Modeling hourly dissolved oxygen concentration (DO)using two different adaptive neuro-fuzzy inference systems(ANFIS): a comparative study

Salim Heddam

Received: 2 March 2013 /Accepted: 23 August 2013 /Published online: 21 September 2013# Springer Science+Business Media Dordrecht 2013

Abstract This article presents a comparison of twoadaptive neuro-fuzzy inference systems (ANFIS)-based neuro-fuzzy models applied for modelingdissolved oxygen (DO) concentration. The two modelsare developed using experimental data collected fromthe bottom (USGS station no: 420615121533601) andtop (USGS station no: 420615121533600) stations atKlamath River at site KRS12a nr Rock Quarry, Oregon,USA. The input variables used for the ANFIS modelsare water pH, temperature, specific conductance, andsensor depth. Two ANFIS-based neuro-fuzzy systemsare presented. The two neuro-fuzzy systems are: (1) gridpartition-based fuzzy inference system, namedANFIS_GRID, and (2) subtractive-clustering-basedfuzzy inference system, named ANFIS_SUB. In bothmodels, 60 % of the data set was randomly assigned tothe training set, 20 % to the validation set, and 20 % tothe test set. The ANFIS results are compared with mul-tiple linear regression models. The system proposed inthis paper shows a novelty approach with regard to theusage of ANFIS models for DO concentrationmodeling.

Keyword Modeling . Dissolved oxygen . DO .

ANFIS . Grid partition . Subtractive clustering .MLR

Introduction

Dissolved oxygen (DO) plays an important role in thedetermination of the water quality. DO levels indicateshow well the water is aerated and it is a commonlymeasured parameter because it is an immediate indica-tor; inadequate oxygen levels will quickly affect aquaticlife and will be threatening vitality for all aquatic life(Singaraja et al. 2011). Oxygen is considered to be a keyparameter for the evaluation of water quality in aquaticecosystems (Jamieson et al. 2013). Dissolved oxygencontent is one of the easiest and most basic water qualityparameters to measure and is a good indicator of overallstream health. Because of resource deficiencies, budgetconstraints, and the sheer number of river miles to beassessed, there is a need for cost-efficient and effectivemethods that can estimate DO for a large number ofriver miles using limited monitoring data (Money et al.2009). Concentration of dissolved oxygen in water bod-ies is the result of four basic processes: the physicalprocesses of aeration and diffusion carrying oxygenfrom the air into the water and from the water into theair; and the biological processes of oxygen productionvia photosynthesis and oxygen consumption via respi-ratory processes and chemical reactions during degra-dation of organic matter (Nakova et al. 2009).

The rapid development of numerical models pro-vides a large number of models to be used in engineer-ing problems or environmental problems. Up to now, avariety of water quality models are available and thetechniques become quite mature (Chau 2006). Themain objective of developing a dissolved oxygen

Environ Monit Assess (2014) 186:597–619DOI 10.1007/s10661-013-3402-1

S. Heddam (*)Faculty of Science, Agronomy Department, HydraulicsDivision University 20 Août 1955,Route EL HADAIK, BP 26, Skikda, Algeriae-mail: heddamsalim@yahoo.fr

model is to make projections for lakes with differentmorphometries and tropics levels, at different latitudesand altitudes, and to extrapolate to possible futureclimate scenarios (Stefan and fang 1994). During thelast few decades, many authors have carried out model-ing work on dissolved oxygen concentration in differ-ent ecosystems.

Kayombo et al. (2000) developed dissolved oxygensub-model in order to depict the combined influence oflight, pH, temperature, and carbon dioxide on the pro-cesses of dissolved oxygen production and utilization insecondary facultative waste stabilization ponds.Radwan et al. (2003) modeled dissolved oxygen andbiochemical oxygen demand (BOD) dynamics in a riverin Flanders (Belgium). Boano et al. (2006) developed astochastic model for the evolution of dissolved oxygenand biochemical oxygen demand components along ariver with independent BOD point inputs. Benoit et al.(2006) implemented a laterally integrated, two-dimensional model of the dissolved oxygen distributionfor the bottom waters of the Laurentian Channel inEastern Canada. Facchini et al. (2007) proposed a blackbox approach for the analysis of the dissolved oxygentime series in a lagoon, based on techniques related tononlinear dynamics concepts and deterministic chaos.Hull et al. (2008) analyzed and discussed the seasonaland daily dynamics of dissolved oxygen and a continu-ous simulation model is developed to represent them forthe salt lake of Fogliano (Italy). Williams and Boorman(2012) applied the QUESTOR model in a stepwisemanner (process complexity and additional data wereadded sequentially to the model) to simulate the flow,water temperature, and dissolved oxygen concentrationsalong a stretch of a chalk stream in southern England.

Misra (2011) proposed a non-linear mathematicalmodel for depletion of dissolved oxygen due to algalbloom in a lake. The model is formulated by consideringfour variables namely, cumulative concentration of nu-trients, density of algal population, density of detritus,and concentration of dissolved oxygen. Misra et al.(2011) investigated the effect of time delay involved inthe formation of nutrients from detritus on the concen-tration of dissolved oxygen. A dynamic model ofdissolved oxygen in relation with nature of solar irradi-ance, temperature, salinity, and mineralization of partic-ulate organic matter, at Sagar island of Hooghly–Matlaestuarine complex, India, with the help of single-dimension differential equation is proposed by Mandalet al. (2012). Martin et al. (2013) proposed a water

quality model for an 800-km reach of the AthabascaRiver in Alberta, Canada, for DO and the factors thatdetermine its concentration. After validation, the modelwas used to assess the assimilative capacity of the riverand mitigation measures that could be deployed.Recently, Jamieson et al. (2013) investigated the use ofrepeated measures of stable isotopes of O2, in conjunc-tion with measurement of DO concentration, to quantifygas transfer coefficients in the Grand River, Ontario,Canada.

