894gt 0 119894 =
1 119901 in the definition of the set Θ is a specific feature ofthe modeling adsorption isotherms and meets the criterionfor the isotherm to be positive increasing and concave onthe set [0infin) namely to be ldquoLrdquo or ldquoHrdquo type isotherm
221 Weighting Schemes in the Method of Least SquaresIn the method of OLS the observations are assumed tobe homoscedastic and all of the points are assigned equalweights 119908
119894= 1 119894 = 1 119899 In the absence of more
complete information it is commonly accepted that uniformweighting is satisfactory and OLS are widely used in modelfitting [21] If the assumption of constant standard deviationsof measurement errors is relaxed the heteroscedasticity ischaracterized by an 119899 element vector 120590
120576= (1205901205761 1205901205762 120590
120576119899)
where each 120590120576119894is population standard deviation of the noise
at 119910119894 120576119894sim 119873(0 120590
120576119894) Weights calculated by
119908119894=1
120590120576119894
(4)
are introduced into (3) in order to account for inconstantvariance and the method is referred to as weighted leastsquares (WLS) or sometimes ldquochi-square fittingrdquo [22] Theassumption that the weights are known exactly is not valid inreal applications so estimated weights must be used instead[23]
Ideally observation weights should be estimated accord-ing to individual estimates of measurement error such that119908119894= 1119904119889119910
119894 where 119904119889119910
119894is the standard deviation of 119894th
measurement These are called instrumental weightsWhen individual error estimates are unavailable other
empirical weights may provide a simple approximation ofstandard deviation For the peculiar case of heteroscedas-ticity important in many analytical methods relative stan-dard deviations are reasonably constant over a considerabledynamic rangeThus 120590
120576119894is proportional to 119910
119894and the weights
can be estimated as 119908119894= 1119910
119894
However the error structure in real data usually liessomewhere on a continuous between a constant absoluteerror (homoscedastic) at one extreme and a constant percent-age error at the other Between these two there is an error forwhich the standard deviation is proportional to the squareroot of the expected value119908
119894= 1119910
119894
05This type of weights iscalled Poisson weights or hybrid weights and they should beappliedwhen the shot noise is present Shot noise is dominantwhen a finite number of particles that carry energy (ionselectrons and photons) are counted at the detector part of theinstrumentThe characteristic expressions for each weightingtype are presented in Table 2
ISOFIT a software package for fitting sorption isothermsto experimental data by weighted least squares supports
4 The Scientific World Journal
Table 1 Adsorption isotherm models
No Type Type offunction Nonlinear form Linear form
Type of weights ExpressionAbsolute weights 1Poisson weights 1119910
05
119894
Assumption of constant percentage error 1119910119894
Instrumental weights 1sd119910119894
three alternatives uniform weighting sorbed relative whereweights are inversely proportional to sorbed concentrationsand solute relative where weights are inversely proportionalto measured solute concentrations [24]
23 Orthogonal Distance Regression Methods In a moregeneral situation considerable errors can occur in bothvariables It is stated that if the errors in 119909
119894are greater than
one-tenth of the errors in 119910119894 then the overall error is signif-
icantly increased Moreover the regression parameters andtheir confidence intervals are then biased using (ordinary)weighted least squares [25]
Let the considerable error be also present in the measure-ments of the independent variable
119909119894= 119883119894+ 120575119894 (5)
where 120575119894 sim 119873(0 120590120575119894)
Again the model will not fit the observed data points(119909119894 119910119894) 119894 = 1 119899 exactly so the corresponding set of points
(119909119894 119910119894) 119894 = 1 119899 that do fit the model exactly and that are
at the same time the closest to experimental data points is tobe considered For each data point the value of independentvariable 119909
119894is expressed introducing an error term 120593
119894
119909119894= 119909119894+ 120593119894 (6)
The values 119910119894are predicted by the model function
119910119894= 119891 (119909
119894+ 120593119894 ) (7)
A reasonable way to estimate the unknown parameters in thiscase is to minimize the weighted sum of squares of all errorsby minimizing the functional
ErrFun ( ) =119899
sum
119894=1
1199082
119894(120595119894
2
+ 1198892
1198941205932
119894) (8)
on the set Θ times Φ where 119908119894and 119889
119894are data weights in the 119910
and 119909 directions respectively and
Φ = 120593 = (1205931 1205932 120593
119899) isin R119899 119909
119894+ 120593119894ge 0 119894 = 1 119899
(9)
Commonly (8) is expressed in its expanded form where thedifference between calculated and experimentally observedvalue of 119910 is emphasized
ErrFun ( ) =119899
sum
119894= 1
1199082
119894[(119891 (119909
119894+ 120593119894 ) minus 119910
119894)2
+ 1198892
1198941205932
119894]
(10)
This approach is known as errors in variables or orthogonaldistance regression or total least squares Condition 119909
119894+
120593119894ge 0 119894 = 1 119899 in the defining relation (9) is set in
order to meet the natural condition that concentration is anonnegative value
231Weighting Schemes in theMethod of Orthogonal DistanceRegression In orthogonal distance regression analysis ofsorption data units of the variables on the axes are notthe same It is necessary to introduce weights as constantsselected to scale each type of variable 119910
119894or 119909119894 This is done
in order to put the model errors (120595119894and 120593
119894) on a comparable
basis so it will be meaningful to add them all togetherinto the sum of error function in (8) Typically weights arechosen as estimates of the population standard deviation ofthe experimental measurements of each variable type 119908
119894=
1119904119889119910119894and 119908
119894119889119894= 1119904119889119909
119894 Another way is to assign weights
to be proportional to the inverse of experimental values119908119894=
1119910119894and 119908
119894119889119894= 1119909
119894 The effect of such weights is that at
the same time heteroscedasticity is accounted for and scaledmodel errors in 119910 and 119909 direction are dimensionless
The Scientific World Journal 5
In Figure 1 differences between different OLS and orthog-onal distance regressionmethodswith andwithout weightingare presented
Geometrically if the data pairs (119909119894 119910119894) and the curve 119910 =
119891(119909 ) are presented in the Cartesian coordinates in ordinaryleast squares minimization of the error function correspondsto minimization of the shortest distances from data pointsto a line in a direction that is parallel to the vertical axisFigure 1(a) is a standard geometric illustration of the leastsquares method Instead of the vertical offsets the shortestdistances from points to the line are considered in themethod of orthogonal distance regression If the data arehomoscedastic and the units of 119909 and 119910 are the same allthe weights 119908
119894and 119889
119894are equal to one Equation (8) is then
simplified to a formula that possesses a meaning of the sumof the areas of the circles shown in Figure 1(b)
ErrFun ( ) =119899
sum
119894=1
[(119910119894minus 119910119894)2
+ (119909119894minus 119909119894)2
] (11)
In this case the radii of these circles are equal to distancesbetween the points (119909
119894 119910119894) and the fitting line Put in other
words the fitting line is a tangent line to all circlesGeometrical representation of a case when 119909 and 119910 are
variables that do not have the same units or the data isheteroscedastic is presented in Figure 1(c) Weights are intro-duced and half axes of the ellipses in Figure 1(c) correspondto the combined measure of the distance expressed in (8)While the global minimum of this error function is uniquethis kind of straightforward geometrical representation is nolonger meaningful
Orthogonal distance regression methods have been usedin the fields of science such as economy [26] automaticcontrol [27] and pharmacology [28] and a significant workhas been done for the development of stable and efficientalgorithm for ODR estimation of parameters
3 Materials and Methods
31 Numerical Experiments Numerical experiments weredesigned to be as close as possible representation of a typicalexperimental setup in adsorption studies It was adoptedthat batch experiments are performed in laboratory bakerscontaining mass of sorbent 119898 and volume 119881 of sorbatesolution Initial concentrations of sorbate solutions (119862
119900119894true)are chosen to be 05 10 50 100 500 and 1000 All units areignored since they are irrelevant Further on it was assumedthat the theoretical adsorption isotherm expressed in termsof its true parameters is exactly matching the adsorptionprocess Values of the true parameters were arbitrarily set toget the operative expression 119902
119890true = 119891(119862119890true) where 119862119890trueis errorless equilibrium sorbate concentration and 119902
119890true iserrorless equilibrium sorbate loading At the same time massbalance expressed in (12) is satisfied
119902119890119894true =
(119862119900119894true minus 119862119890119894true)
119898119881 (12)
The true equilibrium concentration is then calculated solvingthe equation
119891 (119862119890119894true 120579) minus
(119862119900119894true minus 119862119890119894true)
119898119881 = 0 (13)
It is assumed that simple univariate chemical measurementsystem with additive zero mean white Gaussian measure-ment noise is used as an analytical tool to determine 119862
119900119894trueand 119862
119890119894true Thus random noise (120575119894119900 sim 119873(0 120590
119900
120575119894) and
120575119894119890 sim 119873(0 120590120575
119894119890) resp) was added to these values to obtain
The rest of the procedure was identical as if the experimentswere performed in laboratory The equilibrium sorbent load-ing was calculated from (12) and the collected data wassubjected to fitting routines
Flow chart of laboratory experiment and a matchingnumerical experiment is presented in Figure 2
32 Noise Precision Models Since a wide variety of sub-stances (toxic metals organic pollutants etc) are in focusof adsorption research community also a wide variety ofanalytical techniques are used for initial and equilibrium con-centrations determination Accordingly the measurementerrors they introduce differ in type and magnitude Thereare different mathematical models named noise precisionmodels (NPMs) that have been proposed to estimate thechange of analytical precision as a function of the analyteconcentration List of such models for specific analyticalmethods together with explanations of error sources can befound in literature [23] In this paper the NPMs were chosento be the simplest physically plausible [29] Firstly the sixdifferent magnitudes of homoscedastic noise were used (H1ndashH6) Noise population standard deviations were from verylow (001 002) and medium (005 01) to high (02 04) Thenoise of the data was therefore between 001 and 04 of thedata range Secondly the five types of heteroscedastic noiselinear (Lin) quadratic (Quad) hockey stick (HS) constantrelative standard deviation of 5 (RSD5) and constantrelative standard deviation of 10 (RSD10) were involvedExpressions of the NPMs used in the study and their relevantparameters are listed in Table 3
33 Methods for Fitting Adsorption Isotherms Althoughthere are some experiments where it is reasonable to assumethat one variable (119909) is largely free from errors such anassumption is manifestly not true in cases of adsorptionisotherms The value on the 119909-axis is an equilibrium con-centration 119862
119890exp which is determined by chemical analysisand thus inevitably affected by the measurement error Equi-librium loading 119902
119890exp (which is on the 119910-axis) is calculatedfrom the equilibrium concentration and as a result an errorin this concentration appears in both coordinates Particularattention must be given to equations in which one variable
6 The Scientific World Journal
x
y(xi yi)
(xi f(xi ))
(a)
x
y(xi yi)
(xi + 120593i f(xi + 120593i ))
(b)
x
y
(xi yi)
(xi + 120593i f(xi + 120593i ))
(c)
Figure 1 Geometric illustration of differences among different regression methods (a) OLS without weighting (b) orthogonal distanceregression without weighting and (c) orthogonal distance regression with weighting
is involved on both sides since the independence of errorsis not fulfilled [30]The use of orthogonal distance regressionmodeling procedurewould be statistically correct in this case
Since this study is based on simulated data populationstandard deviations of measurement errors are known and
(10) can be transformed into theoretical orthogonal distanceregression criterion (TODR) introducing the relations
119908119894=1
120590120576119894
119908119894119889119894=1
120590120575119894
(15)
for the weights on the 119910- and 119909-axes respectively It can benoticed that the population standard deviation of the erroron the 119909-axis is actually the population standard deviation ofthe equilibrium concentration
120590120575119894= 120590119890
120575119894 (16)
For the 119910 axis 120590120576119894is calculated based on (12) According to
the error propagation law
1205902
120576119894= 1198962
[(120590119890
120575119894)2
+ (120590119900
120575119894)2
] (17)
where 119896 = 119881119898 it is assumed that only the error of the119862119900exp and 119862119890exp determination is significant while mass and
Figure 2 Flow chart illustrating the steps in adsorption equilibrium experiment and a matching numerical experiment
Final formulation of the TODR error function which isminimized in case of theoretical fitting of adsorption data ispresented as
TODR =119899
sum
119894=1
[(119902119890119894exp minus 119902119890119894cal
120590120576119894
)
2
+ (119862119890119894exp minus 119862119890119894cal
120590119890
120575119894
)
2
]
(18)
Although TODR cannot be used outside the theoreticaldomain it was included in this study to serve as a golden stan-dard It is expected to represent the best possible results thatcan be achieved with the certain observations in possession
A typical isotherm data set in the experimental domainconsists of 5ndash10 points Very often researches perform theirexperiments in triplicate [31ndash33] but in cases when thereagents are expensive or toxic there are no replications Bythese two different methods different levels of data qualityare obtained The objective of this paper was to take both ofthese cases into consideration
Data obtained in only one numerical experiment (match-ing the case when the laboratory experiments are performedwith no replication) were fitted by the use of four errorfunctions OLS ODRMPSD andHYBRDTheir expressionsare presented in Table 4 Least squares fitting of the linearized
8 The Scientific World Journal
Table 4 Definitions of error functions in cases when there are no replicated concentration measurements
+ (119862119890119894exp minus 119862119890119894cal
119862119890119894exp
)
2
]
]
data (LIN) was additionally applied only on the Langmuirand Freundlich equations
Formulae of the error functions in Table 4 are a bitdifferent from the ones reviewed in [4]The key reason is thatthe number of data points is constant all the time and therewill be no isotherm ranking based on this error functionsso the constants n-p were removed for the sake of simplicityParameter estimates obtained with these slightly modifiederror functions are the same because the multiplication oferror function with a constant nonzero value does not affectthe position of a global minimum
OLS MPSD and HYBRD are basically least squaresmethods with different types of weights included OLS is theapproach with all weights equal to one In case of MPSDassumption of constant percentage error is accepted andweighting by the equilibrium loading is applied For theHYBRD error functions weights are of the Poisson typeODR abbreviation in this context is used for the orthogonaldistance regression analog of the MPSD Assumption ofconstant percentage error is accepted for both of the axesand the weights are 1119902
119890exp and 1119862119890exp for the119910- and 119909-axesrespectively
The second group of calculations matched the case whenlaboratory experiments are performed in triplicate Means ofequilibrium sorbent loading (119902
119890119894exp) and equilibrium sorbateconcentration (119862
119890119894exp) from three subsequent numericalexperiments were passed to the following error functionsexperimental weighted orthogonal distance regression
