Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 102 (2013) 319–326

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Spectrochimica Acta Part A: Molecular andBiomolecular Spectroscopy

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Applying constraints on model-based methods: Estimation of rate constantsin a second order consecutive reaction

Mohsen Kompany-Zareh ⇑, Maryam KhoshkamDepartment of Chemistry, Institute for Advanced Studies in Basic Sciences (IASBS), GavaZang 45137-66731, Zanjan, Iran

h i g h l i g h t s

" In this study two rate constants of asecond order consecutive reactionwere estimated.

" Two types of model basedtechniques were used to estimatethe rate constants of reaction.

" It was shown that different rateconstants were obtained whendifferent methods were applied.

" It was shown that proper constraintscan reduce the range of acceptableresults.

1386-1425/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.saa.2012.10.030

⇑ Corresponding author. Tel.: +98 241 4153123; faxE-mail address: kompanym@iasbs.ac.ir (M. Kompa

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Received 11 June 2012Received in revised form 7 October 2012Accepted 19 October 2012Available online 1 November 2012

Keywords:Second order kineticConsecutive reactionHard modelingNonlinear fittingDiazonium saltsOrtho-Amino benzoeic acid

a b s t r a c t

This paper describes estimation of reaction rate constants and pure ultraviolet/visible (UV–vis) spectra ofthe component involved in a second order consecutive reaction between Ortho-Amino benzoeic acid (o-ABA) and Diazoniom ions (DIAZO), with one intermediate. In the described system, o-ABA was notabsorbing in the visible region of interest and thus, closure rank deficiency problem did not exist. Con-centration profiles were determined by solving differential equations of the corresponding kinetic model.In that sense, three types of model-based procedures were applied to estimate the rate constants of thekinetic system, according to Levenberg/Marquardt (NGL/M) algorithm. Original data-based, Score-basedand concentration-based objective functions were included in these nonlinear fitting procedures. Resultsshowed that when there is error in initial concentrations, accuracy of estimated rate constants stronglydepends on the type of applied objective function in fitting procedure. Moreover, flexibility in applicationof different constraints and optimization of the initial concentrations estimation during the fitting proce-dure were investigated. Results showed a considerable decrease in ambiguity of obtained parameters byapplying appropriate constraints and adjustable initial concentrations of reagents.

� 2012 Elsevier B.V. All rights reserved.

Introduction

The spectra measured from a chemical reaction can render atwo way data matrix, which contains both the reaction kineticinformation (such as rate constants) and the pure spectrum of eachcomponent involved in the reaction. Usually it is of interest to

ll rights reserved.

: +98 241 4153232.ny-Zareh).

know end points in the reaction, time needed to obtain an opti-mum yield, time to reach 90% completion of the reaction, informa-tion on presence of impurities and deviation of the reaction fromthe expected mechanism. Methods have been proposed and usedto estimate reaction rate constants from spectral data of chemicalreactions [1–9]. Univariate methods, often involve using singlewavelength or measurement to determine the rate constants ofthe kinetic system by fitting to a known model. There are somelimitations when using such methods: firstly, it is often not easyto find selective wavelength for all component (especially in

320 M. Kompany-Zareh, M. Khoshkam / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 102 (2013) 319–326

UV–vis and NIR spectroscopy, but Raman for example is quiteselective), secondly, it is necessary to know the exact initial con-centration of reactants and finally, the reactions and chemicals in-volved are highly affected by physical and chemical parameterslike temperature or pH [10,11].

Some chemometrics methods apply multivariate data analysisfor solving the kinetic models taking into account both kineticand spectroscopic information [12]. Multivariate data, allow softor model free analysis of the measurements [13–15]. Many softmethods have been applied to estimate reaction rate constantsand pure spectra of involved component in the reaction [2,3,16].All these methods attempt to analyze the data based on very fewconstraints, such as non-negativity of concentration and spectralprofiles or bilinearity [13,15,17–19]. The main advantage of softmethods is that once analyzing without applying a specific kineticmodel, possible errors caused by assumption of an uncorrectedmodel will be avoided [13]. However, the disadvantage of modelfree analysis is that obtained spectral and concentration profilesare not unique and show intensity and rotational ambiguities[3,4]. In some cases, selectivity in the data matrix removes rota-tional ambiguity. It is proved that closure constraint removesintensity ambiguity and by imposing kinetic model uniqueness isachieved [1,4,20]. Besides, application of constraints such as non-negativity of spectral and concentration profiles, unimodality orclosure in soft methods can decrease ambiguities to a large extent,but not completely [17,18].