Neural networks have attracted a lot of interest for thepast decade for their ability to capture nonlinear behav-iors from input–output data of a process. Various authorshave emphasized the capability of artificial neural net-works (ANNs) for modeling. Diamantopoulou et al.(2007) developed a three-layer cascade correlation arti-ficial neural network models to predict the monthlyvalues of six water quality parameters nitrates, specificconductivity, dissolved oxygen, sodium, magnesium,and calcium, of Axios river, at the Axioupolis stationand of three water quality parameters nitrates, specificconductivity, and dissolved oxygen, of Strymon river, atthe Sidirokastro stat ion. A model based onbackpropagation (BP) neural network and Levenberg-Marquant algorithm, using measurements of pH, tem-perature, and outflow, is described by Li-Hua and Li(2008) and applied for the estimation of the dissolvedoxygen of the Yellow River in Lanzhou city, China.Ranković et al. (2010) developed a feedforward neuralnetwork model to predict the dissolved oxygen in theGruźa Reservoir, Serbia, using experimental data whichare collected for 3 years. Basant et al. (2010) used linearand nonlinear modeling approaches for simultaneousprediction of the dissolved oxygen and biochemicaloxygen demand levels in river water, in India, usingthe set of independent measured variables. The partialleast squares (PLS2) regression and feed forwardbackpropagation artificial neural networks (FFBPANNs) modeling methods were applied to predict theDO and BOD levels using 11 input variables measuredmonthly in the river water at eight different sites over aperiod of 10 years. Chen et al. (2010) developed a back-propagation algorithm neural network to synchronouslysimulate concentrations of total nitrogen (TN), totalphosphorus (TP), and dissolved oxygen (DO) in re-sponse to agricultural non-point source pollution forany month and location in the Changle River, southeastChina. In a recent study, Liu et al. (2012) used Elmanneural networks for predicting dissolved oxygen of

598 Environ Monit Assess (2014) 186:597–619

Hyriopsis Cumingii ponds (China), using water qualitydata as input variables.

Few studies have been directed toward the use offuzzy logic for dissolved oxygen modeling. Using fuzzypattern recognition, Giusti and Marsili-Libelli (2009)modeled the dissolved oxygen dynamics in Orbetellolagoon (Italy) as a function of the physico-chemical andecological system variables, including the submergedvegetation, nutrients, and hydrodynamics. A final as-sessment of the model validity is obtained by incorpo-rating the whole DO dynamics (model, fuzzy patternrecognition, and parameter combination) into the gener-al lagoon model and producing a consistently correctseries of DO daily distributions over a yearly cycle.

Based on these results, we can conclude that theperformances of these models vary depending on thetype of data, and it is necessary to compare the perfor-mance of the reported models. Thus, it is difficult to tellwhich model will be more suitable for a particularapplication. Most of the models discussed above needseveral different input variables measured at long time-step (monthly and daily) and are not easily accessible.On the other hand, the river water quality fluctuationsduring a season, a week, or a day are known to be verysignificant. Ranković et al. (2010) have chosen thefollowing input variables as an input to the DO model:water pH, water temperature, chloride, total phosphate,nitrites, nitrates, ammonia, iron, manganese, and elec-trical conductivity, at monthly time step. Correlationcoeff ic ient was 0.874 in the test ing phase.Diamantopoulou et al. (2007) selected nine input var-iables: nitrates (NO3

−), specific conductivity (ECw),water temperature, pH, sodium (Na+), discharge (Q),total phosphorus (TP), sulfates (SO4

2−), and chlorides(Cl−), at monthly time step. It has been observed thatcorrelation coefficient (CC) between observed and cal-culated is relatively strong (CC=0.8741). In the modelof Liu et al. (2012), the authors have used the followinginput variables: water temperature (WT), solar radia-tion (SR), wind speed (WS), and pH, at hourly timestep, only at the summer season (from July 2 to August4). However, no significant correlation coefficient be-tween observed and calculated dissolved oxygen wasobserved (CC=0.13/0.23). Li-Hua and Li (2008) com-puted DO using three input variable factors influencingthe DO of river including the outflow, the water tem-perature, and the pH as input to the BP neural networkmodel. Comparison of dissolved oxygen concentrationbetween the models forecasted value and the actual

value shows that the relative error is less than 7.70 %and relative standard deviation is less than 7.81 %.Basant et al. (2010) selected 11 input variables mea-sured monthly in the river water: water pH, total alka-linity, total hardness, total solids, chemical oxygendemand, ammonical nitrogen, nitrate nitrogen, chlo-ride, phosphate, potassium, and sodium. It has beenobserved that correlation coefficient between observedand calculated is relatively strong (CC=0.860).

These models can be differentiated from each otheron two aspects. The first is in data types used as inputfor the models. The second aspect relates to the timestep (daily and monthly). The reviewed models showthe following drawbacks: (1) the large number of dif-ferent input variables (nine, 10, 11, etc.); (2) the timestep chosen for the development of the model (dailyand monthly), Pinto et al. (2012) reported that thecomplexity and uncertainty associated with the assess-ment of water quality can be alleviated if a reliableprediction tool is available that uses measurementswhich can be carried out in a short time (in few hoursrather than days); (3) additionally, there was a relative-ly strong correlation between observed and calculatedDO (0.860/0.870). Under such drawbacks, the predic-tion of dissolved oxygen concentrations (DO) in riverneeds a new innovative method such as the adaptiveneuro-fuzzy inference systems (ANFIS) model, atshort hourly time step, with high correlation coeffi-cient, rather than 0.99, and using a few numbers ofinput water quality variables. Furthermore, to the best ofauthor’s knowledge, there is no such study in the liter-ature which examines the use of adaptive neuro-fuzzyinference systems technique for dissolved oxygenmodeling in rivers and lakes. Therefore, in this studyusing adaptive neuro-fuzzy inference system, a newmodeling approach is developed to predict dissolvedoxygen concentration. Two adaptive neuro-fuzzy infer-ence systems models are developed and compared: (1)grid-partition-based fuzzy inference system, namedANFIS_GRID, and (2) subtractive-clustering-based,named ANFIS_SUB. The ANFIS presented in this stud-y was trained with the backpropagation gradient descentmethod in combination with the least squares method(LSM). The performance and predictive capabilityof ANFIS models are investigated and comparedwith multiple linear regression (MLR) models. Thepredictive ability of the ANFIS model and thelevel of importance of the water quality variablesare presented and discussed.

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Methodology

Description of study area and data set

The Klamath River flows from Upper Klamath Lake insouth central Oregon past a series of dams into northernCalifornia where the river eventually empties into thePacific Ocean near the town of Klamath. As a result of awide range of influences such as dam construction,landscape modifications, altered hydrologic conditions,population growth, and agriculture, the Klamath Riverdoes not meet certain water-quality standards as speci-fied by the States of Oregon and California (Rounds andSullivan 2009). In this study, the time series of hourlyrecords for water temperature (TE, °C), dissolved oxy-gen (DO, mg/l), pH (standard unit), specific conductance(SC, μS/cm), and sensor depth (SD, meters) at twostations were used. They were provided by the UnitedStates Geological Survey (USGS station no:420615121533601 [bottom] and 420615121533600[top], latitude 42°06′15″, longitude −121°53′36″) atKlamath River at site KRS12a nr Rock Quarry,Oregon, USA. Information on the USGS database isavailable at: http://water.usgs.gov. Figure 1 shows thelocations of the stations in study area.