(E3WODR) triplicate orthogonal distance regression(E3ODR) and weighted least squares (WLS) Definitionsof error functions for replicated measurements are listed inTable 5
It is important to say that E3WODR is the experimentalrealization of TODR Estimates of standard deviation of thevariables on the 119910- and 119909-axes are calculated as described inSection 231 incorporating the following equations
119904119889119910119894= radic3
sum
119895=1
(119902119890119894exp minus 119902119890119894exp)
2
119895
2 (19)
119904119889119909119894= radic3
sum
119895=1
(119862119890119894exp minus 119862119890119894exp)
2
119895
2 (20)
Weighting in the E3ODRmethod is based on themean valuesof equilibrium sorbent loading and equilibrium sorbateconcentration
119908119894=
1
119902119890119894exp
119908119894119889119894=
1
119862119890119894exp
(21)
For the WLS method instrumental weights are calculatedbased on (19)
The Scientific World Journal 9
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Candidate theoretical models are subsequently fitted to theexperimental data (parameters of themodel functions 119902
119890exp =119891(119862119890exp) are determined) and finally the best fitting model is
chosen to represent the experimental system Two differentsteps of the described procedure can be noticed firstly themethod used for obtaining parameter values and secondlythe method used for the isotherm selection
Themost commonly used empirical adsorption isothermmodels are the Langmuir and Freundlich isotherms [3]In the past decades the equations of these two parameterfunctions were routinely linearized and the parameters weredirectly obtained by linear regression The preferred oneamong the linearized equations would have been chosen bythe coefficient of determination (1198772) closer to one Nonlinearregression being an iterative procedure gained popularityin the era of microcomputers Parameter estimates in thismethod are obtained through minimization of the quadraticerror between experimental data 119902
119890exp and model outputs119902119890calc for all sample points Literature survey summarized ina review paper of Foo and Hameed [4] showed that besidesordinary least squares (OLS) researchers use many othererror functions namely hybrid fractional error function(HYBRD) Marquardtrsquos percent standard deviation (MPSD)average relative error (ARE) and sum of absolute errors(EABS) The coefficient of determination (1198772) root meansquared error (RMSE) all of the mentioned error functionsand sometimes Akaike information criterion [5] are calcu-lated to measure the goodness of fit and as a criterion for theselection of optimum isotherm
However El-Khaiary noticed that both dependent andindependent variables used for construction of isothermequations are affected by experimental errors and first usedthe method known as orthogonal distance regression (ODR)for the isotherm parameter estimation [6]
Having so many options open the researcher has todecide which one to apply The paper of El-Khaiary andMalash [7] contains the insightful analysis of the probablymost common error misuse of linearization Studies com-paring the accuracy of different error functions in predictingthe isotherm parameters and the optimum isotherm arepresented in the literature [8ndash10] An important limitationto these earlier studies is that they have been conductedprimarily on experimental data However there are a fewdrawbacks of such approach the true underlying isothermfunction is not known and the final conclusion about whichof the applied criteria has properly discovered it cannotbe drawn Also the values of the true parameters are notknown and it is not possible to decide which of the modelingapproaches achieved the most accurate parameter estimatesYet another problem is that even proving the validity ofsome method in just one particular case one cannot easilygeneralize the conclusion and suggest the use of the methodwithout sound background theory
Valuable information can be obtained when laboratoryexperiments are simulated through extensive Monte Carlocalculations This technique allows for both complete spec-ification and absolute control of all relevant parameters acondition that real experiments never approximate well An
advantage of Monte Carlo simulations is that they can berepeated thousands of times in a reasonable time and at verylow cost
This study was performed with the aim to answer thequestion which modeling approach should be applied inparticular case A few aspects of the problem were addressedDo the isotherm equation type and number of parametersmake the difference How do the properties of the analyticalmethod for the initial and equilibrium concentrations deter-mination affect the parameter estimation procedure Whatis the preferred method if one has some information aboutthe measurement error structure And what is the winningmethod in the case when the only available information isthe isotherm dataset that consists of 5ndash10 points with noreplication
The Monte Carlo technique was used as a tool to testthe differences between nonlinear and orthogonal distanceregressionmethods Tendencies withinmodeling approacheswere revealed on a large number of generated datasetsallowing the precision and accuracy of parameter estimatesto be determined by comparison with true parameter valuesFive isotherm models in the presence of five noise precisionmodels (NPMs)were analyzed by eightmodeling approachesThree levels of reality were distinguishedmdashtheoretical levelat the one side when the noise structure is exactly knownand the two experimental levels at the other side one in theabsence of data about noise structure and the second whenthe estimates of standard deviations could be obtained
As a result of this investigation a clear strategy for datareduction in the field of adsorption is presented
2 Theoretical Background
21 Adsorption Isotherms Over the years a wide varietyof equilibrium isotherm models have been formulated Ingeneral an adsorption isotherm is the relationship betweenquantity of the component retained on a solid phase (119902
119890)
and the remaining sorbate concentration in the fluid phase(119862119890) mathematically expressed as 119902
119890= 119891(119862
119890) The main
drawback of the isotherms is that the isotherm does notprovide automatically any information about the reactionsinvolved and mechanistic interpretations must be carefullyverified [11] Additionally they cannot take into account theeffect of ionic strength pH of the solution composition of themedia and temperature Despite these limitations isothermsare largely employed to describe sorption phenomena Gileset al [12] classified isotherms as ldquoCrdquo ldquoLrdquo ldquoHrdquo and ldquoSrdquo basedon the 4 main shapes of isotherms commonly observedAccording to this classification ldquoCrdquo isotherm is a line ofzero-origin and ldquoLrdquo and ldquoHrdquo are concave curves supportedby the fact that the ratio between the concentration of thecompound remaining in solution and adsorbed on the solidincreases when the solute concentration increases The ldquoHrdquotype isotherm is only a particular case of the ldquoLrdquo isothermwhere the initial slope is very high Progressive saturationof the solid is supported by these concave isotherms andtwo possibilities are distinguished the curve reaches a strictasymptotic plateau (the solid has a limited sorption capacity)
The Scientific World Journal 3
and the curve does not reach any plateau (the solid doesnot show clearly a limited sorption capacity) ldquoSrdquo type ofadsorption isotherm is sigmoidal-shaped and thus has got apoint of inflection It is always a result of at least two oppositemechanisms Compared to the ldquoLrdquo and ldquoHrdquo isotherms the ldquoSrdquoclass occurs less frequently [13] and it will not be addressedin this paper
From a mathematical point of view isotherm equationscan be grouped into rational power and transcendentalfunctions [3] Important for the convergence propertiesand computational difficulty is the number of parametersMost of the isotherms used for liquid-phase adsorptiondescription are two or three parameter isotherms while forthe adsorption of gases hybrid isotherms with significantlyhigher number of parameters are also present in the literature[14] The equations of five adsorption isotherms addressed inthis paper are listed in Table 1
They were chosen to be widely used and to repre-sent different types of mathematical functions (LangmuirRedlich-Peterson and Sips isotherms are rational functionsFreundlich isotherm is a power function and Jovanovicisotherm is a transcendental function) and different numberof parameters (Langmuir Freundlich and Jovanovic are two-parameter isotherms and Redlich-Peterson and Sips arethree-parameter isotherms) To avoid unnecessary repeti-tions detailed characteristics of the isotherms are not pre-sented Additional information can be found in the literature
22 Method of Least Squares Let independent data pairs(119909119894 119910119894) 119894 = 1 119899 be observed from the underlying
true values (119883119894119884119894) and accept the assumption that only
dependent variable 119910119894is affected by measurement error
119909119894= 119883119894
119910119894= 119884119894+ 120576119894
(1)
where 120576119894is additive zero mean white Gaussian noise
The noise is assumed to be homoscedastic with constantpopulation standard deviation 120590
120576 written in short notation
120576119894sim 119873(0 120590
120576) Although (1) is not absolutely satisfied in
practice and it is often the case that 119909119894have errors these
errors can be safely ignored if they are much smaller than thecorresponding errors in 119910
119894[6]
Assume the smooth function119884 = 119891(119883 120579) is a truemodelwhere 120579 isin 119877119901 is a vector of true parameters With more datapoints than parameters (119899 gt 119901) it is not possible to solvethe model and calculate the values of the true parametersInstead the question is how to obtain the best compromiseso that the model predictions (119910
119894) are on the whole as close as
possible to the observed data values Closeness for any singleobservation may be measured by the vertical distance (120595
119894)
from the data point to the fitted curve
119910119894= 119910119894+ 120595119894 (2)
Closeness averaged over the entire data set is often measuredby the sum of the squares of the individual distances Any
point = (1205791 1205792 120579
119901) in Θ = 120579 = (120579
1 1205792 120579
119901) isin R119901
120579119894gt 0 119894 = 1 119901 which minimizes the functional
ErrFun () =119899
sum
119894=1
1199082
119894(119891 (119909
119894 ) minus 119910
119894)2
(3)
where 119908119894are data weights and equation for the fitted curve
reads 119910119894= 119891(119909
119894 ) is called the least squares estimate of the
unknown parameters if it exists [20] Condition 120579119894gt 0 119894 =
1 119901 in the definition of the set Θ is a specific feature ofthe modeling adsorption isotherms and meets the criterionfor the isotherm to be positive increasing and concave onthe set [0infin) namely to be ldquoLrdquo or ldquoHrdquo type isotherm
221 Weighting Schemes in the Method of Least SquaresIn the method of OLS the observations are assumed tobe homoscedastic and all of the points are assigned equalweights 119908
119894= 1 119894 = 1 119899 In the absence of more
complete information it is commonly accepted that uniformweighting is satisfactory and OLS are widely used in modelfitting [21] If the assumption of constant standard deviationsof measurement errors is relaxed the heteroscedasticity ischaracterized by an 119899 element vector 120590
120576= (1205901205761 1205901205762 120590
120576119899)
where each 120590120576119894is population standard deviation of the noise
at 119910119894 120576119894sim 119873(0 120590
120576119894) Weights calculated by
119908119894=1
120590120576119894
(4)
are introduced into (3) in order to account for inconstantvariance and the method is referred to as weighted leastsquares (WLS) or sometimes ldquochi-square fittingrdquo [22] Theassumption that the weights are known exactly is not valid inreal applications so estimated weights must be used instead[23]
Ideally observation weights should be estimated accord-ing to individual estimates of measurement error such that119908119894= 1119904119889119910
119894 where 119904119889119910
119894is the standard deviation of 119894th
measurement These are called instrumental weightsWhen individual error estimates are unavailable other
empirical weights may provide a simple approximation ofstandard deviation For the peculiar case of heteroscedas-ticity important in many analytical methods relative stan-dard deviations are reasonably constant over a considerabledynamic rangeThus 120590
120576119894is proportional to 119910
119894and the weights
can be estimated as 119908119894= 1119910
119894
However the error structure in real data usually liessomewhere on a continuous between a constant absoluteerror (homoscedastic) at one extreme and a constant percent-age error at the other Between these two there is an error forwhich the standard deviation is proportional to the squareroot of the expected value119908
119894= 1119910
119894
05This type of weights iscalled Poisson weights or hybrid weights and they should beappliedwhen the shot noise is present Shot noise is dominantwhen a finite number of particles that carry energy (ionselectrons and photons) are counted at the detector part of theinstrumentThe characteristic expressions for each weightingtype are presented in Table 2
ISOFIT a software package for fitting sorption isothermsto experimental data by weighted least squares supports
4 The Scientific World Journal
Table 1 Adsorption isotherm models
No Type Type offunction Nonlinear form Linear form
Type of weights ExpressionAbsolute weights 1Poisson weights 1119910
05
119894
Assumption of constant percentage error 1119910119894
Instrumental weights 1sd119910119894
three alternatives uniform weighting sorbed relative whereweights are inversely proportional to sorbed concentrationsand solute relative where weights are inversely proportionalto measured solute concentrations [24]
23 Orthogonal Distance Regression Methods In a moregeneral situation considerable errors can occur in bothvariables It is stated that if the errors in 119909
119894are greater than
one-tenth of the errors in 119910119894 then the overall error is signif-
icantly increased Moreover the regression parameters andtheir confidence intervals are then biased using (ordinary)weighted least squares [25]
Let the considerable error be also present in the measure-ments of the independent variable
119909119894= 119883119894+ 120575119894 (5)
where 120575119894 sim 119873(0 120590120575119894)
Again the model will not fit the observed data points(119909119894 119910119894) 119894 = 1 119899 exactly so the corresponding set of points
(119909119894 119910119894) 119894 = 1 119899 that do fit the model exactly and that are
at the same time the closest to experimental data points is tobe considered For each data point the value of independentvariable 119909
119894is expressed introducing an error term 120593
119894
119909119894= 119909119894+ 120593119894 (6)
The values 119910119894are predicted by the model function
119910119894= 119891 (119909
119894+ 120593119894 ) (7)
A reasonable way to estimate the unknown parameters in thiscase is to minimize the weighted sum of squares of all errorsby minimizing the functional
ErrFun ( ) =119899
sum
119894=1
1199082
119894(120595119894
2
+ 1198892
1198941205932
119894) (8)
on the set Θ times Φ where 119908119894and 119889
119894are data weights in the 119910
and 119909 directions respectively and
Φ = 120593 = (1205931 1205932 120593
119899) isin R119899 119909
119894+ 120593119894ge 0 119894 = 1 119899
(9)
Commonly (8) is expressed in its expanded form where thedifference between calculated and experimentally observedvalue of 119910 is emphasized
ErrFun ( ) =119899
sum
119894= 1
1199082
119894[(119891 (119909
119894+ 120593119894 ) minus 119910
119894)2
+ 1198892
1198941205932
119894]
(10)
This approach is known as errors in variables or orthogonaldistance regression or total least squares Condition 119909
119894+
120593119894ge 0 119894 = 1 119899 in the defining relation (9) is set in
order to meet the natural condition that concentration is anonnegative value
231Weighting Schemes in theMethod of Orthogonal DistanceRegression In orthogonal distance regression analysis ofsorption data units of the variables on the axes are notthe same It is necessary to introduce weights as constantsselected to scale each type of variable 119910
119894or 119909119894 This is done
in order to put the model errors (120595119894and 120593
119894) on a comparable