Model-based or hard modeling techniques were applied basedon the postulation of a chemical model. Most model-based meth-ods are based on type of the objective function available and havebeen described in previous work [19]. All these methods are basedon fitting the nonlinear parameters (rate constants in kinetic case)of the chemical reactions while the linear parameters are elimi-nated (spectra of pure components). These methods present somebenefits but also some disadvantages [5,21–27]. The main disad-vantage of model-based methods is errors caused by assumingincorrect model for the chemical system in hand. Choosing the cor-rect model depends on the previous information available aboutthe experimental system [23]. The main advantage of these meth-ods is uniqueness of the obtained parameters and one can expectto obtain the same parameters by applying any algorithm of mod-el-based methods, if the postulated model is correct. Neuhold et al.showed that uncertainties in initial concentrations and dosing rateof reactants have a major contribution in the error of estimatedrate constants. To calculate the contribution of these factors, theyincorporated these uncertainties during hard model optimizationprocedure [28].

In this work, reaction between o-ABA and DIAZO was studiedand different Hard modeling methods based on different objectivefunctions, were applied to estimate two rate constants of thechemical reaction being studied. Molecular structure of the re-agents is shown in Fig. 1a and possible reaction mechanism isshown in Fig. 1b.

Effect of errors in estimating initial concentrations of reagentson distribution of estimated rate constants was investigated. Rateconstants of a second order consecutive reaction were estimated,firstly using fixed values of initial concentration of the reagents,and secondly by optimizing them during fitting. For estimatingthe rate constants, non-negativity was the only constraint appliedto estimate pure spectra of the absorbing component (using fastnon-negativity least squares proposed by Bro et al. [29]). In othercases, additional constraints (e.g. local rank information) were ob-tained from Fixed Size Moving Window-Evolving Factor Analysis(FSMWFA) and were imposed during the fitting procedure [30–33]. Effects of applied constraints and optimization of the valueof initial concentrations during the fitting procedure, on distribu-tion of the estimated rate constants from different methods were

investigated. The goal was to investigate reducing effect of apply-ing suitable constraints and adjustable initial concentration of re-agents on ambiguity of the obtained kinetic parameters.

Materials and methods

Reagents, solutions and procedure

o-ABA, citric acid and sodium nitrite were purchased fromMerck and used without further purification. Sulfanilamide, sulfa-mic acid and Sodium Dodecyl Sulfate (SDS) were purchased fromFluka. To buffer the pH, a mixture containing 0.25 M citric acidand sodium hydroxide were prepared and pH was adjusted to6.5 in all the experiments. Stock solution of o-ABA was preparedby dissolving 17.5 mg o-ABA in 1 ml ethanol, and diluted withwater until 50 ml. A stock solution of sulfanilamide 4 � 10�2 Mwas prepared in HCl 0.3 M. Solutions of sodium nitrite 0.2 M, sul-famic acid 0.5 M and SDS 20% were also prepared. To prepare astock solution of DIAZO 1 � 10�2 M, 12.5 ml sulfanilamide and15 ml sodium nitrite were added into a 50 ml volumetric flaskand were allowed to react for 10 min. Then 15 ml sulfamic acidwas added to remove the excess nitrite. After 15 min the solutionwas made up to mark with water. The fresh DIAZO solution wasprepared daily. In the considered system o-ABA was not an absorb-ing component but DIAZO, the intermediate and the product wereabsorbing [34–36]. Different ratios from o-ABA and DIAZO wereused in experiments as listed in Table 1.