A brief data summary is presented herein to high-light background water quality, concentration ranges,and notable differences between concentrations at thetop and bottom of the water column. Concentrations ofdissolved oxygen (DO) at the top and bottom of thewater column were not similar with significant varia-tion. DO concentrations were generally higher at thetop. It can be seen from Table 1, for the top station,dissolved oxygen concentrations ranged over threeorders of magnitude, with minimum and maximumvalues of 0.1 and nearly 17 mg/L (16.60 mg/L). Themean of all observations was 7.80 mg/L. At the bottomstation, DO concentrations ranged over three orders ofmagnitude, with minimum and maximum values of 0.4and nearly 12 mg/L (11.90 mg/L). The mean of allobservations was 6.31 mg/L. Specific conductanceranged from 126 to 359 μS/cm, with a mean value of180 μS/cm (Table 1), at the top station. At the bottomstation, the minimum value of specific conductance,122 μS/cm, was slightly less than the value at the top.The maximum for specific conductance was slightlyless than 571 μS/cm, and the mean of all observationswas 200 μS/cm higher than the value at the bottom. ThepH of water ranged over three orders of magnitude, with

minimum and maximum values of 7.10 and 9.6. Themean of all observations was between 7.80 and 8.10.The pH values were generally similar with slightlydifferences. Figure 2 shows the evolution of the waterquality variables used in this study.

Temperature is the important factor that affects DOconcentration, the volume of the DO in water at anygiven time is dependent upon the temperature of thewater (Mandal et al. 2012). The concentration ofdissolved oxygen in surface water is controlled bytemperature and has both a seasonal and a daily cycle.Cold water can hold more dissolved oxygen than warmwater. In winter and early spring, when the watertemperature is low, the dissolved oxygen concentrationis high. In summer and fall, when thewater temperature ishigh, the dissolved oxygen concentration is low (USGS2013). Temperature inversely related to the concentrationof dissolved oxygen in water; as temperature increases,dissolved oxygen decrease. Conversely, a temperaturedecline causes the oxygen concentration to increase. Itcan be seen in Table 1, for the bottom station, the tem-perature of water ranged over three orders of magnitude,with minimum and maximum values of 0.5 and nearly25 °C (24.70 °C). The mean of all observations was10.93 °C. At the top station, temperature of water rangedover three orders of magnitude, with minimum and max-imum values of 0.5and nearly 28 °C (27.30 °C). Themean of all observations was 11.77 °C.

Pearson correlation coefficients were calculated toidentify the statistically significant correlation betweenthe variables as shown in Table 2. Pearson correlationcoefficients among the variables showed a number ofstrong, moderate, and weakly (positive and negative)associations. It can be seen from Table 2, for the bottomstation, the strong negative correlation coefficient be-tween water temperature and DO is (CC=−0.87), imply-ing that any model built using water temperature willcertainly be able to compute the DO concentrationssatisfactorily, this two parameters are highly interrelatedwith each other, the negative correlation coefficientindicates that as one variable increases, the other de-creases, and vice versa. At the top station, as seen fromTable 2, the correlation coefficient between water tem-perature and DO is (CC=−0.46). It is observed fromTable 2, for the top station, parameters such pH, SC,and SD are correlated with DO with coefficient of 0.43,0.49, and −0.23, respectively. Also, at the bottom, pH,SC, and SD are correlated with DO with coefficient of0.26, 0.49, and 0.08, respectively. Pearson method of

600 Environ Monit Assess (2014) 186:597–619

correlation showed that a weakly relationship wasobtained between the SD and DO. It is observed fromTable 2 for the top station that parameters such as pH andTE exhibit strong positive relationship with correlationcoefficient of 0.47. pH, SC, and SD are correlated withTEwith coefficient of 0.47, −0.39, and 0.03, respectively.SC and SD show weakly correlation with pH with aPearson correlation coefficient of 0.14 and −0.26, respec-tively. SC and SD show moderate negative correlationwith correlation coefficient of −0.31. Although there arestrong correlation between TE and SC (CC=−0.43), re-sults in Table 2 reveal the weakly correlations between allthe water quality variables at the bottom station.

Division of data

The dataset selected had a total 2,945 patterns for eachstation (Table 1). In the table, the Xmean, Xmax, Xmin, Sx,and Cv denote the mean, maximum, minimum, stan-dard deviation and coefficient of variation, respective-ly. The 2,945 hourly samples of four input variables(TE, pH, SC, and SD) and one output variable (DO)were used for the models. Among these, 1,768 input–output pairs (60 % of the total), randomly chosen fromthe data sequence, were used in the training set, 588input–output pairs (20 % of the total), were used in thevalidation set and the remaining 588 days (20 % of thetotal) of the available data set were reserved for testingthe developed models. Note that both the ANFIS_SUB,ANFIS_GRID, and MLR models employ the sametraining, validation, and test data sets for an appropriateperformance comparison. Because the five variablesdescribed above had different dimensions, and therewas major difference among values, it was consideredto be necessary to standardize the primary data in orderto enhance the training speed and the precision of neuralnetworks. Input data were entered into the models afternormalization. For this purpose, Eq. 1 was utilized.

z ¼ x− xσ

ð1Þ

While z presents the normalized data, x is referred tothe original data, x stands for the average of data, and σshows the standard deviation of data. Output results willbe prescribed as predicted values after denormalization.All the input and output variables were normalized tohave zero mean and unit variance.

Adaptive neuro-fuzzy inference systems

An adaptive network, as its name implies, is a networkstructure consisting of nodes and directional linksthrough which the nodes are connected. Moreover,parts or all of the nodes are adaptive, which meanseach output of these nodes depends on the parameterspertaining to this node and the learning rule specifieshow these parameters should be changed to minimize aprescribed error measure (Jang 1993). ANFIS is amultilayer feed-forward network where each node per-forms a particular function on incoming signals. Bothsquare and circle node symbols are used to representdifferent properties of adaptive learning. To performdesired input–output characteristics, adaptive learningparameters are updated based on gradient learningrules (Jang 1993; Jang and Gulley 1996). For simplic-ity, we assume the fuzzy inference system under con-sideration has two inputs, x and y, and one output z.