basis so it will be meaningful to add them all togetherinto the sum of error function in (8) Typically weights arechosen as estimates of the population standard deviation ofthe experimental measurements of each variable type 119908
119894=
1119904119889119910119894and 119908
119894119889119894= 1119904119889119909
119894 Another way is to assign weights
to be proportional to the inverse of experimental values119908119894=
1119910119894and 119908
119894119889119894= 1119909
119894 The effect of such weights is that at
the same time heteroscedasticity is accounted for and scaledmodel errors in 119910 and 119909 direction are dimensionless
The Scientific World Journal 5
In Figure 1 differences between different OLS and orthog-onal distance regressionmethodswith andwithout weightingare presented
Geometrically if the data pairs (119909119894 119910119894) and the curve 119910 =
119891(119909 ) are presented in the Cartesian coordinates in ordinaryleast squares minimization of the error function correspondsto minimization of the shortest distances from data pointsto a line in a direction that is parallel to the vertical axisFigure 1(a) is a standard geometric illustration of the leastsquares method Instead of the vertical offsets the shortestdistances from points to the line are considered in themethod of orthogonal distance regression If the data arehomoscedastic and the units of 119909 and 119910 are the same allthe weights 119908
119894and 119889
119894are equal to one Equation (8) is then
simplified to a formula that possesses a meaning of the sumof the areas of the circles shown in Figure 1(b)
ErrFun ( ) =119899
sum
119894=1
[(119910119894minus 119910119894)2
+ (119909119894minus 119909119894)2
] (11)
In this case the radii of these circles are equal to distancesbetween the points (119909
119894 119910119894) and the fitting line Put in other
words the fitting line is a tangent line to all circlesGeometrical representation of a case when 119909 and 119910 are
variables that do not have the same units or the data isheteroscedastic is presented in Figure 1(c) Weights are intro-duced and half axes of the ellipses in Figure 1(c) correspondto the combined measure of the distance expressed in (8)While the global minimum of this error function is uniquethis kind of straightforward geometrical representation is nolonger meaningful
Orthogonal distance regression methods have been usedin the fields of science such as economy [26] automaticcontrol [27] and pharmacology [28] and a significant workhas been done for the development of stable and efficientalgorithm for ODR estimation of parameters
3 Materials and Methods
31 Numerical Experiments Numerical experiments weredesigned to be as close as possible representation of a typicalexperimental setup in adsorption studies It was adoptedthat batch experiments are performed in laboratory bakerscontaining mass of sorbent 119898 and volume 119881 of sorbatesolution Initial concentrations of sorbate solutions (119862
119900119894true)are chosen to be 05 10 50 100 500 and 1000 All units areignored since they are irrelevant Further on it was assumedthat the theoretical adsorption isotherm expressed in termsof its true parameters is exactly matching the adsorptionprocess Values of the true parameters were arbitrarily set toget the operative expression 119902
119890true = 119891(119862119890true) where 119862119890trueis errorless equilibrium sorbate concentration and 119902
119890true iserrorless equilibrium sorbate loading At the same time massbalance expressed in (12) is satisfied
119902119890119894true =
(119862119900119894true minus 119862119890119894true)
119898119881 (12)
The true equilibrium concentration is then calculated solvingthe equation
119891 (119862119890119894true 120579) minus
(119862119900119894true minus 119862119890119894true)
119898119881 = 0 (13)
It is assumed that simple univariate chemical measurementsystem with additive zero mean white Gaussian measure-ment noise is used as an analytical tool to determine 119862
119900119894trueand 119862
119890119894true Thus random noise (120575119894119900 sim 119873(0 120590
119900
120575119894) and
120575119894119890 sim 119873(0 120590120575
119894119890) resp) was added to these values to obtain
The rest of the procedure was identical as if the experimentswere performed in laboratory The equilibrium sorbent load-ing was calculated from (12) and the collected data wassubjected to fitting routines
Flow chart of laboratory experiment and a matchingnumerical experiment is presented in Figure 2
32 Noise Precision Models Since a wide variety of sub-stances (toxic metals organic pollutants etc) are in focusof adsorption research community also a wide variety ofanalytical techniques are used for initial and equilibrium con-centrations determination Accordingly the measurementerrors they introduce differ in type and magnitude Thereare different mathematical models named noise precisionmodels (NPMs) that have been proposed to estimate thechange of analytical precision as a function of the analyteconcentration List of such models for specific analyticalmethods together with explanations of error sources can befound in literature [23] In this paper the NPMs were chosento be the simplest physically plausible [29] Firstly the sixdifferent magnitudes of homoscedastic noise were used (H1ndashH6) Noise population standard deviations were from verylow (001 002) and medium (005 01) to high (02 04) Thenoise of the data was therefore between 001 and 04 of thedata range Secondly the five types of heteroscedastic noiselinear (Lin) quadratic (Quad) hockey stick (HS) constantrelative standard deviation of 5 (RSD5) and constantrelative standard deviation of 10 (RSD10) were involvedExpressions of the NPMs used in the study and their relevantparameters are listed in Table 3
33 Methods for Fitting Adsorption Isotherms Althoughthere are some experiments where it is reasonable to assumethat one variable (119909) is largely free from errors such anassumption is manifestly not true in cases of adsorptionisotherms The value on the 119909-axis is an equilibrium con-centration 119862
119890exp which is determined by chemical analysisand thus inevitably affected by the measurement error Equi-librium loading 119902
119890exp (which is on the 119910-axis) is calculatedfrom the equilibrium concentration and as a result an errorin this concentration appears in both coordinates Particularattention must be given to equations in which one variable
6 The Scientific World Journal
x
y(xi yi)
(xi f(xi ))
(a)
x
y(xi yi)
(xi + 120593i f(xi + 120593i ))
(b)
x
y
(xi yi)
(xi + 120593i f(xi + 120593i ))
(c)
Figure 1 Geometric illustration of differences among different regression methods (a) OLS without weighting (b) orthogonal distanceregression without weighting and (c) orthogonal distance regression with weighting
is involved on both sides since the independence of errorsis not fulfilled [30]The use of orthogonal distance regressionmodeling procedurewould be statistically correct in this case
Since this study is based on simulated data populationstandard deviations of measurement errors are known and
(10) can be transformed into theoretical orthogonal distanceregression criterion (TODR) introducing the relations
119908119894=1
120590120576119894
119908119894119889119894=1
120590120575119894
(15)
for the weights on the 119910- and 119909-axes respectively It can benoticed that the population standard deviation of the erroron the 119909-axis is actually the population standard deviation ofthe equilibrium concentration
120590120575119894= 120590119890
120575119894 (16)
For the 119910 axis 120590120576119894is calculated based on (12) According to
the error propagation law
1205902
120576119894= 1198962
[(120590119890
120575119894)2
+ (120590119900
120575119894)2
] (17)
where 119896 = 119881119898 it is assumed that only the error of the119862119900exp and 119862119890exp determination is significant while mass and
Figure 2 Flow chart illustrating the steps in adsorption equilibrium experiment and a matching numerical experiment
Final formulation of the TODR error function which isminimized in case of theoretical fitting of adsorption data ispresented as
TODR =119899
sum
119894=1
[(119902119890119894exp minus 119902119890119894cal
120590120576119894
)
2
+ (119862119890119894exp minus 119862119890119894cal
120590119890
120575119894
)
2
]
(18)
Although TODR cannot be used outside the theoreticaldomain it was included in this study to serve as a golden stan-dard It is expected to represent the best possible results thatcan be achieved with the certain observations in possession
A typical isotherm data set in the experimental domainconsists of 5ndash10 points Very often researches perform theirexperiments in triplicate [31ndash33] but in cases when thereagents are expensive or toxic there are no replications Bythese two different methods different levels of data qualityare obtained The objective of this paper was to take both ofthese cases into consideration
Data obtained in only one numerical experiment (match-ing the case when the laboratory experiments are performedwith no replication) were fitted by the use of four errorfunctions OLS ODRMPSD andHYBRDTheir expressionsare presented in Table 4 Least squares fitting of the linearized
8 The Scientific World Journal
Table 4 Definitions of error functions in cases when there are no replicated concentration measurements
+ (119862119890119894exp minus 119862119890119894cal
119862119890119894exp
)
2
]
]
data (LIN) was additionally applied only on the Langmuirand Freundlich equations
Formulae of the error functions in Table 4 are a bitdifferent from the ones reviewed in [4]The key reason is thatthe number of data points is constant all the time and therewill be no isotherm ranking based on this error functionsso the constants n-p were removed for the sake of simplicityParameter estimates obtained with these slightly modifiederror functions are the same because the multiplication oferror function with a constant nonzero value does not affectthe position of a global minimum
OLS MPSD and HYBRD are basically least squaresmethods with different types of weights included OLS is theapproach with all weights equal to one In case of MPSDassumption of constant percentage error is accepted andweighting by the equilibrium loading is applied For theHYBRD error functions weights are of the Poisson typeODR abbreviation in this context is used for the orthogonaldistance regression analog of the MPSD Assumption ofconstant percentage error is accepted for both of the axesand the weights are 1119902
119890exp and 1119862119890exp for the119910- and 119909-axesrespectively
The second group of calculations matched the case whenlaboratory experiments are performed in triplicate Means ofequilibrium sorbent loading (119902
119890119894exp) and equilibrium sorbateconcentration (119862
119890119894exp) from three subsequent numericalexperiments were passed to the following error functionsexperimental weighted orthogonal distance regression
(E3WODR) triplicate orthogonal distance regression(E3ODR) and weighted least squares (WLS) Definitionsof error functions for replicated measurements are listed inTable 5
It is important to say that E3WODR is the experimentalrealization of TODR Estimates of standard deviation of thevariables on the 119910- and 119909-axes are calculated as described inSection 231 incorporating the following equations
119904119889119910119894= radic3
sum
119895=1
(119902119890119894exp minus 119902119890119894exp)
2
119895
2 (19)
119904119889119909119894= radic3
sum
119895=1
(119862119890119894exp minus 119862119890119894exp)
2
119895
2 (20)
Weighting in the E3ODRmethod is based on themean valuesof equilibrium sorbent loading and equilibrium sorbateconcentration
119908119894=
1
119902119890119894exp
119908119894119889119894=
1
119862119890119894exp
(21)
For the WLS method instrumental weights are calculatedbased on (19)
The Scientific World Journal 9
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
and the curve does not reach any plateau (the solid doesnot show clearly a limited sorption capacity) ldquoSrdquo type ofadsorption isotherm is sigmoidal-shaped and thus has got apoint of inflection It is always a result of at least two oppositemechanisms Compared to the ldquoLrdquo and ldquoHrdquo isotherms the ldquoSrdquoclass occurs less frequently [13] and it will not be addressedin this paper
From a mathematical point of view isotherm equationscan be grouped into rational power and transcendentalfunctions [3] Important for the convergence propertiesand computational difficulty is the number of parametersMost of the isotherms used for liquid-phase adsorptiondescription are two or three parameter isotherms while forthe adsorption of gases hybrid isotherms with significantlyhigher number of parameters are also present in the literature[14] The equations of five adsorption isotherms addressed inthis paper are listed in Table 1
They were chosen to be widely used and to repre-sent different types of mathematical functions (LangmuirRedlich-Peterson and Sips isotherms are rational functionsFreundlich isotherm is a power function and Jovanovicisotherm is a transcendental function) and different numberof parameters (Langmuir Freundlich and Jovanovic are two-parameter isotherms and Redlich-Peterson and Sips arethree-parameter isotherms) To avoid unnecessary repeti-tions detailed characteristics of the isotherms are not pre-sented Additional information can be found in the literature
22 Method of Least Squares Let independent data pairs(119909119894 119910119894) 119894 = 1 119899 be observed from the underlying
true values (119883119894119884119894) and accept the assumption that only
dependent variable 119910119894is affected by measurement error
119909119894= 119883119894
119910119894= 119884119894+ 120576119894
(1)
where 120576119894is additive zero mean white Gaussian noise
The noise is assumed to be homoscedastic with constantpopulation standard deviation 120590
120576 written in short notation
120576119894sim 119873(0 120590
120576) Although (1) is not absolutely satisfied in
practice and it is often the case that 119909119894have errors these
errors can be safely ignored if they are much smaller than thecorresponding errors in 119910
119894[6]
Assume the smooth function119884 = 119891(119883 120579) is a truemodelwhere 120579 isin 119877119901 is a vector of true parameters With more datapoints than parameters (119899 gt 119901) it is not possible to solvethe model and calculate the values of the true parametersInstead the question is how to obtain the best compromiseso that the model predictions (119910
119894) are on the whole as close as
possible to the observed data values Closeness for any singleobservation may be measured by the vertical distance (120595
119894)
from the data point to the fitted curve
119910119894= 119910119894+ 120595119894 (2)
Closeness averaged over the entire data set is often measuredby the sum of the squares of the individual distances Any
point = (1205791 1205792 120579
119901) in Θ = 120579 = (120579
1 1205792 120579
119901) isin R119901
120579119894gt 0 119894 = 1 119901 which minimizes the functional
ErrFun () =119899
sum
119894=1
1199082
119894(119891 (119909
119894 ) minus 119910
119894)2
(3)
where 119908119894are data weights and equation for the fitted curve
reads 119910119894= 119891(119909
119894 ) is called the least squares estimate of the
unknown parameters if it exists [20] Condition 120579119894gt 0 119894 =
1 119901 in the definition of the set Θ is a specific feature ofthe modeling adsorption isotherms and meets the criterionfor the isotherm to be positive increasing and concave onthe set [0infin) namely to be ldquoLrdquo or ldquoHrdquo type isotherm
221 Weighting Schemes in the Method of Least SquaresIn the method of OLS the observations are assumed tobe homoscedastic and all of the points are assigned equalweights 119908
119894= 1 119894 = 1 119899 In the absence of more
complete information it is commonly accepted that uniformweighting is satisfactory and OLS are widely used in modelfitting [21] If the assumption of constant standard deviationsof measurement errors is relaxed the heteroscedasticity ischaracterized by an 119899 element vector 120590
120576= (1205901205761 1205901205762 120590
120576119899)
where each 120590120576119894is population standard deviation of the noise
at 119910119894 120576119894sim 119873(0 120590
120576119894) Weights calculated by
119908119894=1
120590120576119894
(4)
are introduced into (3) in order to account for inconstantvariance and the method is referred to as weighted leastsquares (WLS) or sometimes ldquochi-square fittingrdquo [22] Theassumption that the weights are known exactly is not valid inreal applications so estimated weights must be used instead[23]
Ideally observation weights should be estimated accord-ing to individual estimates of measurement error such that119908119894= 1119904119889119910
119894 where 119904119889119910