Apparatus

The experiments were spectrometrically monitored in a 1 cm-length glass cuvette, using S-2150 diode-array-UV/Vis spectropho-tometer (SCINCO Co. Ltd) at room temperature. The time intervalbetween recording two successive spectra was 15 s. The spectrawere digitized with one data point per nanometer and the wave-length range was 300–550 nm. The pH values were adjusted withcitrate buffer and controlled with a Metrohm 713 pH- meter pro-vided with combined glass electrode. Data processing was per-formed using MATLAB version 6.5 (The Mathworks, Natick, MA)on a Pentium IV personal computer equipped with 256 Mb RAM.For the case considered in this study, concentration profiles ofthe components were obtained by numerical solution of differen-tial equations, using ODE functions in MATLAB [1,37–42]. Nonlin-ear fitting was performed by Newton–Gauss–Levenberg/Marquardt (NGL/M) algorithm [28,43,44].

Theory

Collected spectra were arranged in a matrix, X. Table 2 showsthe applied methods for estimation of the rate constants of thekinetic system. Details of the methods are described in previouspublication [19]. As shown in the table, three sets of objective func-tions were applied in nonlinear fitting procedure: (1) X-basedmethods consisted of mixX, hrdX for individual data matrices andaugX for augmented data matrix. These methods use kX� bXk asobjective function. where X is data matrix. (2) A T-based method,such as pcrT, that uses kT� bTk as objective function (T is score ma-trix). (3) C-based methods that include mixD, pcrC and pcrD forindividual data matrices and augC for augmented data matrix.Objective function for C-based methods is kC� bCk or kD� bDk,where C and D are concentration profiles of component and pseudocomponent respectively. pcrC and pcrD are known as targettransform fitting methods [22–24].

Each one of these three sets of methods has both advantagesand disadvantages. In the case of linear dependencies between col-

Fig. 1. (a) Molecular structure of reagents; I: amino benzoic acid; II: diazonium ion of sulfanilamide. (b) Mechanism of reaction between ortho aminobenzoic acid anddiazonium ion.

Table 1Different initial concentrations of reactants in experiments.

Ratio A0/D0 o-ABA (A) (lM) DIAZO (D) (lM) Number of recorded spectra

Dataset 1 0.20 50 247 150Dataset 2 0.18 50 280 151Dataset 3 0.61 150 247 150Dataset 4 0.54 150 280 150Dataset 5 1.11 250 225 100Dataset 6 1.01 250 247 131

Table 2Applied methods in this study and the type of projection in each method.

Method Objective function Type of projection

Mixed spectra mixX jjX� bXjj2 � kX� kDkDþXk2 X into D

mixD kkD� bDk2 � kkD� XXþkDk2 D into X

Hard modeling hrdX kX � bXjj2 � kX� kCkCþXk2 X into C

hrdC kkC� bCk2 � kkC� XXþkCk2 C into X

Target testing (PCR) pcrT kT� bTk2 � kT� kCkCþTk2 T into C

pcrC kkC � PCRCk2 � kkC � TT+kCk2 C into TpcrD kkD � PCRDk2 � kkD � TT+kDk2 D into T

Augmentation augX kXaug � bXaugk2 � kXaug � kCaugkCþaug Xaugk2 X into C

augC kkCaug � bCaugk2 � kkCaug � Xaug XþaugkCaugk2 C into X

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umns of X, singularity problem will arise when calculating inverseof (XTX), to estimate pinv(X). In such cases, it is preferred to use Tbased methods instead of X based ones. The main disadvantage ofX and T based methods, is that they are not convenient when deal-ing with nonlinear data matrices. If it is the case, C-based methodsare preferred to X and T based methods. In the presence of shiftand drift in the data, X space includes other variations in additionto concentration changes. In situations like this, projection of Conto X results in proper regression models and C based methodsare advantageous.