Suppose that the rule base contains two fuzzy if–then rules of Takagi and Sugeno’s type:

Rule1 ¼ If x isA1ð Þand y isB1ð ÞThen f 1 ¼ p1xþ q1yþ r1ð Þ ð2Þ

Rule2 ¼ If x isA2ð Þand y isB2ð ÞThen f 2 ¼ p2xþ q2yþ r2ð Þ ð3ÞWhere x and y are the inputs, Ai and Bi are the fuzzy

sets; fi are the outputs within the fuzzy region specifiedby the fuzzy rule; and pi, qi, and ri are the designparameters that are determined during the training pro-cess. The ANFIS architecture to implement these tworules is shown in Fig. 3, in which a circle indicates afixed node, whereas a square indicates an adaptive node.

Layer 1: Every node i in this layer is a square nodewith a node function:

Ο1i ¼ μΑi

xð Þ; i ¼ 1; 2; ð4Þ

Ο1i ¼ μΒi−2

yð Þ; i ¼ 3; 4; ð5Þ

Where x (or y) is the input to node i, and Ai (or Bi−2)is the linguistic label (small, large, etc.) associatedwith this node function, and where μΑi

xð Þ and

μΒi−2yð Þ can adopt any fuzzy membership function

(MF). Assuming a Gaussian function as a member-ship function, Ai can be computed as

Environ Monit Assess (2014) 186:597–619 601

μAixð Þ ¼ exp −0:5 x− cið Þ=σif g2

h i; ð6Þ Where (σi, ci) are parameter sets. Parameters in

this layer are referred to as premise parameters.

Fig. 1 Study area located at the Klamath River at site KRS12a nr Rock Quarry, Oregon, USA (adopted from [Sullivan et al. 2012])

602 Environ Monit Assess (2014) 186:597–619

Layer 2: Every node i in this layer is a fixed node,marked by a circle node, labeled ∏, whichmultiplies the incoming signals and outputsthe product.

Ο2i ¼ wi ¼ μ

ΑiμΒi; i ¼ 1; 2; ð7Þ

The output signal wi denotes the firingstrength of a rule. The number of nodes ofthis layer equals the number of fuzzy “if–then” rules in the fuzzy inference system.

Layer 3: Every node i in this layer is a fixed node,marked by a circle node, labeled N. The ith

Table 1 Hourly statistical parameters of data set

Station Data set Unit Xmean Xmax Xmin Sx Cv Correlationwith DO

KRS12a [top] TE °C 11.77 27.30 0.50 7.81 0.66 −0.467pH / 8.10 9.60 7.20 0.62 0.07 0.435

SC μS/cm 179.85 359.00 126.00 32.47 0.18 0.493

SD m 0.94 1.25 0.67 0.11 0.12 −0.234DO mg/l 7.80 16.60 0.10 3.11 0.39 1.000

KRS12a [bottom] TE °C 10.93 24.70 0.50 7.30 0.66 −0.870pH / 7.79 9.10 7.10 0.40 0.05 0.267

SC μS/cm 199.97 571.00 122.00 72.04 0.36 0.495

SD m 3.85 4.19 3.47 0.21 0.05 0.085

DO mg/l 6.31 11.90 0.40 3.37 0.53 1.000

Xmean mean, Xmax maximum, Xmin minimum, Sx standard deviation, Cv coefficient of variation

Tem

per

atu

re (

°C)

Fig. 2 Graphs showing time series of temperature (°C), pH, specific conductance (μS/cm), sensor depth (m), and dissolved oxygenconcentration (mg/L), top and bottom, with hourly time step at Klamath River at site KRS12a nr Rock Quarry, Oregon, USA

Environ Monit Assess (2014) 186:597–619 603

node calculates the ratio of the ith rule’s firingstrength to the sum of all rule’s firing strengths:

Ο3i ¼ wi ¼ wi

.w1 þ w2ð Þ

� �i ¼ 1; 2; ð8Þ

For convenience, outputs of this layer arecalled normalized firing strengths.

Layer 4: In this layer, the nodes are adaptive nodes.The output of each node in this layer issimply the product of the normalized firingstrength and a first-order polynomial (for afirst-order Sugeno model). Thus, the out-puts of this layer are given by

Ο4i ¼ wi f i ¼ wi pi xþ qi yþ rið Þi ¼ 1; 2

ð9ÞWhere wi is the output of layer 3 and

({pi, qi, and ri}) is the parameter set of thisnode parameters in this layer will be re-ferred to as consequent parameters.

Layer 5: The single node in this layer is a fixed nodelabeled Σ, which computes the overall out-put as the summation of all incoming sig-nals, i.e.,

Ο5i ¼

Xi¼1

wi f i ¼Xi¼1

wi f i.

w1 þ w2ð Þ !

: ð10Þ

Explicitly, this layer sums the node’s out-put in the previous layer to be the output ofthe whole network.

It can be observed that there are two adap-tive layers in this ANFIS architecture, namelythe first layer and the fourth layer. In the firstlayer, there are twomodifiable parameters ({σi,ci}), which are related to the input membership

functions. These parameters are the so-calledpremise parameters. In the fourth layer, thereare also three modifiable parameters ({pi, qi,and ri}), pertaining to the first-order polynomi-al. These parameters are so-called consequentparameters (Jang 1993).