119894is the standard deviation of 119894th
measurement These are called instrumental weightsWhen individual error estimates are unavailable other
empirical weights may provide a simple approximation ofstandard deviation For the peculiar case of heteroscedas-ticity important in many analytical methods relative stan-dard deviations are reasonably constant over a considerabledynamic rangeThus 120590
120576119894is proportional to 119910
119894and the weights
can be estimated as 119908119894= 1119910
119894
However the error structure in real data usually liessomewhere on a continuous between a constant absoluteerror (homoscedastic) at one extreme and a constant percent-age error at the other Between these two there is an error forwhich the standard deviation is proportional to the squareroot of the expected value119908
119894= 1119910
119894
05This type of weights iscalled Poisson weights or hybrid weights and they should beappliedwhen the shot noise is present Shot noise is dominantwhen a finite number of particles that carry energy (ionselectrons and photons) are counted at the detector part of theinstrumentThe characteristic expressions for each weightingtype are presented in Table 2
ISOFIT a software package for fitting sorption isothermsto experimental data by weighted least squares supports
4 The Scientific World Journal
Table 1 Adsorption isotherm models
No Type Type offunction Nonlinear form Linear form
Type of weights ExpressionAbsolute weights 1Poisson weights 1119910
05
119894
Assumption of constant percentage error 1119910119894
Instrumental weights 1sd119910119894
three alternatives uniform weighting sorbed relative whereweights are inversely proportional to sorbed concentrationsand solute relative where weights are inversely proportionalto measured solute concentrations [24]
23 Orthogonal Distance Regression Methods In a moregeneral situation considerable errors can occur in bothvariables It is stated that if the errors in 119909
119894are greater than
one-tenth of the errors in 119910119894 then the overall error is signif-
icantly increased Moreover the regression parameters andtheir confidence intervals are then biased using (ordinary)weighted least squares [25]
Let the considerable error be also present in the measure-ments of the independent variable
119909119894= 119883119894+ 120575119894 (5)
where 120575119894 sim 119873(0 120590120575119894)
Again the model will not fit the observed data points(119909119894 119910119894) 119894 = 1 119899 exactly so the corresponding set of points
(119909119894 119910119894) 119894 = 1 119899 that do fit the model exactly and that are
at the same time the closest to experimental data points is tobe considered For each data point the value of independentvariable 119909
119894is expressed introducing an error term 120593
119894
119909119894= 119909119894+ 120593119894 (6)
The values 119910119894are predicted by the model function
119910119894= 119891 (119909
119894+ 120593119894 ) (7)
A reasonable way to estimate the unknown parameters in thiscase is to minimize the weighted sum of squares of all errorsby minimizing the functional
ErrFun ( ) =119899
sum
119894=1
1199082
119894(120595119894
2
+ 1198892
1198941205932
119894) (8)
on the set Θ times Φ where 119908119894and 119889
119894are data weights in the 119910
and 119909 directions respectively and
Φ = 120593 = (1205931 1205932 120593
119899) isin R119899 119909
119894+ 120593119894ge 0 119894 = 1 119899
(9)
Commonly (8) is expressed in its expanded form where thedifference between calculated and experimentally observedvalue of 119910 is emphasized
ErrFun ( ) =119899
sum
119894= 1
1199082
119894[(119891 (119909
119894+ 120593119894 ) minus 119910
119894)2
+ 1198892
1198941205932
119894]
(10)
This approach is known as errors in variables or orthogonaldistance regression or total least squares Condition 119909
119894+
120593119894ge 0 119894 = 1 119899 in the defining relation (9) is set in
order to meet the natural condition that concentration is anonnegative value
231Weighting Schemes in theMethod of Orthogonal DistanceRegression In orthogonal distance regression analysis ofsorption data units of the variables on the axes are notthe same It is necessary to introduce weights as constantsselected to scale each type of variable 119910
119894or 119909119894 This is done
in order to put the model errors (120595119894and 120593
119894) on a comparable
basis so it will be meaningful to add them all togetherinto the sum of error function in (8) Typically weights arechosen as estimates of the population standard deviation ofthe experimental measurements of each variable type 119908
119894=
1119904119889119910119894and 119908
119894119889119894= 1119904119889119909
119894 Another way is to assign weights
to be proportional to the inverse of experimental values119908119894=
1119910119894and 119908
119894119889119894= 1119909
119894 The effect of such weights is that at
the same time heteroscedasticity is accounted for and scaledmodel errors in 119910 and 119909 direction are dimensionless
The Scientific World Journal 5
In Figure 1 differences between different OLS and orthog-onal distance regressionmethodswith andwithout weightingare presented
Geometrically if the data pairs (119909119894 119910119894) and the curve 119910 =
119891(119909 ) are presented in the Cartesian coordinates in ordinaryleast squares minimization of the error function correspondsto minimization of the shortest distances from data pointsto a line in a direction that is parallel to the vertical axisFigure 1(a) is a standard geometric illustration of the leastsquares method Instead of the vertical offsets the shortestdistances from points to the line are considered in themethod of orthogonal distance regression If the data arehomoscedastic and the units of 119909 and 119910 are the same allthe weights 119908
119894and 119889
119894are equal to one Equation (8) is then
simplified to a formula that possesses a meaning of the sumof the areas of the circles shown in Figure 1(b)
ErrFun ( ) =119899
sum
119894=1
[(119910119894minus 119910119894)2
+ (119909119894minus 119909119894)2
] (11)
In this case the radii of these circles are equal to distancesbetween the points (119909
119894 119910119894) and the fitting line Put in other
words the fitting line is a tangent line to all circlesGeometrical representation of a case when 119909 and 119910 are
variables that do not have the same units or the data isheteroscedastic is presented in Figure 1(c) Weights are intro-duced and half axes of the ellipses in Figure 1(c) correspondto the combined measure of the distance expressed in (8)While the global minimum of this error function is uniquethis kind of straightforward geometrical representation is nolonger meaningful
Orthogonal distance regression methods have been usedin the fields of science such as economy [26] automaticcontrol [27] and pharmacology [28] and a significant workhas been done for the development of stable and efficientalgorithm for ODR estimation of parameters
3 Materials and Methods
31 Numerical Experiments Numerical experiments weredesigned to be as close as possible representation of a typicalexperimental setup in adsorption studies It was adoptedthat batch experiments are performed in laboratory bakerscontaining mass of sorbent 119898 and volume 119881 of sorbatesolution Initial concentrations of sorbate solutions (119862
119900119894true)are chosen to be 05 10 50 100 500 and 1000 All units areignored since they are irrelevant Further on it was assumedthat the theoretical adsorption isotherm expressed in termsof its true parameters is exactly matching the adsorptionprocess Values of the true parameters were arbitrarily set toget the operative expression 119902
119890true = 119891(119862119890true) where 119862119890trueis errorless equilibrium sorbate concentration and 119902
119890true iserrorless equilibrium sorbate loading At the same time massbalance expressed in (12) is satisfied
119902119890119894true =
(119862119900119894true minus 119862119890119894true)
119898119881 (12)
The true equilibrium concentration is then calculated solvingthe equation
119891 (119862119890119894true 120579) minus
(119862119900119894true minus 119862119890119894true)
119898119881 = 0 (13)
It is assumed that simple univariate chemical measurementsystem with additive zero mean white Gaussian measure-ment noise is used as an analytical tool to determine 119862
119900119894trueand 119862
119890119894true Thus random noise (120575119894119900 sim 119873(0 120590
119900
120575119894) and
120575119894119890 sim 119873(0 120590120575
119894119890) resp) was added to these values to obtain
The rest of the procedure was identical as if the experimentswere performed in laboratory The equilibrium sorbent load-ing was calculated from (12) and the collected data wassubjected to fitting routines
Flow chart of laboratory experiment and a matchingnumerical experiment is presented in Figure 2
32 Noise Precision Models Since a wide variety of sub-stances (toxic metals organic pollutants etc) are in focusof adsorption research community also a wide variety ofanalytical techniques are used for initial and equilibrium con-centrations determination Accordingly the measurementerrors they introduce differ in type and magnitude Thereare different mathematical models named noise precisionmodels (NPMs) that have been proposed to estimate thechange of analytical precision as a function of the analyteconcentration List of such models for specific analyticalmethods together with explanations of error sources can befound in literature [23] In this paper the NPMs were chosento be the simplest physically plausible [29] Firstly the sixdifferent magnitudes of homoscedastic noise were used (H1ndashH6) Noise population standard deviations were from verylow (001 002) and medium (005 01) to high (02 04) Thenoise of the data was therefore between 001 and 04 of thedata range Secondly the five types of heteroscedastic noiselinear (Lin) quadratic (Quad) hockey stick (HS) constantrelative standard deviation of 5 (RSD5) and constantrelative standard deviation of 10 (RSD10) were involvedExpressions of the NPMs used in the study and their relevantparameters are listed in Table 3
33 Methods for Fitting Adsorption Isotherms Althoughthere are some experiments where it is reasonable to assumethat one variable (119909) is largely free from errors such anassumption is manifestly not true in cases of adsorptionisotherms The value on the 119909-axis is an equilibrium con-centration 119862
119890exp which is determined by chemical analysisand thus inevitably affected by the measurement error Equi-librium loading 119902
119890exp (which is on the 119910-axis) is calculatedfrom the equilibrium concentration and as a result an errorin this concentration appears in both coordinates Particularattention must be given to equations in which one variable
6 The Scientific World Journal
x
y(xi yi)
(xi f(xi ))
(a)
x
y(xi yi)
(xi + 120593i f(xi + 120593i ))
(b)
x
y
(xi yi)
(xi + 120593i f(xi + 120593i ))
(c)
Figure 1 Geometric illustration of differences among different regression methods (a) OLS without weighting (b) orthogonal distanceregression without weighting and (c) orthogonal distance regression with weighting
is involved on both sides since the independence of errorsis not fulfilled [30]The use of orthogonal distance regressionmodeling procedurewould be statistically correct in this case
Since this study is based on simulated data populationstandard deviations of measurement errors are known and
(10) can be transformed into theoretical orthogonal distanceregression criterion (TODR) introducing the relations
119908119894=1
120590120576119894
119908119894119889119894=1
120590120575119894
(15)
for the weights on the 119910- and 119909-axes respectively It can benoticed that the population standard deviation of the erroron the 119909-axis is actually the population standard deviation ofthe equilibrium concentration
120590120575119894= 120590119890
120575119894 (16)
For the 119910 axis 120590120576119894is calculated based on (12) According to
the error propagation law
1205902
120576119894= 1198962
[(120590119890
120575119894)2
+ (120590119900
120575119894)2
] (17)
where 119896 = 119881119898 it is assumed that only the error of the119862119900exp and 119862119890exp determination is significant while mass and
Figure 2 Flow chart illustrating the steps in adsorption equilibrium experiment and a matching numerical experiment
Final formulation of the TODR error function which isminimized in case of theoretical fitting of adsorption data ispresented as
TODR =119899
sum
119894=1
[(119902119890119894exp minus 119902119890119894cal
120590120576119894
)
2
+ (119862119890119894exp minus 119862119890119894cal
120590119890
120575119894
)
2
]
(18)
Although TODR cannot be used outside the theoreticaldomain it was included in this study to serve as a golden stan-dard It is expected to represent the best possible results thatcan be achieved with the certain observations in possession
A typical isotherm data set in the experimental domainconsists of 5ndash10 points Very often researches perform theirexperiments in triplicate [31ndash33] but in cases when thereagents are expensive or toxic there are no replications Bythese two different methods different levels of data qualityare obtained The objective of this paper was to take both ofthese cases into consideration
Data obtained in only one numerical experiment (match-ing the case when the laboratory experiments are performedwith no replication) were fitted by the use of four errorfunctions OLS ODRMPSD andHYBRDTheir expressionsare presented in Table 4 Least squares fitting of the linearized
8 The Scientific World Journal
Table 4 Definitions of error functions in cases when there are no replicated concentration measurements
+ (119862119890119894exp minus 119862119890119894cal
119862119890119894exp
)
2
]
]
data (LIN) was additionally applied only on the Langmuirand Freundlich equations
Formulae of the error functions in Table 4 are a bitdifferent from the ones reviewed in [4]The key reason is thatthe number of data points is constant all the time and therewill be no isotherm ranking based on this error functionsso the constants n-p were removed for the sake of simplicityParameter estimates obtained with these slightly modifiederror functions are the same because the multiplication oferror function with a constant nonzero value does not affectthe position of a global minimum
OLS MPSD and HYBRD are basically least squaresmethods with different types of weights included OLS is theapproach with all weights equal to one In case of MPSDassumption of constant percentage error is accepted andweighting by the equilibrium loading is applied For theHYBRD error functions weights are of the Poisson typeODR abbreviation in this context is used for the orthogonaldistance regression analog of the MPSD Assumption ofconstant percentage error is accepted for both of the axesand the weights are 1119902
119890exp and 1119862119890exp for the119910- and 119909-axesrespectively
The second group of calculations matched the case whenlaboratory experiments are performed in triplicate Means ofequilibrium sorbent loading (119902
119890119894exp) and equilibrium sorbateconcentration (119862
119890119894exp) from three subsequent numericalexperiments were passed to the following error functionsexperimental weighted orthogonal distance regression
(E3WODR) triplicate orthogonal distance regression(E3ODR) and weighted least squares (WLS) Definitionsof error functions for replicated measurements are listed inTable 5
It is important to say that E3WODR is the experimentalrealization of TODR Estimates of standard deviation of thevariables on the 119910- and 119909-axes are calculated as described inSection 231 incorporating the following equations
119904119889119910119894= radic3
sum
119895=1
(119902119890119894exp minus 119902119890119894exp)
2
119895
2 (19)
119904119889119909119894= radic3
sum
119895=1
(119862119890119894exp minus 119862119890119894exp)
2
119895
2 (20)
Weighting in the E3ODRmethod is based on themean valuesof equilibrium sorbent loading and equilibrium sorbateconcentration
119908119894=
1
119902119890119894exp
119908119894119889119894=
1
119862119890119894exp
(21)
For the WLS method instrumental weights are calculatedbased on (19)
The Scientific World Journal 9
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Type of weights ExpressionAbsolute weights 1Poisson weights 1119910
05
119894
Assumption of constant percentage error 1119910119894
Instrumental weights 1sd119910119894
three alternatives uniform weighting sorbed relative whereweights are inversely proportional to sorbed concentrationsand solute relative where weights are inversely proportionalto measured solute concentrations [24]
23 Orthogonal Distance Regression Methods In a moregeneral situation considerable errors can occur in bothvariables It is stated that if the errors in 119909