It can be concluded that when space of data, X, is different fromspace of C, different results are achieved from X based and C basedmethods and, if these approaches are applied to determine param-eters of a hard model by fitting, obtained parameters from the twoapproaches will be different. In case of imposing an improper con-straint or using incorrect parameter values for construction of dataduring the fitting procedure, an unwanted restriction is driven tothe data and results of X based and C based procedures are ex-pected to be different.

To check this, imposing the constraints, all the mentionedmethods were applied to the experimental data to determine mod-el parameters. Although experimental data sets were not rank defi-cient (o-ABA is not spectrally absorbing component), mixD andpcrD were applied as described in previous work [19].

Results and discussion

Number of significant components

UV–vis absorption spectra of six samples containing six differ-ent initial concentrations of DIAZO and o-ABA (according to Ta-ble 1) were recorded as a function of time in room temperature,to get second order consecutive reaction (a mesh of sixth datasetsis shown in Fig. 2). Corresponding logarithm of eigenvalues vs. PCnumber from six column wise augmented datasets are shown inFig. 3. It indicates that there are three significant PCs in consideredsystem. Eigenvalue plot of each individual dataset is similar to thatpresented in Fig. 3 (results were not shown).

Fig. 2. Mesh plot of UV–Vis spectrum recorded during the reaction time in dataset 6. The ratios of o-ABA and DIAZO is 1.01.

Fig. 3. Plot of logarithm eigenvalues vs. the number of principal components foraugmented data.

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Presence of three principal components (the chemical rank ofindividual and augmented datasets) shows presence of threeabsorbing component as described in experimental section andthere are no interferences, shifts or drifts in each dataset. Thus,there is no rank deficiency problem in the investigated system.All the methods, except hrdX and hrdC, can be used in a rank defi-cient system as described in previous work [19].

Errors calculations in estimated parameters

Knowing Jacobian matrix (J) the error value for ith estimatedparameter, ri, can be calculated as follows:

ri ¼ rY

ffiffiffiffiffihij

qð1Þ

in which hij is an element of Hessian matrix H = JT J and rY is stan-dard deviation of elements of residual matrix, R, which can becalculated from the following equation:

rY ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RSSðnt � nkÞ � ððnc � nkÞ þ npÞ

sð2Þ

In Eq. (2), RSS is the sum of squares of R matrix elements andthe dominator of equation is degrees of freedom and equals tothe number of residual elements which represent the number ofexperimental parameters (nt � nk) minus the number of fittedparameters. Scalars nt, nk, nc, and np indicate the number of timepoints, wavelengths, components, and significant principal compo-nents, respectively. error calculations are described completely byPuxty et al. [41].

Fixed initial concentration

No constraint (condition 1)Since all the six reactions were performed under the same con-

dition and the only difference between them was the difference be-tween initial concentrations of o-ABA and DIAZO, the two rateconstants were expected to be the same in six datasets. In thisway, the average of k1 and k2 values from the six experiments wereconsidered. The averages were estimated by applying differenthard models on each set of data.

The averages of estimated rate constants from six datasets incondition 1 are listed in Table 3. Condition 1 means fixed initialconcentrations of the reactants and applying no constraints duringoptimization of the rate constants. Distributions of estimated rateconstants are shown in Fig. 4a and b. Fig. 4a shows that the esti-mated rate constants from mixD, pcrC, pcrD, and hrdC methods (re-sulted from the projection of concentration matrix onto dataspace) and those from mixX, pcrT and hrdX methods (resulted fromthe projection of data or score matrix on concentration matrixspace), appear in different regions and clusters of the plots.Fig. 4b includes the same data, but they are organized accordingto data sets. Different data sets are not clustered in different re-gions of the plot. Fig. 4b illustrates that distribution of the resultsare not due to the difference in experimental conditions. As thereis no interference or shift or drift in data, (which can be understoodfrom the rank of the augmented data) C and X methods should givethe same results. So, observed differences between the estimatedrate constants obtained from different methods and experimentsare not acceptable.

To obtain more information about the system, fixed size movingwindow evolving factor analysis (FSMWEFA) was applied to theexperimental data. FSMWEFA gives information about selectivewavelength regions and allows estimating the number of absorb-

Table 3Average of estimated rate constants by applying different methods in different conditions.