Learning algorithm of ANFIS

The task of the learning algorithm for this architectureis to tune all the modifiable parameters, namely ({σi,ci}) and ({pi, qi, and ri}), to make the ANFIS outputmatch the training data. When the premise parametersσi and ci of the membership function are fixed, theoutput of the ANFIS model can be written as

f ¼ w1= w1 þ w2ð Þð Þ f 1 þ w1= w1 þ w2ð Þð Þ f 2 ð11Þ

Substituting Eq. 9 into Eq. 11 yields

f ¼ w1 f 1 þ w2 f 2 ð12ÞBy substituting the fuzzy if–then rules into Eq. 12, it

becomes

f ¼ w1 p1xþ q1yþ r1ð Þ þ w2 p2xþ q2yþ r2ð Þ ð13ÞAfter rearrangement, the output can be expressed as

f ¼ w1x� �

p1 þ w1y� �

q1 þ w1

� �r1

þ w2x� �

p2 þ w2y� �

q2 þ w2

� �r2 ð14Þ

Which is a linear combination of the modifiableconsequent parameters p1, q1, r1, p2, q2, and r2. TheLSM can be used to identify the optimal values of these

Table 2 Pearson correlation coefficients between and among physical water-quality parameters, and dissolved oxygen concentration

KRS12a [top] KRS12a [bottom]

TE (°C) pH SC (μS/cm) SD (m) DO (mg/L) TE (°C) pH SC (μS/cm) SD (m) DO (mg/L)

TE (°C) 1.00 1.00

pH 0.47 1.00 0.05 1.00

SC (μS/cm) −0.39 0.14 1.00 −0.43 0.28 1.00

SD (m) 0.03 −0.26 −0.31 1.00 −0.18 0.08 0.14 1.00

DO (mg/L) −0.46 0.43 0.49 −0.23 1.00 −0.87 0.26 0.49 0.08 1.00

°C degree Celsius, μS/cm microseimens per centimeter, m meter, mg/L milligrams per liter

604 Environ Monit Assess (2014) 186:597–619

parameters easily, when the premise parameters are notfixed, the search space becomes larger and the conver-gence of the training becomes slower. A hybrid algo-rithm combining the least squares method and thegradient descent method is adopted to solve this prob-lem. The hybrid algorithm is composed of a forwardpass and a backward pass. The least squares method(forward pass) is used to optimize the consequentparameters with the premise parameters fixed. Oncethe optimal consequent parameters are found, the back-ward pass starts immediately. The gradient descentmethod (backward pass) is used to adjust optimallythe premise parameters corresponding to the fuzzy setsin the input domain. The output of the ANFIS iscalculated by employing the consequent parametersfound in the forward pass. The output error is used toadapt the premise parameters by means of a standardbackpropagation algorithm. It has been proven that thishybrid algorithm is highly efficient in training theANFIS (Jang 1993).

Performance indices

Various statistical measures have been developed and usedin the literature. To assess the fitting and predictive accu-racy of the models, the data sets were mathematicallyevaluated by calculating the following evaluation criteria:coefficient of correlation (CC), root mean squared error(RMSE), and mean absolute error (MAE). In addition, a

graphical comparison was performed to illustrate the ac-curacy of the proposed models.

CC ¼1

Ν

XΟi−Οmð Þ Ρi−Ρmð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

Ν

Xi¼1

n

Οi−Οmð Þ2s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

Ν

Xi¼1

n

Ρi−Ρmð Þ2s ð15Þ

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Ν

Xi¼1

Ν

Οi−Ρið Þ2vuut ð16Þ

MAE ¼ 1

Ν

Xi¼1

Ν

Οi−Ρij j ð17Þ

Where N is the number of data points, Oi is somemeasured value, and Pi is the corresponding model predic-tion. Om and Pm are the average values of Oi and Pi.

Results and discussions

Establishment of fuzzy rule bases

Fuzzy modeling is a system identification task, whichinvolves two phases: structure identification and pa-rameter estimation. Fuzzy modeling for nonlinear

Table 3 The two neuro fuzzymodels Model ANFIS_SUB ANFIS_GRID

Structure identification Subtractive clustering Grid partitioning

Parameter estimation Backpropagation andlinear least squares

Backpropagation andlinear least squares

x yf (x, y)

Layer1

x

y

A1

A2

B1

x yy

B2

N

N

w2

w1

11fw

22 fw

1w

2w

Input Layer Layer2 Layer3 Layer4 Layer5

Output Layer

Consequent Parameters (pi, qi, ri) Premise Parameters ( i, ci)

∑

σFig. 3 The architecture ofANFIS model

Environ Monit Assess (2014) 186:597–619 605

system is based on a set of IF–THEN rules which uselinguistic propositions of human thinking. Fuzzy ruleextraction process is called structure identification fornonlinear systems modeling (Yu and Li 2009). In thispaper, two commonly used methods are applied forstructure identification: the grid partition (GRID) andthe subtractive clustering (SUB) methods. The notationof the two models, the structure identification andparameter estimation techniques are summarized inTable 3.

ANFIS_GRID

Fuzzy grid is a well-known method in which input–output data are partitioned into grids (Jang 1993). TheANFIS_GRID fuzzy inference system is the combina-tion of grid partition and ANFIS. Grid partition dividesthe data space into rectangular sub-spaces using axis-paralleled partition based on pre-defined number ofmembership functions and their types in each dimen-sion. Premise fuzzy sets and parameters are calculatedusing the least square estimate method based on thepartition and MF. It is obvious that the wide applicationof grid partition is threatened by the large number of

rules (Wei et al. 2007). Grid partition is only suitable forcases with small number of input variables (e.g., lessthan 6) (http://www.cs.nthu.edu.tw/∼jang/an.sfaq.htm).

In the ANFIS_GRID model, each input variable,which varies within a range, might be clustered intoseveral class values in layer 1 to build up fuzzy rules,and each fuzzy rule would be constructed throughseveral parameters of membership function in layer 2.Accordingly, the number of parameters which need to bedetermined is enormous as the rule is increased. Themodels identified in this paper on the basis of this gridpartitioning of antecedent variables use Gaussian mem-bership functions. With grid partition method for theANFIS model, the total number of modifiable parameters(TNP) can be calculated as follows (Heddam et al. 2012):

TNP ¼ PP þ CP ð18Þ

Where PP is the premise parameters numbers andCP is the consequent parameters numbers. PP and CPare calculated as follows:

ΡΡ ¼ NM � NMF � 2 ð19Þ

Table 4 Performances of the two ANFIS and MLR models for the two stations, using only water temperature (TE) as input variable

Model Regression equation/number of rules Training Validation Testing

MAE RMSE CC MAE RMSE CC MAE RMSE CC

MLRa DO=−0.227×TE+11.071 2.237 2.751 0.460 2.235 2.781 0.447 2.270 2.743 0.505

MLRb DO=−0.364×TE+10.723 1.395 1.671 0.867 1.339 1.605 0.881 1.411 1.694 0.868

ANFIS_SUBa 6 1.625 2.136 0.725 1.625 2.081 0.740 1.752 2.236 0.717

ANFIS_GRIDa 3 2.000 2.383 0.638 1.986 2.404 0.641 2.006 2.422 0.652

ANFIS_SUBb 6 0.955 1.234 0.930 0.893 1.211 0.934 0.936 1.226 0.934

ANFIS_GRIDb 3 1.046 1.297 0.922 1.005 1.262 0.928 1.034 1.285 0.927

a KRS12a [top], b KRS12a [bottom]