119894are greater than
one-tenth of the errors in 119910119894 then the overall error is signif-
icantly increased Moreover the regression parameters andtheir confidence intervals are then biased using (ordinary)weighted least squares [25]
Let the considerable error be also present in the measure-ments of the independent variable
119909119894= 119883119894+ 120575119894 (5)
where 120575119894 sim 119873(0 120590120575119894)
Again the model will not fit the observed data points(119909119894 119910119894) 119894 = 1 119899 exactly so the corresponding set of points
(119909119894 119910119894) 119894 = 1 119899 that do fit the model exactly and that are
at the same time the closest to experimental data points is tobe considered For each data point the value of independentvariable 119909
119894is expressed introducing an error term 120593
119894
119909119894= 119909119894+ 120593119894 (6)
The values 119910119894are predicted by the model function
119910119894= 119891 (119909
119894+ 120593119894 ) (7)
A reasonable way to estimate the unknown parameters in thiscase is to minimize the weighted sum of squares of all errorsby minimizing the functional
ErrFun ( ) =119899
sum
119894=1
1199082
119894(120595119894
2
+ 1198892
1198941205932
119894) (8)
on the set Θ times Φ where 119908119894and 119889
119894are data weights in the 119910
and 119909 directions respectively and
Φ = 120593 = (1205931 1205932 120593
119899) isin R119899 119909
119894+ 120593119894ge 0 119894 = 1 119899
(9)
Commonly (8) is expressed in its expanded form where thedifference between calculated and experimentally observedvalue of 119910 is emphasized
ErrFun ( ) =119899
sum
119894= 1
1199082
119894[(119891 (119909
119894+ 120593119894 ) minus 119910
119894)2
+ 1198892
1198941205932
119894]
(10)
This approach is known as errors in variables or orthogonaldistance regression or total least squares Condition 119909
119894+
120593119894ge 0 119894 = 1 119899 in the defining relation (9) is set in
order to meet the natural condition that concentration is anonnegative value
231Weighting Schemes in theMethod of Orthogonal DistanceRegression In orthogonal distance regression analysis ofsorption data units of the variables on the axes are notthe same It is necessary to introduce weights as constantsselected to scale each type of variable 119910
119894or 119909119894 This is done
in order to put the model errors (120595119894and 120593
119894) on a comparable
basis so it will be meaningful to add them all togetherinto the sum of error function in (8) Typically weights arechosen as estimates of the population standard deviation ofthe experimental measurements of each variable type 119908
119894=
1119904119889119910119894and 119908
119894119889119894= 1119904119889119909
119894 Another way is to assign weights
to be proportional to the inverse of experimental values119908119894=
1119910119894and 119908
119894119889119894= 1119909
119894 The effect of such weights is that at
the same time heteroscedasticity is accounted for and scaledmodel errors in 119910 and 119909 direction are dimensionless
The Scientific World Journal 5
In Figure 1 differences between different OLS and orthog-onal distance regressionmethodswith andwithout weightingare presented
Geometrically if the data pairs (119909119894 119910119894) and the curve 119910 =
119891(119909 ) are presented in the Cartesian coordinates in ordinaryleast squares minimization of the error function correspondsto minimization of the shortest distances from data pointsto a line in a direction that is parallel to the vertical axisFigure 1(a) is a standard geometric illustration of the leastsquares method Instead of the vertical offsets the shortestdistances from points to the line are considered in themethod of orthogonal distance regression If the data arehomoscedastic and the units of 119909 and 119910 are the same allthe weights 119908
119894and 119889
119894are equal to one Equation (8) is then
simplified to a formula that possesses a meaning of the sumof the areas of the circles shown in Figure 1(b)
ErrFun ( ) =119899
sum
119894=1
[(119910119894minus 119910119894)2
+ (119909119894minus 119909119894)2
] (11)
In this case the radii of these circles are equal to distancesbetween the points (119909
119894 119910119894) and the fitting line Put in other
words the fitting line is a tangent line to all circlesGeometrical representation of a case when 119909 and 119910 are
variables that do not have the same units or the data isheteroscedastic is presented in Figure 1(c) Weights are intro-duced and half axes of the ellipses in Figure 1(c) correspondto the combined measure of the distance expressed in (8)While the global minimum of this error function is uniquethis kind of straightforward geometrical representation is nolonger meaningful
Orthogonal distance regression methods have been usedin the fields of science such as economy [26] automaticcontrol [27] and pharmacology [28] and a significant workhas been done for the development of stable and efficientalgorithm for ODR estimation of parameters
3 Materials and Methods
31 Numerical Experiments Numerical experiments weredesigned to be as close as possible representation of a typicalexperimental setup in adsorption studies It was adoptedthat batch experiments are performed in laboratory bakerscontaining mass of sorbent 119898 and volume 119881 of sorbatesolution Initial concentrations of sorbate solutions (119862
119900119894true)are chosen to be 05 10 50 100 500 and 1000 All units areignored since they are irrelevant Further on it was assumedthat the theoretical adsorption isotherm expressed in termsof its true parameters is exactly matching the adsorptionprocess Values of the true parameters were arbitrarily set toget the operative expression 119902
119890true = 119891(119862119890true) where 119862119890trueis errorless equilibrium sorbate concentration and 119902
119890true iserrorless equilibrium sorbate loading At the same time massbalance expressed in (12) is satisfied
119902119890119894true =
(119862119900119894true minus 119862119890119894true)
119898119881 (12)
The true equilibrium concentration is then calculated solvingthe equation
119891 (119862119890119894true 120579) minus
(119862119900119894true minus 119862119890119894true)
119898119881 = 0 (13)
It is assumed that simple univariate chemical measurementsystem with additive zero mean white Gaussian measure-ment noise is used as an analytical tool to determine 119862
119900119894trueand 119862
119890119894true Thus random noise (120575119894119900 sim 119873(0 120590
119900
120575119894) and
120575119894119890 sim 119873(0 120590120575
119894119890) resp) was added to these values to obtain
The rest of the procedure was identical as if the experimentswere performed in laboratory The equilibrium sorbent load-ing was calculated from (12) and the collected data wassubjected to fitting routines
Flow chart of laboratory experiment and a matchingnumerical experiment is presented in Figure 2
32 Noise Precision Models Since a wide variety of sub-stances (toxic metals organic pollutants etc) are in focusof adsorption research community also a wide variety ofanalytical techniques are used for initial and equilibrium con-centrations determination Accordingly the measurementerrors they introduce differ in type and magnitude Thereare different mathematical models named noise precisionmodels (NPMs) that have been proposed to estimate thechange of analytical precision as a function of the analyteconcentration List of such models for specific analyticalmethods together with explanations of error sources can befound in literature [23] In this paper the NPMs were chosento be the simplest physically plausible [29] Firstly the sixdifferent magnitudes of homoscedastic noise were used (H1ndashH6) Noise population standard deviations were from verylow (001 002) and medium (005 01) to high (02 04) Thenoise of the data was therefore between 001 and 04 of thedata range Secondly the five types of heteroscedastic noiselinear (Lin) quadratic (Quad) hockey stick (HS) constantrelative standard deviation of 5 (RSD5) and constantrelative standard deviation of 10 (RSD10) were involvedExpressions of the NPMs used in the study and their relevantparameters are listed in Table 3
33 Methods for Fitting Adsorption Isotherms Althoughthere are some experiments where it is reasonable to assumethat one variable (119909) is largely free from errors such anassumption is manifestly not true in cases of adsorptionisotherms The value on the 119909-axis is an equilibrium con-centration 119862
119890exp which is determined by chemical analysisand thus inevitably affected by the measurement error Equi-librium loading 119902
119890exp (which is on the 119910-axis) is calculatedfrom the equilibrium concentration and as a result an errorin this concentration appears in both coordinates Particularattention must be given to equations in which one variable
6 The Scientific World Journal
x
y(xi yi)
(xi f(xi ))
(a)
x
y(xi yi)
(xi + 120593i f(xi + 120593i ))
(b)
x
y
(xi yi)
(xi + 120593i f(xi + 120593i ))
(c)
Figure 1 Geometric illustration of differences among different regression methods (a) OLS without weighting (b) orthogonal distanceregression without weighting and (c) orthogonal distance regression with weighting
is involved on both sides since the independence of errorsis not fulfilled [30]The use of orthogonal distance regressionmodeling procedurewould be statistically correct in this case
Since this study is based on simulated data populationstandard deviations of measurement errors are known and
(10) can be transformed into theoretical orthogonal distanceregression criterion (TODR) introducing the relations
119908119894=1
120590120576119894
119908119894119889119894=1
120590120575119894
(15)
for the weights on the 119910- and 119909-axes respectively It can benoticed that the population standard deviation of the erroron the 119909-axis is actually the population standard deviation ofthe equilibrium concentration
120590120575119894= 120590119890
120575119894 (16)
For the 119910 axis 120590120576119894is calculated based on (12) According to
the error propagation law
1205902
120576119894= 1198962
[(120590119890
120575119894)2
+ (120590119900
120575119894)2
] (17)
where 119896 = 119881119898 it is assumed that only the error of the119862119900exp and 119862119890exp determination is significant while mass and
Figure 2 Flow chart illustrating the steps in adsorption equilibrium experiment and a matching numerical experiment
Final formulation of the TODR error function which isminimized in case of theoretical fitting of adsorption data ispresented as
TODR =119899
sum
119894=1
[(119902119890119894exp minus 119902119890119894cal
120590120576119894
)
2
+ (119862119890119894exp minus 119862119890119894cal
120590119890
120575119894
)
2
]
(18)
Although TODR cannot be used outside the theoreticaldomain it was included in this study to serve as a golden stan-dard It is expected to represent the best possible results thatcan be achieved with the certain observations in possession
A typical isotherm data set in the experimental domainconsists of 5ndash10 points Very often researches perform theirexperiments in triplicate [31ndash33] but in cases when thereagents are expensive or toxic there are no replications Bythese two different methods different levels of data qualityare obtained The objective of this paper was to take both ofthese cases into consideration
Data obtained in only one numerical experiment (match-ing the case when the laboratory experiments are performedwith no replication) were fitted by the use of four errorfunctions OLS ODRMPSD andHYBRDTheir expressionsare presented in Table 4 Least squares fitting of the linearized
8 The Scientific World Journal
Table 4 Definitions of error functions in cases when there are no replicated concentration measurements
+ (119862119890119894exp minus 119862119890119894cal
119862119890119894exp
)
2
]
]
data (LIN) was additionally applied only on the Langmuirand Freundlich equations
Formulae of the error functions in Table 4 are a bitdifferent from the ones reviewed in [4]The key reason is thatthe number of data points is constant all the time and therewill be no isotherm ranking based on this error functionsso the constants n-p were removed for the sake of simplicityParameter estimates obtained with these slightly modifiederror functions are the same because the multiplication oferror function with a constant nonzero value does not affectthe position of a global minimum
OLS MPSD and HYBRD are basically least squaresmethods with different types of weights included OLS is theapproach with all weights equal to one In case of MPSDassumption of constant percentage error is accepted andweighting by the equilibrium loading is applied For theHYBRD error functions weights are of the Poisson typeODR abbreviation in this context is used for the orthogonaldistance regression analog of the MPSD Assumption ofconstant percentage error is accepted for both of the axesand the weights are 1119902
119890exp and 1119862119890exp for the119910- and 119909-axesrespectively
The second group of calculations matched the case whenlaboratory experiments are performed in triplicate Means ofequilibrium sorbent loading (119902
119890119894exp) and equilibrium sorbateconcentration (119862
119890119894exp) from three subsequent numericalexperiments were passed to the following error functionsexperimental weighted orthogonal distance regression
(E3WODR) triplicate orthogonal distance regression(E3ODR) and weighted least squares (WLS) Definitionsof error functions for replicated measurements are listed inTable 5
It is important to say that E3WODR is the experimentalrealization of TODR Estimates of standard deviation of thevariables on the 119910- and 119909-axes are calculated as described inSection 231 incorporating the following equations
119904119889119910119894= radic3
sum
119895=1
(119902119890119894exp minus 119902119890119894exp)
2
119895
2 (19)
119904119889119909119894= radic3
sum
119895=1
(119862119890119894exp minus 119862119890119894exp)
2
119895
2 (20)
Weighting in the E3ODRmethod is based on themean valuesof equilibrium sorbent loading and equilibrium sorbateconcentration
119908119894=
1
119902119890119894exp
119908119894119889119894=
1
119862119890119894exp
(21)
For the WLS method instrumental weights are calculatedbased on (19)
The Scientific World Journal 9
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
In Figure 1 differences between different OLS and orthog-onal distance regressionmethodswith andwithout weightingare presented
Geometrically if the data pairs (119909119894 119910119894) and the curve 119910 =
119891(119909 ) are presented in the Cartesian coordinates in ordinaryleast squares minimization of the error function correspondsto minimization of the shortest distances from data pointsto a line in a direction that is parallel to the vertical axisFigure 1(a) is a standard geometric illustration of the leastsquares method Instead of the vertical offsets the shortestdistances from points to the line are considered in themethod of orthogonal distance regression If the data arehomoscedastic and the units of 119909 and 119910 are the same allthe weights 119908
119894and 119889
119894are equal to one Equation (8) is then
simplified to a formula that possesses a meaning of the sumof the areas of the circles shown in Figure 1(b)
ErrFun ( ) =119899
sum
119894=1
[(119910119894minus 119910119894)2
+ (119909119894minus 119909119894)2
] (11)
In this case the radii of these circles are equal to distancesbetween the points (119909
119894 119910119894) and the fitting line Put in other
words the fitting line is a tangent line to all circlesGeometrical representation of a case when 119909 and 119910 are
variables that do not have the same units or the data isheteroscedastic is presented in Figure 1(c) Weights are intro-duced and half axes of the ellipses in Figure 1(c) correspondto the combined measure of the distance expressed in (8)While the global minimum of this error function is uniquethis kind of straightforward geometrical representation is nolonger meaningful
Orthogonal distance regression methods have been usedin the fields of science such as economy [26] automaticcontrol [27] and pharmacology [28] and a significant workhas been done for the development of stable and efficientalgorithm for ODR estimation of parameters
3 Materials and Methods
31 Numerical Experiments Numerical experiments weredesigned to be as close as possible representation of a typicalexperimental setup in adsorption studies It was adoptedthat batch experiments are performed in laboratory bakerscontaining mass of sorbent 119898 and volume 119881 of sorbatesolution Initial concentrations of sorbate solutions (119862