Condition Average of rateconstants

Mixed spectra Target testing (PCR) Hard modeling Augmentation

mixX mixD pcrT pcrC pcrD hrdX hrdC augX augC

1a k1avb 3.95 (±0.31) 1.79 (±0.45) 3.94 (±0.30) 1.25 (±0.57) 1.25 (±0.70) 3.96 (±0.30) 1.26 (±0.56) 2.94 (±0.02) 1.79 (±0.01)

k2avb � 103 3.10 (±0.39) 3.82 (±0.78) 3.11 (±0.40) 5.62 (±1.70) 5.67 (±1.74) 3.10 (±0.39) 5.47 (±1.40) 3.72 (±0.01) 3.74 (±0.032)

2 k1av 3.89 (±0.21) 3.71 (± 0.28) 3.95 (±0.17) 3.12 (±0.36) 3.78 (±0.28) 3.96 (±0.24) 3.97 (±0.15) 3.53 (±0.000) 3.53 (±0.000)k2av � 103 3.13 (±0.54) 2.90 (±0.69) 3.12 (±0.34) 2.20 (±1.17) 2.68 (±0.70) 3.11 (±0.50) 2.02 (±0.67) 3.40 (±0.003) 2.31 (±0.008)

3 k1av 3.80 (±0.19) 3.57 (±0.03) 3.82 (±0.18) 3.66 (±0.003) 3.59 (±0.03) 3.83 (±0.19) 3.67 (±0.01) 3.71 (±4.23) 3.54 (±2.65)k2av � 103 3.02 (±0.37) 5.79 (±1.31) 3.02 (±0.37) 3.46 (±0.10) 6.16 (±1.64) 3.02 (±0.37) 4.13 (±0.65) 2.58 (±1.07) 3.578 (±0.01)

4 k1av 3.77 (±0.11) 3.76 (±0.04) 3.73 (±0.10) 3.78 (±0.04) 3.76 (±0.07) 3.78 (±0.056) 3.16 (±0.42) 3.67 (±0.07) 3.67 (±0.01)k2av � 103 3.01 (±0.59) 3.42 (±0.45) 3.11 (±0.49) 3.06 (±0.19) 3.41 (±0.58) 3.78 (±0.03) 3.30 (±0.38) 2.99 (±0.74) 2.76 (±0.13)

a Condition 1: No constraint and fixed initial concentration. Condition 2: Selectivity constraint and fixed initial concentration. Condition 3: No constraint and optimizationof initial concentration. Condition 4: Spectral similarity constraint and optimization of initial concentration.

b k1av (mol�1 L S�1) and k2av (S�1) in each row, is the average of six estimated rate constants that obtained from the six experimental data sets, with ratios of A0/D0 equal to:0.20, 0.18, 0.61, 0.54, 1.11 and 1.

M. Kompany-Zareh, M. Khoshkam / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 102 (2013) 319–326 323

ing components coexisting in a certain wavelength region. In thefirst step of a FSMWEFA analysis, singular value decomposition(SVD) was applied in a window containing the first m wavelengthsof data matrix. In the second step, the window was shifted onewavelength. Thus, this second window contains wavelengths from2 to m + 1. The window shifts until it contains the last m wave-length. Each window is decomposed by SVD and m singular valuesare plotted as a function of the wavelengths. By applying FSMWE-FA on dataset 6 in wavelength direction, as shown in Fig. 4c in theregion from wavelength 480 nm to 500 nm, rank of the experimen-tal data is one. This region of rank one, was also observed for theother sets of experimental data. The rank of the considered regionfor column wise augmented data was one.

Investigation of first row of experimental datasets, where theprogress of reaction was not significant, showed that the initialsubstance, DIAZO, has no absorbance in the region of rank one.Thus by presence of rank one region in datasets, two cases are pos-sible: (1) presence of selective region for one of the componentsspectrum (intermediate or product) and (2) presence of spectralsimilarity (linear dependency) between intermediate and productin this region.