Table 5 Combinations of inputvariables considered in develop-ing models

Ι denotes included variable, ●denotes excluded variable

Variable Models

M10 M9 M8 M7 M6 M5 M4 M3 M2 M1

DO Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι

TE Ι Ι Ι ● Ι Ι Ι Ι ● ●

pH Ι Ι Ι Ι ● Ι ● ● Ι Ι

SC Ι Ι ● Ι Ι ● Ι ● Ι ●

SD Ι ● Ι Ι Ι ● ● Ι ● Ι

606 Environ Monit Assess (2014) 186:597–619

NFR ¼ NMFNM ð20Þ

CP ¼ NFR� NM þ OPð Þ ð21ÞWhere NM is the number of the input variable, NMF

is the number of membership function per each input,(2) is the numbers of modifiable parameters for eachmembership function (Gaussian MFs), NFR is the num-bers of Fuzzy rules that will be created by all inputs, andOP is the system output which is always equal to one(hourly dissolved oxygen concentration).

Grid partitioning method represents the whole rangeof the input space, and if the complex optimization ofantecedent parameters can be properly addressed, theresulting FIS may perform better than subtractive clus-tering (SUB) methods. This is precisely one of theadvantages of the ANFIS (Nayak et al. 2005). It has

been often claimed that grid-type fuzzy partitions cannothandle high-dimensional problems with many inputvariables due to the curse of dimensionality. That is,when we use the grid-type fuzzy partition, the numberof fuzzy if–then rules exponentially increases as thenumber of input variables increases (Ishibuchi et al.1999). Moreover, the optimization of antecedent param-eter becomes complex if grid partitioning is employed(Nayak et al. 2005). For example, if we have four inputvariables (TE, pH, SC, and SD) and because there is aneed for considerable membership functions (MFs) foreach of them (five MFs), the rules will be 54 rules (625rules) and the calculation of the parameters of this modelwill be very difficult. However, as the number of MFsincreases, the time taken for model training also in-creases considerably. The ANFIS is a good model evenwhen the MFs is as small as 3 (Jang 1993). Typically,the hourly dissolved oxygen modeling problem has

Table 6 Total number of pa-rameters for the two ANFISmodels developed

Station Designation Models

ANFIS_SUB ANFIS_GRID

KRS12a [bottom] Number of linear parameters 75 405

Number of nonlinear parameters 120 24

Total number of parameters 195 429

Number of fuzzy rules 15 81

KRS12a [top] Number of linear parameters 65 405

Number of nonlinear parameters 104 24

Total number of parameters 169 429

Number of fuzzy rules 13 81

Table 7 Performances of the MLR models in different phases for KRS12a [bottom] station

Model Training Validation Testing

MAE RMSE CC MAE RMSE CC MAE RMSE CC

M10 1.027 1.207 0.933 1.014 1.210 0.934 0.990 1.204 0.935

M9 1.080 1.273 0.925 1.059 1.259 0.928 1.037 1.252 0.930

M8 1.028 1.211 0.932 1.019 1.217 0.933 0.990 1.205 0.935

M7 2.486 2.907 0.500 2.498 2.871 0.534 2.479 2.912 0.524

M6 1.299 1.581 0.882 1.217 1.499 0.897 1.326 1.611 0.881

M5 1.076 1.276 0.925 1.060 1.264 0.928 1.028 1.251 0.930

M4 1.323 1.610 0.877 1.237 1.529 0.893 1.350 1.640 0.877

M3 1.380 1.649 0.871 1.326 1.668 0.884 1.39 1.583 0.872

M2 2.486 2.907 0.500 2.499 2.871 0.534 2.480 2.912 0.524

M1 2.710 3.244 0.258 2.770 3.266 0.275 2.727 3.243 0.322

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exactly four input variables, in the current study, thenumber of MFs assigned to each input has been 3,generating only a small number of fuzzy if–then rules(34=81 rules). In this case the use of ANFIS_GRID isjustified and can be a good choice. We successfullyovercome the curse of dimensionality problem.

ANFIS_SUB

The ANFIS_SUB fuzzy inference system combines thesubtractive clustering method and ANFIS (Wei et al.2007). The subtractive clustering method is proposed byChiu (1994) by extending the mountain clustering methodproposed byYager and Filev (1994). Subtractive clusteringis based on a measure of the density of data points in thefeature space (Chiu 1996). The subtractive clustering ap-proach is used to determine the number of rules andantecedent membership functions by considering eachcluster center (Di) as a fuzzy rule. In this approach eachdata point of a set of N data points {x1. . . xN} in a p-dimensional space is considered as the candidate for clustercenters. After normalizing and scaling data points in each

direction, a density measure at data point xi is computed onthe basis of its locationwith respect to other data points andexpressed as (Lohani et al. 2006):

Di ¼Xj¼1

N

exp −2

ra

� �2

: xi−x j�� ��2 !

ð22Þ

Where ra is a positive constant called cluster radius.A data point is considered as a cluster center whenmore data points are closer to it. Therefore, the datapoint (x1

*) with highest density measure (D1*) is con-

sidered as first cluster center. Now excluding the influ-ence of the first cluster center, the density measure ofall other data points is recalculated as:

Di ¼ Di−D*i ⋅μ x*i� ð23Þ

μ x*i� ¼ exp −

xi−x j�� ��2rb.2

� �20B@

1CA ð24Þ

where rb (rb>ra) is a positive constant that results in ameasurable reduction in densitymeasures of neighborhood

Table 8 Performances of the two ANFIS models in different phases for KRS12a [bottom] station