119900119894true)are chosen to be 05 10 50 100 500 and 1000 All units areignored since they are irrelevant Further on it was assumedthat the theoretical adsorption isotherm expressed in termsof its true parameters is exactly matching the adsorptionprocess Values of the true parameters were arbitrarily set toget the operative expression 119902
119890true = 119891(119862119890true) where 119862119890trueis errorless equilibrium sorbate concentration and 119902
119890true iserrorless equilibrium sorbate loading At the same time massbalance expressed in (12) is satisfied
119902119890119894true =
(119862119900119894true minus 119862119890119894true)
119898119881 (12)
The true equilibrium concentration is then calculated solvingthe equation
119891 (119862119890119894true 120579) minus
(119862119900119894true minus 119862119890119894true)
119898119881 = 0 (13)
It is assumed that simple univariate chemical measurementsystem with additive zero mean white Gaussian measure-ment noise is used as an analytical tool to determine 119862
119900119894trueand 119862
119890119894true Thus random noise (120575119894119900 sim 119873(0 120590
119900
120575119894) and
120575119894119890 sim 119873(0 120590120575
119894119890) resp) was added to these values to obtain
The rest of the procedure was identical as if the experimentswere performed in laboratory The equilibrium sorbent load-ing was calculated from (12) and the collected data wassubjected to fitting routines
Flow chart of laboratory experiment and a matchingnumerical experiment is presented in Figure 2
32 Noise Precision Models Since a wide variety of sub-stances (toxic metals organic pollutants etc) are in focusof adsorption research community also a wide variety ofanalytical techniques are used for initial and equilibrium con-centrations determination Accordingly the measurementerrors they introduce differ in type and magnitude Thereare different mathematical models named noise precisionmodels (NPMs) that have been proposed to estimate thechange of analytical precision as a function of the analyteconcentration List of such models for specific analyticalmethods together with explanations of error sources can befound in literature [23] In this paper the NPMs were chosento be the simplest physically plausible [29] Firstly the sixdifferent magnitudes of homoscedastic noise were used (H1ndashH6) Noise population standard deviations were from verylow (001 002) and medium (005 01) to high (02 04) Thenoise of the data was therefore between 001 and 04 of thedata range Secondly the five types of heteroscedastic noiselinear (Lin) quadratic (Quad) hockey stick (HS) constantrelative standard deviation of 5 (RSD5) and constantrelative standard deviation of 10 (RSD10) were involvedExpressions of the NPMs used in the study and their relevantparameters are listed in Table 3
33 Methods for Fitting Adsorption Isotherms Althoughthere are some experiments where it is reasonable to assumethat one variable (119909) is largely free from errors such anassumption is manifestly not true in cases of adsorptionisotherms The value on the 119909-axis is an equilibrium con-centration 119862
119890exp which is determined by chemical analysisand thus inevitably affected by the measurement error Equi-librium loading 119902
119890exp (which is on the 119910-axis) is calculatedfrom the equilibrium concentration and as a result an errorin this concentration appears in both coordinates Particularattention must be given to equations in which one variable
6 The Scientific World Journal
x
y(xi yi)
(xi f(xi ))
(a)
x
y(xi yi)
(xi + 120593i f(xi + 120593i ))
(b)
x
y
(xi yi)
(xi + 120593i f(xi + 120593i ))
(c)
Figure 1 Geometric illustration of differences among different regression methods (a) OLS without weighting (b) orthogonal distanceregression without weighting and (c) orthogonal distance regression with weighting
is involved on both sides since the independence of errorsis not fulfilled [30]The use of orthogonal distance regressionmodeling procedurewould be statistically correct in this case
Since this study is based on simulated data populationstandard deviations of measurement errors are known and
(10) can be transformed into theoretical orthogonal distanceregression criterion (TODR) introducing the relations
119908119894=1
120590120576119894
119908119894119889119894=1
120590120575119894
(15)
for the weights on the 119910- and 119909-axes respectively It can benoticed that the population standard deviation of the erroron the 119909-axis is actually the population standard deviation ofthe equilibrium concentration
120590120575119894= 120590119890
120575119894 (16)
For the 119910 axis 120590120576119894is calculated based on (12) According to
the error propagation law
1205902
120576119894= 1198962
[(120590119890
120575119894)2
+ (120590119900
120575119894)2
] (17)
where 119896 = 119881119898 it is assumed that only the error of the119862119900exp and 119862119890exp determination is significant while mass and
Figure 2 Flow chart illustrating the steps in adsorption equilibrium experiment and a matching numerical experiment
Final formulation of the TODR error function which isminimized in case of theoretical fitting of adsorption data ispresented as
TODR =119899
sum
119894=1
[(119902119890119894exp minus 119902119890119894cal
120590120576119894
)
2
+ (119862119890119894exp minus 119862119890119894cal
120590119890
120575119894
)
2
]
(18)
Although TODR cannot be used outside the theoreticaldomain it was included in this study to serve as a golden stan-dard It is expected to represent the best possible results thatcan be achieved with the certain observations in possession
A typical isotherm data set in the experimental domainconsists of 5ndash10 points Very often researches perform theirexperiments in triplicate [31ndash33] but in cases when thereagents are expensive or toxic there are no replications Bythese two different methods different levels of data qualityare obtained The objective of this paper was to take both ofthese cases into consideration
Data obtained in only one numerical experiment (match-ing the case when the laboratory experiments are performedwith no replication) were fitted by the use of four errorfunctions OLS ODRMPSD andHYBRDTheir expressionsare presented in Table 4 Least squares fitting of the linearized
8 The Scientific World Journal
Table 4 Definitions of error functions in cases when there are no replicated concentration measurements
+ (119862119890119894exp minus 119862119890119894cal
119862119890119894exp
)
2
]
]
data (LIN) was additionally applied only on the Langmuirand Freundlich equations
Formulae of the error functions in Table 4 are a bitdifferent from the ones reviewed in [4]The key reason is thatthe number of data points is constant all the time and therewill be no isotherm ranking based on this error functionsso the constants n-p were removed for the sake of simplicityParameter estimates obtained with these slightly modifiederror functions are the same because the multiplication oferror function with a constant nonzero value does not affectthe position of a global minimum
OLS MPSD and HYBRD are basically least squaresmethods with different types of weights included OLS is theapproach with all weights equal to one In case of MPSDassumption of constant percentage error is accepted andweighting by the equilibrium loading is applied For theHYBRD error functions weights are of the Poisson typeODR abbreviation in this context is used for the orthogonaldistance regression analog of the MPSD Assumption ofconstant percentage error is accepted for both of the axesand the weights are 1119902
119890exp and 1119862119890exp for the119910- and 119909-axesrespectively
The second group of calculations matched the case whenlaboratory experiments are performed in triplicate Means ofequilibrium sorbent loading (119902
119890119894exp) and equilibrium sorbateconcentration (119862
119890119894exp) from three subsequent numericalexperiments were passed to the following error functionsexperimental weighted orthogonal distance regression
(E3WODR) triplicate orthogonal distance regression(E3ODR) and weighted least squares (WLS) Definitionsof error functions for replicated measurements are listed inTable 5
It is important to say that E3WODR is the experimentalrealization of TODR Estimates of standard deviation of thevariables on the 119910- and 119909-axes are calculated as described inSection 231 incorporating the following equations
119904119889119910119894= radic3
sum
119895=1
(119902119890119894exp minus 119902119890119894exp)
2
119895
2 (19)
119904119889119909119894= radic3
sum
119895=1
(119862119890119894exp minus 119862119890119894exp)
2
119895
2 (20)
Weighting in the E3ODRmethod is based on themean valuesof equilibrium sorbent loading and equilibrium sorbateconcentration
119908119894=
1
119902119890119894exp
119908119894119889119894=
1
119862119890119894exp
(21)
For the WLS method instrumental weights are calculatedbased on (19)
The Scientific World Journal 9
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Figure 1 Geometric illustration of differences among different regression methods (a) OLS without weighting (b) orthogonal distanceregression without weighting and (c) orthogonal distance regression with weighting
is involved on both sides since the independence of errorsis not fulfilled [30]The use of orthogonal distance regressionmodeling procedurewould be statistically correct in this case
Since this study is based on simulated data populationstandard deviations of measurement errors are known and
(10) can be transformed into theoretical orthogonal distanceregression criterion (TODR) introducing the relations
119908119894=1
120590120576119894
119908119894119889119894=1
120590120575119894
(15)
for the weights on the 119910- and 119909-axes respectively It can benoticed that the population standard deviation of the erroron the 119909-axis is actually the population standard deviation ofthe equilibrium concentration
120590120575119894= 120590119890
120575119894 (16)
For the 119910 axis 120590120576119894is calculated based on (12) According to
the error propagation law
1205902
120576119894= 1198962
[(120590119890
120575119894)2
+ (120590119900
120575119894)2
] (17)
where 119896 = 119881119898 it is assumed that only the error of the119862119900exp and 119862119890exp determination is significant while mass and
Figure 2 Flow chart illustrating the steps in adsorption equilibrium experiment and a matching numerical experiment
Final formulation of the TODR error function which isminimized in case of theoretical fitting of adsorption data ispresented as
TODR =119899
sum
119894=1
[(119902119890119894exp minus 119902119890119894cal
120590120576119894
)
2
+ (119862119890119894exp minus 119862119890119894cal
120590119890
120575119894
)
2
]
(18)
Although TODR cannot be used outside the theoreticaldomain it was included in this study to serve as a golden stan-dard It is expected to represent the best possible results thatcan be achieved with the certain observations in possession
A typical isotherm data set in the experimental domainconsists of 5ndash10 points Very often researches perform theirexperiments in triplicate [31ndash33] but in cases when thereagents are expensive or toxic there are no replications Bythese two different methods different levels of data qualityare obtained The objective of this paper was to take both ofthese cases into consideration
Data obtained in only one numerical experiment (match-ing the case when the laboratory experiments are performedwith no replication) were fitted by the use of four errorfunctions OLS ODRMPSD andHYBRDTheir expressionsare presented in Table 4 Least squares fitting of the linearized
8 The Scientific World Journal
Table 4 Definitions of error functions in cases when there are no replicated concentration measurements
+ (119862119890119894exp minus 119862119890119894cal
119862119890119894exp
)
2
]
]
data (LIN) was additionally applied only on the Langmuirand Freundlich equations
Formulae of the error functions in Table 4 are a bitdifferent from the ones reviewed in [4]The key reason is thatthe number of data points is constant all the time and therewill be no isotherm ranking based on this error functionsso the constants n-p were removed for the sake of simplicityParameter estimates obtained with these slightly modifiederror functions are the same because the multiplication oferror function with a constant nonzero value does not affectthe position of a global minimum
OLS MPSD and HYBRD are basically least squaresmethods with different types of weights included OLS is theapproach with all weights equal to one In case of MPSDassumption of constant percentage error is accepted andweighting by the equilibrium loading is applied For theHYBRD error functions weights are of the Poisson typeODR abbreviation in this context is used for the orthogonaldistance regression analog of the MPSD Assumption ofconstant percentage error is accepted for both of the axesand the weights are 1119902
119890exp and 1119862119890exp for the119910- and 119909-axesrespectively
The second group of calculations matched the case whenlaboratory experiments are performed in triplicate Means ofequilibrium sorbent loading (119902
119890119894exp) and equilibrium sorbateconcentration (119862
119890119894exp) from three subsequent numericalexperiments were passed to the following error functionsexperimental weighted orthogonal distance regression
(E3WODR) triplicate orthogonal distance regression(E3ODR) and weighted least squares (WLS) Definitionsof error functions for replicated measurements are listed inTable 5
It is important to say that E3WODR is the experimentalrealization of TODR Estimates of standard deviation of thevariables on the 119910- and 119909-axes are calculated as described inSection 231 incorporating the following equations
119904119889119910119894= radic3
sum
119895=1
(119902119890119894exp minus 119902119890119894exp)
2
119895
2 (19)
119904119889119909119894= radic3
sum
119895=1
(119862119890119894exp minus 119862119890119894exp)
2
119895
2 (20)
Weighting in the E3ODRmethod is based on themean valuesof equilibrium sorbent loading and equilibrium sorbateconcentration
119908119894=
1
119902119890119894exp
119908119894119889119894=
1
119862119890119894exp
(21)
For the WLS method instrumental weights are calculatedbased on (19)
The Scientific World Journal 9
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Figure 2 Flow chart illustrating the steps in adsorption equilibrium experiment and a matching numerical experiment
Final formulation of the TODR error function which isminimized in case of theoretical fitting of adsorption data ispresented as
TODR =119899
sum
119894=1
[(119902119890119894exp minus 119902119890119894cal
120590120576119894
)
2
+ (119862119890119894exp minus 119862119890119894cal
120590119890
120575119894
)
2
]
(18)
Although TODR cannot be used outside the theoreticaldomain it was included in this study to serve as a golden stan-dard It is expected to represent the best possible results thatcan be achieved with the certain observations in possession
A typical isotherm data set in the experimental domainconsists of 5ndash10 points Very often researches perform theirexperiments in triplicate [31ndash33] but in cases when thereagents are expensive or toxic there are no replications Bythese two different methods different levels of data qualityare obtained The objective of this paper was to take both ofthese cases into consideration
Data obtained in only one numerical experiment (match-ing the case when the laboratory experiments are performedwith no replication) were fitted by the use of four errorfunctions OLS ODRMPSD andHYBRDTheir expressionsare presented in Table 4 Least squares fitting of the linearized
8 The Scientific World Journal
Table 4 Definitions of error functions in cases when there are no replicated concentration measurements
+ (119862119890119894exp minus 119862119890119894cal
119862119890119894exp
)
2
]
]
data (LIN) was additionally applied only on the Langmuirand Freundlich equations
Formulae of the error functions in Table 4 are a bitdifferent from the ones reviewed in [4]The key reason is thatthe number of data points is constant all the time and therewill be no isotherm ranking based on this error functionsso the constants n-p were removed for the sake of simplicityParameter estimates obtained with these slightly modifiederror functions are the same because the multiplication oferror function with a constant nonzero value does not affectthe position of a global minimum
OLS MPSD and HYBRD are basically least squaresmethods with different types of weights included OLS is theapproach with all weights equal to one In case of MPSDassumption of constant percentage error is accepted andweighting by the equilibrium loading is applied For theHYBRD error functions weights are of the Poisson typeODR abbreviation in this context is used for the orthogonaldistance regression analog of the MPSD Assumption ofconstant percentage error is accepted for both of the axesand the weights are 1119902