Similarity constraint (condition 2)Both possible cases were examined for all of the datasets and

the rate constants were calculated using all methods. If selectivity(presence of one absorbing component) was the reason for forma-tion of rank one region, the score plot would be proportional to theconcentration profile of product or intermediate. Since the shape ofthe score plot is similar to concentration profile of product, itseems that there is a selective region for product. To estimate therate constants of system, considering case 1, the following con-straints were applied during the optimization:

(i) Non-negativity constraints to estimated spectral profiles.(ii) Zero absorbance for DIAZO in wavelengths 420–500 nm,

according to the first row of datasets.(iii) Maximum correlation between estimated concentration

profile of product and score plot from rank one region.(iv) Zero absorbance for intermediate in rank one region.

By constraints ii to iv one deals with vectors. To apply con-straints on these vectors during the optimization process, each vec-tor was converted to a matrix of the same size as the data orconcentration matrix, depending on the utilized method and opti-mization was performed by allocating proper weight to each con-straint. Using constraints i to iv of case 1, methods augX and augCrendered no reasonable results and negative rate constants were

obtained. Since using the augmented data matrix includes moreinformation about the system and is reliable, it seems that assump-tion of the presence of a selective region in this system (case 1) isnot reasonable. Thus presence of spectral similarity, the secondpossible case, was investigated.

Presence of spectral similarity (case 2) was considered forestimation of rate constants. In this case, during the optimiza-tion constraints i and ii, explained above, were applied butinstead of constraints iii and iv, the following constraint wereapplied:

(v) Linear relation between the estimated spectral profile ofproduct and intermediate in region of rank one.

The distribution of estimated rate constants in case 2 is dis-played in Fig. 4d. A comparison between Fig. 4a and d shows thatthe distribution of obtained results in case 2 is lower than that incase 1 Fig. 4a. This shows improvement in the results when a con-straint, such as similarity of spectral profiles, is applied during amodel-based resolution.

Optimized initial concentration

No constraint (condition 3)When the initial concentrations of the reactants were fixed dur-

ing the optimization, estimated values of the rate constants be-came dependent on the values of initial concentrations.Therefore, any change in the applied input values for initial con-centrations resulted in a change in the estimated rate constants.However by optimization of initial concentration values, in addi-tion to the rate constants, the estimated rate constants becameindependent of the applied input values for the initial concentra-tions [33]. The average of obtained rate constants from optimiza-tion of initial concentrations during the resolution iterations aresummarized in Table 3 as condition 3. Fig. 4e displays the distribu-tion of estimated rate constants by optimization of initial concen-tration of o-ABA and DIAZO. It is clear that the spread of estimatedrate constants is smaller than conditions 1 and 2 in which the ini-tial concentrations were fixed during the optimization, especiallyfor k1. From figure it is clear that the distribution of estimated k2

was not changed by optimization of initial concentrations. The rea-son is that k2 is a first order rate constant and is unaffected by ini-tial concentrations of reactants. Thus optimization of initialconcentrations of reactants has no effect on distribution of k2.But k1 is a second order rate constant and is sensitive to initial con-centrations of reactants.

Fig. 4. Distribution of estimated rate constants in different datasets applying no constraint and fixed initial concentration of reagents: considering the applied methods (a),and considering the utilized data sets (b). Resulted log(eigenvalue)s as a function of wavelength, from application of FSMWFA in wavelength direction on dataset 6 (c).Distribution of estimated rate constants in different datasets: (d) in presence of spectral similarity constraint and fixed initial concentration of reagents, considering theapplied methods; (e) without applying any constraint and by optimizing initial concentrations of reagents; (f) applying spectral similarity constraint and by optimizing initialconcentration of reagents.