Model Number of rules Training Validation Testing

MAE RMSE CC MAE RMSE CC MAE RMSE CC

ANFIS_GRID M10 81 0.144 0.204 0.998 0.251 0.344 0.995 0.233 0.343 0.994

M9 27 0.306 0.415 0.992 0.315 0.413 0.993 0.314 0.415 0.993

M8 27 0.229 0.325 0.995 0.284 0.386 0.993 0.256 0.377 0.993

M7 27 0.919 1.307 0.921 1.026 1.434 0.911 1.047 1.517 0.901

M6 27 0.522 0.733 0.976 0.506 0.700 0.979 0.545 0.788 0.973

M5 9 0.412 0.537 0.987 0.395 0.513 0.989 0.413 0.541 0.987

M4 9 0.658 0.945 0.960 0.625 0.899 0.964 0.652 0.927 0.963

M3 9 0.716 0.987 0.956 0.652 0.927 0.962 0.683 0.973 0.959

M2 9 1.464 2.017 0.800 1.476 2.032 0.802 1.560 2.162 0.776

M1 9 1.369 1.902 0.824 1.484 2.060 0.802 1.409 1.967 0.819

ANFIS_SUB M10 15 0.216 0.300 0.995 0.240 0.317 0.995 0.231 0.320 0.995

M9 9 0.322 0.430 0.992 0.357 0.464 0.991 0.357 0.451 0.991

M8 16 0.220 0.309 0.994 0.247 0.327 0.994 0.232 0.333 0.994

M7 15 0.813 1.193 0.935 0.904 1.314 0.924 0.904 1.302 0.925

M6 11 0.508 0.722 0.977 0.465 0.666 0.981 0.481 0.689 0.980

M5 7 0.405 0.514 0.988 0.403 0.516 0.989 0.412 0.528 0.988

M4 5 0.672 0.938 0.960 0.631 0.891 0.965 0.665 0.934 0.962

M3 11 0.648 0.936 0.960 0.597 0.869 0.967 0.607 0.885 0.966

M2 8 1.261 1.888 0.827 1.325 1.989 0.812 1.398 2.066 0.800

M1 11 1.235 1.789 0.846 1.403 1.987 0.817 1.267 1.837 0.844

608 Environ Monit Assess (2014) 186:597–619

data points so as to avoid closely spaced cluster centers(Chiu 1994).

After the density measure for each data point isrevised (Eq. 23), the data point with the highestremaining density measure is obtained and set as thenext cluster center x2

*and all of the density measuresfor data points are revised again. The process is repeat-ed and the density measures of remaining data pointsafter computation of kth cluster center is revised bysubstituting the location (xk

*) and density measure(Dk

*) of the kth cluster center in Eq. 23. This processis stopped when a sufficient number of cluster centersare generated. A sophisticated stopping criterion forautomatically determining the number of clusters issuggested by Chiu (1994, 1996). The choice of ra playsan important role in determining the number of clus-ters. Large values of ra will generate a limited numberof clusters, while small values of ra will generate alarge number of clusters. Hence, it is important toselect proper influential radius (radii) for clusteringthe data space (Chiu 1994). By the end of clustering,a set of fuzzy rules will be obtained. Each clusterrepresents a rule. However, since the clustering iscarried out in a multidimensional space, the relatedfuzzy sets must be obtained. As each axis refers to avariable, the centers of the member ship functions areobtained by projecting the center of each cluster in thecorresponding axis. The widths are obtained on thebasis of the radius (radii) (Eftekhari and Katebi 2008).

With subtractive clustering method for theANFIS_SUB model, TNP can be calculated as follows:Each membership function (Gaussian MFs) has twomodifiable parameters; therefore, premise parametersnumbers are (NM×NMF×2). When fuzzy systems aredesigned by using subtractive clustering method, each

cluster corresponds to a fuzzy rule. Hence, the numberof clusters determines the number of rules. Therefore,consequent parameter numbers are equal toNMF×(NM+OP). Then, the total number of modifiableparameters is equal to the number of premise parametersplus the number of consequent parameters (Heddamet al. 2012).

Multiple linear regression model

Multiple linear regression is a well-known method ofmathematically modeling the relationship between adependent variable and one or more independent var-iables. In general, response variable Ymay be related ton regressor variables. The following model

Υ ¼ β0 þ β1x1 þ β2x2 þ β3x3 þ ::::þ βnxn ð25Þ

is called a multiple linear regression model with nregressor variables. Where β0 is a constant and βi, i=1… n are regression coefficients. This model describesa hyperplane in n-dimensional space of the regressorvariables xj. The parameter βj represents the expectedchange in response Y per unit change in xj when all theremaining independent variables xi (i≠j) are held con-stantly. In this study, the coefficients β0, β1, β2,…, βnwere determined using least squares method.

The simulation results of the two ANFIS models

Major objectives of the present study included an eval-uation and comparison of three different modelingapproaches, namely, ANFIS_SUB, ANFIS_GRID,and MLR for dissolved oxygen concentrations model-ing, in addition to an evaluation of the effects of using

Fig. 4 Scatterplots of calculated versus observed values of dissolved oxygen concentration (DO), for ANFIS_SUB model, at siteKRS12a [bottom], for a training, b validation, and c testing

Environ Monit Assess (2014) 186:597–619 609

different water quality variables as input to the models.Observations of water temperature (TE), pH, specificconductance (SC), and sensor depth (SD) were used asinput for estimation of DO. Comparisons and the per-formance of the three models at each of the two stationsare shown in this section. As discussed earlier, regard-ing the Pearson correlation coefficients reported inTable 2, the strong negative correlation coefficientsbetween water temperature and DO is (CC=−0.87),implying that any model built using water temperaturewill certainly be able to compute the DO concentra-tions satisfactorily, consequently the first model

developed in this study use only water temperature asinput variable.

Regression model equation and models perfor-mances for dissolved oxygen concentration, using onlywater temperature as input variable at the two stationsare shown in Table 4. For comparison, Table 4 alsopresents models constructed using the ANFIS ap-proaches for the same data set at the two stations. Thetwo ANFIS architectures developed herein was foundto yield better agreement with experimental observa-tions for the training, validation, and testing data setcompared to data predicted by the multiple linear

Table 9 Performances of the ANFIS_SUB model for different values of the radius parameter for KRS12a [bottom]

Radius Number of rules Number of MFs Training Validation Testing

MAE RMSE CC MAE RMSE CC MAE RMSE CC

0.90 04 04 0.369 0.491 0.989 0.349 0.463 0.990 0.368 0.482 0.990

0.80 05 05 0.346 0.472 0.990 0.342 0.454 0.991 0.346 0.456 0.991

0.70 05 05 0.332 0.456 0.990 0.326 0.433 0.991 0.332 0.441 0.991

0.60 07 07 0.295 0.401 0.992 0.305 0.392 0.993 0.295 0.390 0.993

0.35 15 15 0.216 0.300 0.995 0.240 0.317 0.995 0.231 0.320 0.995

The table entries in bold show superior results among other ANFIS_SUB structure

Table 10 Final Gaussian membership function parameters (MF) of the inputs data for ANFIS_SUB model for KRS12a [bottom] station