119890exp and 1119862119890exp for the119910- and 119909-axesrespectively
The second group of calculations matched the case whenlaboratory experiments are performed in triplicate Means ofequilibrium sorbent loading (119902
119890119894exp) and equilibrium sorbateconcentration (119862
119890119894exp) from three subsequent numericalexperiments were passed to the following error functionsexperimental weighted orthogonal distance regression
(E3WODR) triplicate orthogonal distance regression(E3ODR) and weighted least squares (WLS) Definitionsof error functions for replicated measurements are listed inTable 5
It is important to say that E3WODR is the experimentalrealization of TODR Estimates of standard deviation of thevariables on the 119910- and 119909-axes are calculated as described inSection 231 incorporating the following equations
119904119889119910119894= radic3
sum
119895=1
(119902119890119894exp minus 119902119890119894exp)
2
119895
2 (19)
119904119889119909119894= radic3
sum
119895=1
(119862119890119894exp minus 119862119890119894exp)
2
119895
2 (20)
Weighting in the E3ODRmethod is based on themean valuesof equilibrium sorbent loading and equilibrium sorbateconcentration
119908119894=
1
119902119890119894exp
119908119894119889119894=
1
119862119890119894exp
(21)
For the WLS method instrumental weights are calculatedbased on (19)
The Scientific World Journal 9
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
+ (119862119890119894exp minus 119862119890119894cal
119862119890119894exp
)
2
]
]
data (LIN) was additionally applied only on the Langmuirand Freundlich equations
Formulae of the error functions in Table 4 are a bitdifferent from the ones reviewed in [4]The key reason is thatthe number of data points is constant all the time and therewill be no isotherm ranking based on this error functionsso the constants n-p were removed for the sake of simplicityParameter estimates obtained with these slightly modifiederror functions are the same because the multiplication oferror function with a constant nonzero value does not affectthe position of a global minimum
OLS MPSD and HYBRD are basically least squaresmethods with different types of weights included OLS is theapproach with all weights equal to one In case of MPSDassumption of constant percentage error is accepted andweighting by the equilibrium loading is applied For theHYBRD error functions weights are of the Poisson typeODR abbreviation in this context is used for the orthogonaldistance regression analog of the MPSD Assumption ofconstant percentage error is accepted for both of the axesand the weights are 1119902
119890exp and 1119862119890exp for the119910- and 119909-axesrespectively
The second group of calculations matched the case whenlaboratory experiments are performed in triplicate Means ofequilibrium sorbent loading (119902
119890119894exp) and equilibrium sorbateconcentration (119862
119890119894exp) from three subsequent numericalexperiments were passed to the following error functionsexperimental weighted orthogonal distance regression
(E3WODR) triplicate orthogonal distance regression(E3ODR) and weighted least squares (WLS) Definitionsof error functions for replicated measurements are listed inTable 5
It is important to say that E3WODR is the experimentalrealization of TODR Estimates of standard deviation of thevariables on the 119910- and 119909-axes are calculated as described inSection 231 incorporating the following equations
119904119889119910119894= radic3
sum
119895=1
(119902119890119894exp minus 119902119890119894exp)
2
119895
2 (19)
119904119889119909119894= radic3
sum
119895=1
(119862119890119894exp minus 119862119890119894exp)
2
119895
2 (20)
Weighting in the E3ODRmethod is based on themean valuesof equilibrium sorbent loading and equilibrium sorbateconcentration
119908119894=
1
119902119890119894exp
119908119894119889119894=
1
119862119890119894exp
(21)
For the WLS method instrumental weights are calculatedbased on (19)
The Scientific World Journal 9
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
34 Numerical Calculations The present work was carriedout using Windows-based PC with hardware configurationcontaining the dual processor AMD Athlon M320 (21 GHzeach) and with 3GB RAM All calculations were performedusing Matlab R2007b Perturbations were generated usingMersenne Twister random number generator For the pur-pose of fitting built-in Matlab function fminsearch was usedfor OLS MPSD HYBRD and WLS [34] It is based on theNelder-Mead simplex direct search algorithm Orthogonaldistance regression calculations used the dsearchn functionas a tool to find the set of (119909
119894 119910119894) values
It is important to note that one complete numericalexperiment and all associated computations were performedfor each simulation step for 2000 steps per combination (onetype of isotherm and one type of NPM) The reason simu-lations were chosen to have 2000 steps was the compromisebetween the aim to have resulting histograms of quality highenough to facilitate quantitative comparison with theory andto prevent the process from lasting unacceptably long Fewof the simulations took longer than 12 hours with the mostaveraging around 8 hours in length It could be noticed thatsimulations with higher values of noise standard deviationin general lasted longer The explanation is the rise in thenumber of function evaluations and the number of iterationsbefore the convergence is achieved
4 Results and Discussion
41 Postprocessing of the Results This study included 55numerical simulations on the whole (each of the fiveisotherms presented in Table 1 was paired with the 11 noiseprecision models presented in Table 3) and they were indi-vidually subjected to the same postprocessing routine Theraw results of each numerical simulation were 2000 (two orthree element) vectors of estimated parameters per model-ing approach Firstly the cases when the particular fittingalgorithm did not converge were counted and correspondingparameter estimates were removed from further consider-ation The second step in postprocessing procedure was tocalculate percentage errors for all the parameter estimates
where 119897 is the parameter ordinal number in the isothermequation (119897 isin 1 2 for the Langmuir Freundlich andJovanovic isotherms and 119897 isin 1 2 3 for Redlich-Petersonand Sips equations) and 119895 is the number of numerical exper-iments The third step was to identify and remove the outliervalues An 119890
119897119895value was considered as an outlier if it is greater
than 75th percentile plus 15 times the interquartile range or ifit is less than 25th percentile minus 15 times the interquartilerange The outliers were removed to once again match thesimulated and real cases It is common practice to discard thefitting results if they donot correspond to common sense Setsof percentage error on parameters obtained in the describedway were used for statistical evaluation
0 50 100 150 200 250 300 350
00010003
001002005010025050075090095098099
09970999
Normal distribution
Prob
abili
ty
minus100 minus50
KR
aRg
Figure 3 Normal probability plot for the parameters of Redlich-Peterson isotherm and OLS processing
42 Statistical Evaluation The normal probability plots wereused to graphically assess whether the obtained parameterestimates could come from a normal distribution Inspectionof such plots showed that in general they are not linearDistributions other than normal introduce curvature so itwas concluded that nonnormal distribution is involved Onerepresentative example is presented in Figure 3
Thus median of percentage error (mE) was used as ameasure of accuracy of the method based on a particularerror function and mean absolute relative error (MARE)
where 119896119891 is the number of converged fits for the particularmethod in a simulation was used as a measure of precisionThe individual performance of each modeling approach wasevaluated for each isotherm and for each type of noise
Comparison of the methods was done separately forthe two following groups of data observations with noreplications and data from triplicate experiments
Properties of different modeling approaches in case whenthe experiments are performed once are presented in Figures4 5 6 7 and 8 for the Langmuir Freundlich JovanovicRedlich-Peterson and Sips adsorption isotherms respec-tively
Properties of different modeling approaches in case whenthe experiments are performed three times are presentedin Figures A1ndashA5 as in Supplementary Material availableonline at httpdxdoiorg1011552014930879 for the Lang-muir Freundlich Jovanovic Redlich-Peterson and Sipsadsorption isotherms respectively
Due to a huge quantity of results obtained in this studysome rules had to be put on what is going to be presented infigures For every type of isotherm the figures are organizedto have two sections one where mE values are presented(figures labeled (a)) and the other where MARE values arepresented (figures labeled (b)) Each section of the plotcontains 7 subplots In one subplot the results of the applied
10 The Scientific World Journal
0
10
0
20
0
0
20
40
60
80
100Lin Quad HS
mE
()
H2 H5 RSD5 RSD10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
minus100 minus80
0
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
20
10
minus70
minus60
minus50
minus40
minus30
minus20
minus10
0
20
10
minus70
minus60 minus60
minus50
minus40
minus40
minus30
minus20
minus20
minus80
minus60
minus40
minus20 minus10
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
(a)
0
10
20
30
40
50
60
70
MA
RE (
)
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
160
KLqmax KL
qmax KLqmax KL
qmax KLqmax KL
qmax KLqmax
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
TODRODROLSMPSDHYBRDLIN
(b)
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Figure 4 Properties of differentmodeling approaches for determination of parameters in Langmuir isotherm (experiments performed once)(a) mE and (b) MARE
methods for one NPM are summarized Trends were noticedand discussed based on the six levels of homoscedastic noisebut in order tomake figures compact just two out of sixNPMs(H2 as an example of low noise andH5 as an example of highnoise) were presented as the first two subplots The next five(3ndash7) subplots were reserved for heteroscedastic NPMs Anadditional remark is valid for all the figures in the following
paragraph it was not possible to use the same scale in allsubplots due to large differences in the magnitudes of theoutcomes from subplot to subplot Nevertheless it does notintroduce any problem because the comparison of methodsis done in frames of a subplot and cross comparisonsbetween different NPMs (and subplots) are not of substantialimportance
The Scientific World Journal 11
0
005
01
015
02
0
5 0
0
1
2
3
4 0
0
05
1
0
2
4H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KF nF KF nF KF nF KF nF KF nF KF nF KFnF
minus02
minus015
minus01
minus005
minus30
minus25
minus20
minus15
minus10
minus5
minus45
minus4
minus35
minus3
minus25
minus2
minus15
minus1
minus3
minus2
minus1
minus05
minus09
minus08
minus07
minus06
minus05
minus04
minus03
minus02
minus01
minus35
minus3
minus25
minus2
minus2
minus4
minus6
minus8
minus10
minus15
minus1
minus05
(a)
0
05
1
15
2
25
3
35
4
45
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
0
10
20
30
40
50
60
0
10
20
30
40
50
60
70
80
90
MA
RE (
)
KFnF nF nF nF nF nF nF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
KF
TODRODROLSMPSDHYBRDLIN
(b)
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Figure 5 Properties of different modeling approaches for determination of parameters in Freundlich isotherm (experiments performedonce) (a) mE and (b) MARE
421 Experiments with No Replications and HomoscedasticNoise As expected for the very low level of homoscedasticnoise (H1 andH2noise precisionmodels) all of the examinedmethods performed well With the increasing of noise stan-dard deviation the accuracy and precision of the methodsbecame worse and differences between methods started toappear
Generalizing the results of all the five isotherms thefollowing statements can be placed Regardless of the
mathematical type of isotherm equation (rational power orexponential) and the number of parameters (two or three)the OLS method had the best properties In the groupof methods applicable in practice it achieved mE valuesclosest to zero and the lowest values of MARE almostidentical to ones determined by the theoretical methodTODR mE of the other tested methods showed higherdiscrepancy from zero and MARE values were higher ODRand MPSD methods had a very bad performance while
12 The Scientific World Journal
0
02
04
06
0
10
20
30
40
50
0
5
10
15
20
0
5
10
15
0
5
10
15
20
0
5
10
15
20
0
10
20
30
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
minus50
minus40
minus30
minus20
minus10
minus10
minus5
minus10
minus5
minus10
minus5
minus10
minus5
minus20
minus10
minus02
minus04
minus06
minus08
(a)
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
10
20
30
40
50
60
0
10
20
30
40
50
60
0
50
100
150
200
250
MA
RE (
)
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJqJ KJ
qJ KJ
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Figure 6 Properties of differentmodeling approaches for determination of parameters in Jovanovic isotherm (experiments performed once)(a) mE and (b) MARE
the results of the HYBRD method were somewhere inbetween
Closeness of the results of OLS and TODR methodsshowed that in case of homoscedastic noise the presence ofmeasurement error on both axes is not of great importanceas it could be expected What is more the weighting by1119902119890exp and 1119862119890exp for the 119910- and 119909-axes respectively in the
ODRmethod is actually wrong since the population standard
deviations are constant at all concentrations That is thereason why the ODR method is biased and of low precisionIn the recent study the different modeling approaches weretested on a Langmuir isotherm with perturbations of datawith the fixed error 119873(0 005) and with plusmn5 error propor-tional to concentration [35] Surprisingly authors found thatthe ODR gives the most accurate estimates (lowest meanstandard deviation and interquartile range) of the isotherm
The Scientific World Journal 13
0
1
2
minus100
minus50
0
50
0
5
10
0
5
10
0
10
20
0
5
10
0
20
40H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
minus8
minus7
minus6
minus5
minus4
minus3
minus2
minus1
minus35
minus30
minus25
minus20
minus15
minus10
minus5
minus30
minus25
minus20
minus15
minus10
minus5
minus20
minus15
minus10
minus5
minus50
minus40
minus40
minus60
minus80
minus30
minus20
minus20
minus10
(a)
0
5
10
15
20
25
30
35
40
45
50
0
10
20
30
40
50
60
70
80
90
0
20
40
60
80
100
120
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
160
20
40
60
80
100
120
0
50
100
150
200
250
0 0 0
MA
RE (
)
KRaR gKR
aR gKRaR gKR
aR gKRaR gKR
aR gKRaR g
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Figure 7 Properties of differentmodeling approaches for determination of parameters inRedlich-Peterson isotherm (experiments performedonce) (a) mE and (b) MARE
parameters among different methods when the experimentaldata have a fixed error
For the two parameter isotherms (Langmuir and Fre-undlich) linearized models were tested due to their pop-ularity Modeling of linearized Langmuir equation was theonly exception from the rule that the greater the populationstandard deviation of the noise the greater the discrepanciesof parameter estimates from their true values Regardless ofthe level of noise the LIN method presented equally bad
results ThemE was about minus65 andMARE in the range 60ndash85 for both of the parameters For the Freundlich isothermLIN model was more accurate than HYBRD MPSD andODR and had about the same variability as MPSD stillresigning in the group of modeling approaches whose usageis not advised in the case of homoscedastic data
422 Experiments with No Replications and HeteroscedasticNoise Looking at the subplots 3ndash7 of Figures 5ndash8 where
14 The Scientific World Journal
0
2
4
6
0
20
40
60
80
0
10
20
30
0
5
10
15
0
10
20
30
0
5
10
15
20
0
20
40
60
80H2 H5 Lin Quad HS RSD5 RSD10m
E (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
minus20
minus15
minus10
minus5
minus15
minus10
minus5minus2
minus4
minus6 minus40
minus20
minus40
minus20
minus30
minus20
minus10
minus30
minus20
minus10
(a)
0
2
4
6
8
10
12
14
16
18
0
10
20
30
40
50
60
70
80
90
0
5
10
15
20
25
30
35
40
45
0
5
10
15
20
25
30
35
40
0
10
20
30
40
50
60
0
5
10
15
20
25
30
35
40
45
0
20
40
60
80
100
MA
RE (
)
qS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmSqS KS
mSqS KSmS
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
TODRODROLSMPSDHYBRD
(b)
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
Figure 8 Properties of different modeling approaches for determination of parameters in Sips isotherm (experiments performed once) (a)mE and (b) MARE