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Spectral similarity constraint (condition 4)In previous sections, it was shown that the constraint of spectral

similarity decreased the difference between the estimated rateconstants from different methods. In this section, in addition tooptimizing initial concentrations value of the reagents, the con-straint of spectral similarity was applied. Results from applicationof all methods on all data sets are shown in Fig. 4f. Comparison ofFig. 4f–d (condition 2) and Fig. 4e (condition 3) shows that applica-tion of both spectral similarity constraint and optimization of ini-tial concentrations (condition 4) lowers the difference betweenestimated rate constants obtained from different methods morethan conditions 2 and 3. Results from applying both spectral sim-ilarity constraint and optimization of initial concentrations aresummarized in Table 3, as condition 4. As shown in the table, incondition 4 estimated values for the rate constants from single setsand augmented data are similar.

As shown in Table 3 the estimated rate constants from augC andaugX methods, are similar to each other when the spectral similar-ity constraint and optimization of initial concentrations were ap-plied in both methods (condition 4). Fig. 5a and b shows theestimated concentration profiles from augC and augX which aresimilar to each other. In Fig. 5c estimated pure spectra from augCmethod is shown.

The rate constants were estimated from Newton–Gauss–Levn-berg/Marquardt algorithm (NGLM), which is based on estimationof Jacobian matrix. The estimated error values for each parameter

were calculated from Jacobian matrix. The results showed that theestimated rate constants from different methods are not similarwhen only a kinetic model was applied in each method and initialconcentrations were fixed during the optimization. The observeddifferences between the obtained rate constant values from differ-ent methods were significantly decreased when a number ofproper constraints were utilized and the initial concentration val-ues were optimized, as well. It is also shown that if more informa-tion such as selectivity or local rank condition about system existsand is applied as constraint during the optimization process, ob-tained results from different methods will be very similar. Onthe other hand, application of suitable constraints reduces ambi-guity in the obtained parameters using different methods. Com-paring the results in conditions 1 and 4, it can be concludedthat the methods with data matrix based objective functions (X-based and T-based methods) are less sensitive to uncertaintiesin initial concentrations of reactants. They give more acceptableresults than C-based methods when no constraints were applied(Table 3).

Conclusion

The rate constants in the experimental data were estimatedusing X based and C based hard modeling approaches. Whenexperimental data were fitted with flexible initial concentrations

Fig. 5. Estimated concentration profiles from model (solid line) and experimentaldata (dotted line) from methods augX (a) and augC (b). Initial concentrations ofreagents were optimized during fitting and similarity constraint was applied forboth methods. (c) Estimated pure spectra of component from methods augC; (1)reactant (DIAZO); (2) intermediate; (3) product.

M. Kompany-Zareh, M. Khoshkam / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 102 (2013) 319–326 325

of reagents, the results from both C based and X based procedureswere similar. It is because besides t the kinetic parameters, the ini-tial concentrations were also optimized to give the same space forX and C. In addition to non-negativity constraint for estimatedpure spectra of absorbing component during the fitting procedure,additional constraints were also obtained from FSMWEFA andwere applied. Results show that applying adjustable initial concen-trations and proper constraints makes space of X and C similar andthe obtained kinetic parameters from both categories the methodswill be similar. Therefore, optimization of initial concentrationsand applying suitable constraints reduce ambiguity in the obtainedparameters from different models.

Notation

Scalars are represented by italic letters, and vectors by lower-case bold-italic ones. Matrices are represented by uppercasebold-italic letters, and a dash represents the transpose (e.g. Y’).Estimated values are represented by a hat (e.g. bY).

Scalars

m

Number of rows of data matrix selected inFSMWEFA

nt

Total number of rows (time) in data matrix nk Total number of columns (wavelength) in data

matrix

nk Number of principal components np Number of nonlinear parameters in optimization k1 First rate constant of second order consecutive

reaction

k2 Second rate constant of second order consecutive

reaction

RSS Residual sum of squares PC Principal components A0 Initial concentration of component o-ABA D0 Initial concentration of reactant DIAZO ri Error for ith estimated parameter rY Standard deviation of E residuals

Matrices

kC Concentration profiles (I � 4) kD Concentration profiles of pseudo component

(I � 3)

E Residual matrix S Spectral profiles (4 � J) T PCA scores X Data matrix (nt � nk) H Hessian matrix J Jacobian matrix

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