the results of modeling Freundlich Jovanovic Redlich-Peterson and Sips isotherm are presented the pattern of theOLS method performance can be easily recognized For allof the isotherms and all of the tested NPMs this methodwas the poorest of all the methods in both the aspects of itsaccuracy and precision This is the expected result becausethe basic assumptions when the OLS method is valid are notmet It is interesting to note that for the Langmuir isotherm(Figure 4 subplots 3ndash7) OLS method was the most accuratemethod but the precision was still very very poor (always
more than 35 for the 119902max and more than 100 for the119870119871)
The same as for the other isotherm types the use of OLS forthe Langmuir isotherm is not advised in the presence of anytype of heteroscedastic noise
TheODRmethodwas generally as accurate as TODRThegreatest deviation ofmE from zerowasminus32 for the RSD10noise type and119870
119865parameter of Freundlich isotherm 57 for
the Lin noise type and 119870119869parameter of Jovanovic isotherm
minus75 for the HS noise type and 119886119877parameter of Redlich-
Peterson isotherm and minus109 for the HS noise type and 119902119878
The Scientific World Journal 15
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
parameter of Sips isotherm Only for the Langmuir isothermdifference between mE of the ODR and TODR was morepronounced reaching the absolute maximum value of 300for 119870
119871 The MARE of ODR was higher than the MARE
of TODR but the lowest among the other tested methodsTypical values of MARE of Langmuir isotherm were in therange 27ndash39 for 119902max and 397ndash704 for 119870
119871 For the
Freundlich isotherm MARE was below 10 for both of theparameters and all types of NPMs The precision in case ofRedlich-Peterson isotherm was roughly about 20 for 119870
119877
and g and two or three times higher for the 119886119877 For the Sips
isotherm the maximum MARE was 537 for 119886119877in case of
RSD10 but most of the time it was roughly about 30Direct comparison of ODR andMPSD qualifiedMPSD as thesecond best choice method Its accuracy and variability wereat the same level or closely followed the ODR The HYBRDmethod was neither accurate nor precise again performingbetween the worst (OLS) and the best (ODR) methods Thesame can be stated for the LIN method of the Langmuir andFreundlich isotherms
423 Experiments Performed in Triplicate In case when theequilibrium adsorption experiments are performed once andthe estimates of standard deviation of the measurement errorare not available weighting is restricted to be fixed or to besome function of measured variable When the experimentsare done in triplicate this restriction is released since theestimates of standard deviations could be obtained Theadsorption literature surprisingly rarely takes into accountthese important statistical details related to the processingof data in regression analysis with replicate measurementsCommonly but not properly data from triplicate measure-ments are just averaged and their mean values are furtheron processed like OLS Weighted regression is a way ofpreserving the information and thus should be preferred
In Figures A1ndashA5 (in Supplementary Material) theresults of modeling Langmuir Freundlich JovanovicRedlich-Peterson and Sips isotherm are presented At firstglance it can be noticed that the accuracy and precision ofthe methods are better than in case of one experiment perpoint
The E3WODRmethod had the best properties WLS per-formed slightly worse and E3ODR was ranked the thirdTheexception was only the Freundlich equation where MAREvalues tended to lower for E3ODR method in case of het-eroscedastic noise However difference between E3WODRand E3ODR was less than 1 and thus this particularbehavior is not of great importance
5 Conclusions and Recommendations
The accuracy of model parameters will depend on whetherthe appropriate conceptual model was chosen whether theexperimental conditions were representative of environ-mental conditions and whether an appropriate parameterestimation method was used
Recently our group faced the problem of modeling theadsorption isotherms [36 37] and the intention of this study
was to single out the method of parameter estimation whichis suitable for adsorption isotherms A detailed investigationhas been carried out to determine whether the mathematicaltype of isotherm function and the type of measurement noise(homoscedastic or heteroscedastic) are key factors that leadto modeling approach choice Commonly used methodsOLS and least squares of linearized equations methodsthat are far less present in literature HYBRD and MPSDand a method that is hardly ever used in adsorption fieldorthogonal distance regression were compared Evaluationsof these methods were conducted on a large number of datasets allowing precision and accuracy of parameter estimatesto be determined by comparison with true parameter values
It was demonstrated that trends that could be noticed donot show dependence on isotherm type Only the magnitudeof percent errors in parameter estimates classifies someof the equation types and their particular parameters asdifficult to fit (119886
119877in Redlich-Peterson isotherm) It was
shown that the impact of measurement error noise type issignificant Neglecting the information about noise structurecan lead to biased andor imprecise parameter estimatesand the researcher should obtain the information about themeasurement error type of the analytical method used forconcentration determination (Testing the analytical systemfor heteroscedasticity should be a part of validation protocolnecessary for the limit of detection calculationlinebreak[3839]) As expected OLS method performed superior in caseof homoscedastic noise no matter whether the noise is highor lowThe results for heteroscedastic noise types revealed thepotential of using ODRmethod with weighting proportionalto 1119902
119890exp and 1119862119890exp for the 119910- and 119909-axes respectively Inthis situation use of this method resulted in smaller bias andbetter precision Although frequently being used in presentadsorption literature for fitting adsorption isotherms OLSmethod is an unfavorable method for the heteroscedasticdata performing much worse than other nonlinear methods
The accuracy and variability of orthogonal distanceregression-based methods (ODR for experiments performedonce and E3ODR and E3WODR for experiments preformedin triplicate) are closely followed by the analog methods thatdo not take into account the influence of measurement erroron both axes MPSD and WLS
Linearization of isotherm equations was once againdiscarded in this study Since the survey of the literaturepublished in last decade showed that in over 95 of the liquidphase adsorption systems the linearization is the preferredmethod [4] a lot of attention should be put on education andspreading the right principles among researches
Further research that is currently in progress in our groupwill hopefully resolve the issue of adequatemodel selection inadsorption studies
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
concentration119889 Weights on the x-axis119890 Percentage error119892 The exponent in Redlich-Peterson isotherm119894 Indices for running number of data points119895 Running number of numerical experiments119896 Coefficient equal to Vm119896119891 Number of converged fits in a simulation119897 Parameter ordinal number in the isotherm
equation119870119865 Freundlich adsorption constant related to
adsorption capacity119870119869 The exponent in Jovanovic isotherm
119870119871 The equilibrium adsorption constant in Lang-
muir equation119870119877 The Redlich-Peterson isotherm parameter
119870119878 The Sips isotherm parameter
119898 The weight of adsorbentmE Median percentage error119898119878 The Sips model exponent
MARE Mean absolute relative error119899 Number of observations119899119865 Adsorption intensity in Freundlich isotherm
119901 Number of parameters119902119890exp Experimentally determined adsorbent equi-
librium loading119902119890exp Mean of three experimentally determined
adsorbent equilibrium loadings119902119890cal Equilibrium adsorbent loading calculated by
isotherm equation119902max The Langmuir maximum adsorption capacity119902119878 The Sips maximum adsorption capacity
119902119869 The Jovanovic maximum adsorption capacity
119904119889119909 Estimates of population standard deviationfor independent variable
119904119889119910 Estimates of population standard deviationfor dependent variable
119881 The volume of adsorbate solution119908 Weights on the y axis(119909119894 119910119894) Data point on the fitted curve that is the
closest to the observation (xi yi)
Greek Symbols
120576 Measurement error in dependent variable120575 Measurement error in independent variable120590120576 Population standard deviation of measurementerror in dependent variable
120590120575 Population standard deviation of measurementerror in independent variable
120590120575119900 Population standard deviation of measurementerror in initial concentration
120590120575119890 Population standard deviation of measurementerror in equilibrium concentration
120579 Vector of true parameters Vector of estimated parameters120593 Vector of distances in x direction between
the observations and true model Vector of distances in x direction between
the observations and model function withestimated parameters
120595 Vertical distance between observation andmodel function
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This study was supported by the Ministry of EducationScience and Technological Development of the Republic ofSerbia (Project no III 43009)
References
[1] R T Yang Adsorbents Fundamentals and Applications JohnWiley amp Sons Hoboken NJ USA 2003
[2] M Joshi A Kremling and A Seidel-Morgenstern ldquoModelbased statistical analysis of adsorption equilibriumdatardquoChem-ical Engineering Science vol 61 no 23 pp 7805ndash7818 2006
[3] G Alberti V Amendola M Pesavento and R Biesuz ldquoBeyondthe synthesis of novel solid phases review on modelling ofsorption phenomenardquo Coordination Chemistry Reviews vol256 no 1-2 pp 28ndash45 2012
[4] K Y Foo and B H Hameed ldquoInsights into the modeling ofadsorption isotherm systemsrdquo Chemical Engineering Journalvol 156 no 1 pp 2ndash10 2010
[5] O M Akpa and E I Unuabonah ldquoSmall-sample correctedakaike information criterion an appropriate statistical tool forranking of adsorption isothermmodelsrdquoDesalination vol 272no 1ndash3 pp 20ndash26 2011
[6] M I El-Khaiary ldquoLeast-squares regression of adsorption equi-librium data comparing the optionsrdquo Journal of HazardousMaterials vol 158 no 1 pp 73ndash87 2008
[7] M I El-Khaiary and G F Malash ldquoCommon data analysiserrors in batch adsorption studiesrdquo Hydrometallurgy vol 105no 3-4 pp 314ndash320 2011
[8] K V Kumar K Porkodi and F Rocha ldquoIsotherms and thermo-dynamics by linear and non-linear regression analysis for thesorption of methylene blue onto activated carbon comparisonof various error functionsrdquo Journal of Hazardous Materials vol151 no 2-3 pp 794ndash804 2008
[9] K V Kumar K Porkodi and F Rocha ldquoComparison of variouserror functions in predicting the optimum isotherm by linearand non-linear regression analysis for the sorption of basic red9 by activated carbonrdquo Journal of Hazardous Materials vol 150no 1 pp 158ndash165 2008
[10] M C Ncibi ldquoApplicability of some statistical tools to predictoptimum adsorption isotherm after linear and non-linearregression analysisrdquo Journal ofHazardousMaterials vol 153 no1-2 pp 207ndash212 2008
The Scientific World Journal 17
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
[11] G Limousin J P Gaudet L Charlet S Szenknect V Barthesand M Krimissa ldquoSorption isotherms a review on physicalbases modeling and measurementrdquo Applied Geochemistry vol22 no 2 pp 249ndash275 2007
[12] C H Giles D Smith and A Huitson ldquoA general treatment andclassification of the solute adsorption isotherm I TheoreticalrdquoJournal of Colloid And Interface Science vol 47 no 3 pp 755ndash765 1974
[13] C Hinz ldquoDescription of sorption data with isotherm equa-tionsrdquo Geoderma vol 99 no 3-4 pp 225ndash243 2001
[14] K V Kumar C Valenzuela-Calahorro J M Juarez M Molina-Sabio J Silvestre-Albero and F Rodriguez-Reinoso ldquoHybridisotherms for adsorption and capillary condensation of N2 at77K on porous and non-porous materialsrdquo Chemical Engineer-ing Journal vol 162 no 1 pp 424ndash429 2010
[15] I Langmuir ldquoThe constitution and fundamental properties ofsolids and liquidsmdashpart I solidsrdquo The Journal of the AmericanChemical Society vol 38 no 2 pp 2221ndash2295 1916
[16] H M F Freundlich ldquoOver the adsorption in solutionrdquo Journalof Physical Chemistry vol 57 pp 385ndash471 1906
[17] D D Do Adsorption Analysis Equilibria and Kinetics ImperialCollege Press London UK 1998
[18] O Redlich and D L Peterson ldquoA useful adsorption isothermrdquoJournal of Physical Chemistry vol 63 no 6 pp 1024ndash1026 1959
[19] R Sips ldquoOn the structure of a catalyst surfacerdquo Journal ofChemical Physics vol 16 no 5 pp 490ndash495 1948
[20] D Jukic K Sabo and R Scitovski ldquoTotal least squares fittingMichaelis-Menten enzyme kinetic model functionrdquo Journal ofComputational and Applied Mathematics vol 201 no 1 pp230ndash246 2007
[21] J C Miller and J N Miller Statistics and Chemometrics forAnalytical Chemistry Pearson Prentice Hall Edinburgh UK5th edition 2005
[22] N R W H Press S A Teukolsky W T Vetterling andB P Flannery Numerical Recipes in C The Art of ScientificComputing Cambridge University Press Cambridge UK 2ndedition 2002
[23] A G Asuero and G Gonzalez ldquoFitting straight lines withreplicated observations by linear regression III Weightingdatardquo Critical Reviews in Analytical Chemistry vol 37 no 3 pp143ndash172 2007
[24] L S Matott and A J Rabideau ldquoISOFITmdasha program forfitting sorption isotherms to experimental datardquo EnvironmentalModelling and Software vol 23 no 5 pp 670ndash676 2008
[25] A Martınez F J del Rıo J Riu and F X Rius ldquoDetecting pro-portional and constant bias in method comparison studies byusing linear regression with errors in both axesrdquo Chemometricsand Intelligent Laboratory Systems vol 49 no 2 pp 179ndash1931999
[26] B Carmichael and A Coen ldquoAsset pricing models with errors-in-variablesrdquo Journal of Empirical Finance vol 15 no 4 pp778ndash788 2008
[27] T Soderstrom K Mahata and U Soverini ldquoIdentification ofdynamic errors-in-variables models approaches based on two-dimensional ARMAmodeling of the datardquo Automatica vol 39no 5 pp 929ndash935 2003
[28] M Tod A Aouimer and O Petitjean ldquoEstimation of phar-macokinetic parameters by orthogonal regression compari-son of four algorithmsrdquo Computer Methods and Programs inBiomedicine vol 67 no 1 pp 13ndash26 2002
[29] E Voigtman ldquoLimits of detection and decisionmdashpart 3rdquo Spec-trochimica Acta B vol 63 no 2 pp 142ndash153 2008
[30] A Sayago M Boccio and A G Asuero ldquoFitting straightlines with replicated observations by linear regression the leastsquares postulatesrdquo Critical Reviews in Analytical Chemistryvol 34 no 1 pp 39ndash50 2004
[31] F C Wu B L Liu K T Wu and R L Tseng ldquoA new linearform analysis of Redlich-Peterson isotherm equation for theadsorptions of dyesrdquo Chemical Engineering Journal vol 162 no1 pp 21ndash27 2010
[32] R I Yousef B El-Eswed and A H Al-Muhtaseb ldquoAdsorptioncharacteristics of natural zeolites as solid adsorbents for phenolremoval from aqueous solutions kinetics mechanism andthermodynamics studiesrdquo Chemical Engineering Journal vol171 no 3 pp 1143ndash1149 2011
[33] D D Maksin A B Nastasovic A D Milutinovic-Nikolic etal ldquoEquilibrium and kinetics study on hexavalent chromiumadsorption onto diethylene triamine grafted glycidyl methacry-late based copolymersrdquo Journal of Hazardous Materials vol209-210 pp 99ndash110 2012
[34] MATLAB Release The MathWorks Natick Mass USA 2007[35] J Poch and I Villaescusa ldquoOrthogonal distance regression a
good alternative to least squares for modeling sorption datardquoJournal of Chemical andEngineeringData vol 57 no 2 pp 490ndash499 2012
[36] B M Jovanovic V L Vukasinovic-Pesic and L V RajakovicldquoEnhanced arsenic sorption by hydrated iron (III) oxide-coatedmaterialsmdashmechanism and performancesrdquoWater EnvironmentResearch vol 83 no 6 pp 498ndash506 2011
[37] B M Jovanovic V L Vukasinovic-Pesic N Dj Veljovic andL V Rajakovic ldquoArsenic removal from water using low-costadsorbentsmdasha comparative studyrdquo Journal of Serbian ChemicalSociety vol 76 pp 1437ndash1452 2011
[38] L V Rajakovic D D Markovic V N Rajakovic-OgnjanovicandD Z Antanasijevic ldquoReview the approaches for estimationof limit of detection for ICP-MS trace analysis of arsenicrdquoTalanta vol 102 pp 79ndash87 2012
[39] H Huang and G A Sorial ldquoStatistical evaluation of an analyti-cal IC method for the determination of trace level perchloraterdquoChemosphere vol 64 no 7 pp 1150ndash1156 2006