Membership functions Temperature pH Specific conductance Sensor depth

a b a b a b a b

MF1 0.406 −1.103 0.604 −0.722 0.772 0.667 0.416 1.123

MF2 0.406 1.495 0.604 0.987 0.772 −0.904 0.416 0.936

MF3 0.406 −1.035 0.604 −0.233 0.772 −0.111 0.416 0.375

MF4 0.406 1.453 0.604 −0.722 0.772 −0.361 0.416 −1.447MF5 0.406 −0.324 0.604 −0.722 0.772 −0.459 0.416 −1.541MF6 0.406 −0.105 0.604 −0.722 0.772 −0.514 0.416 −0.138MF7 0.406 −0.064 0.604 −0.722 0.772 −0.556 0.416 0.889

MF8 0.406 −0.529 0.604 1.720 0.772 1.126 0.416 −0.933MF9 0.406 0.058 0.604 1.232 0.772 0.027 0.416 0.609

MF10 0.406 −1.377 0.604 −0.477 0.772 −0.139 0.416 −0.559MF11 0.406 1.289 0.604 −0.477 0.772 −0.737 0.416 0.889

MF12 0.406 0.318 0.604 −0.477 0.772 −0.598 0.416 −1.167MF13 0.406 0.099 0.604 0.987 0.772 0.027 0.416 −1.167MF14 0.406 −0.187 0.604 0.743 0.772 3.073 0.416 0.842

MF15 0.406 1.795 0.604 −1.699 0.772 −0.834 0.416 0.656

a represents Gaussian MFs center and b determines Gaussian MFs width

610 Environ Monit Assess (2014) 186:597–619

regression model. According to Table 4, for the topstation, in the training phase, the values of CC, RMSE,and MAE, ranged from 0.460 to 0.725, 2.136 to 2.751,and 1.625 to 2.237, respectively. In addition, in thevalidation phase, the values of CC, RMSE, andMAE, ranged from 0.447 to 0.740, 2.081 to 2.781,and 1.625 to 2.235, respectively. Finally, in the testingphase, the values of CC, RMSE, and MAE, rangedfrom 0.505 to 0.717, 2.236 to 2.743, and 1.752 to

2.270, respectively. It can be observed from Table 4,the ANFIS_SUB model performed better than theANFIS_GRID and MLR estimates for training, valida-tion, and testing phases. It may be seen from Table 4,the RMSE, MAE, and CC values for the multiple linearregression were found to be lower than those for theANFIS models, thereby establishing the superiority ofthe ANFIS models. According to Table 4, for thebottom station, the CC values for all the three models

Table 11 Fuzzy rule base for the ANFIS_SUB model for KRS12a [bottom] station

Rule number Rule description

Premise parameters Consequents parameters

1 If TE is TE_MF1 and pH is pH_MF1and SC is SC_MF1 and SD is SD_MF1

THEN DO=0.65×TE +0.53×pH −0.11×SC −0.22×SD +1.9

2 If TE is TE_MF2 and pH is pH_MF2and SC is SC_MF2 and SD is SD_MF2

THEN DO=0.10×TE +0.36×pH +0.53×SC −0.57×SD −1.16

3 If TE is TE_MF3 and pH is pH_MF3and SC is SC_MF3 and SD is SD_MF3

THEN DO=1.40×TE +0.61×pH 0.07×SC +0.13×SD +2.61

4 If TE is TE_MF4 and pH is pH_MF4and SC is SC_MF4 and SD is SD_MF4

THEN DO=0.67×TE +0.30×pH +0.30×SC +0.51×SD −1.46

5 If TE is TE_MF5 and pH is pH_MF5and SC is SC_MF5 and SD is SD_MF5

THEN DO=−0.52×TE +0.55×pH −0.07×SC −0.04×SD +0.42

6 If TE is TE_MF6 and pH is pH_MF6and SC is SC_MF6 and SD is SD_MF6

THEN DO=0.33×TE +0.81×pH −0.46×SC −0.59×SD +0.39

7 If TE is TE_MF7 and pH is pH_MF7and SC is SC_MF7 and SD is SD_MF7

THEN DO=−1.81×TE +0.61×pH −0.89×SC −0.34×SD −0.19

8 If TE is TE_MF8 and pH is pH_MF8and SC is SC_MF8 and SD is SD_MF8

THEN DO=0.41×TE +0.36×pH −0.007×SC −0.04×SD +0.64

9 If TE is TE_MF9 and pH is pH_MF9and SC is SC_MF9 and SD is SD_MF9

THEN DO=−0.45×TE +0.22×pH +0.002×SC −0.20×SD +0.69

10 If TE is TE_MF10 and pH is pH_MF10and SC is SC_MF10 and SD is SD_MF10

THEN DO=2.29×TE +0.08×pH −0.001×SC +0.33×SD +4.16

11 If TE is TE_MF11 and pH is pH_MF11and SC is SC_MF11 and SD is SD_MF11

THEN DO=−0.62×TE +0.43×pH −0.08×SC +0.11×SD −0.87

12 If TE is TE_MF12 and pH is pH_MF12and SC is SC_MF12 and SD is SD_MF12

THEN DO=−0.69×TE +0.32×pH −0.69×SC +0.11×SD −1.02

13 If TE is TE_MF13 and pH is pH_MF13and SC is SC_MF13 and SD is SD_MF13

THEN DO=−0.63×TE −0.17×pH −0.34×SC −0.40×SD +0.44

14 If TE is TE_MF14 and pH is pH_MF14and SC is SC_MF14 and SD is SD_MF14

THEN DO=0.27×TE +0.64×pH +0.04×SC +0.08×SD +0.01

15 If TE is TE_MF15 and pH is pH_MF15and SC is SC_MF15 and SD is SD_MF15

THEN DO=−1.54×TE +0.29×pH +0.65×SC +0.12×SD +2.12

Fig. 5 Comparisons betweenobserved and calculated data ofdissolved oxygen concentra-tion (DO) for the training, val-idation, and the testing sets ofthe ANFIS_SUBmodel, at siteKRS12a [bottom]

Environ Monit Assess (2014) 186:597–619 611

are reasonably good, being smallest (0.867) for MLRmodel and greatest (0.930) for ANFIS_SUB model.The values of other model performances such asRMSE, and MAE indicate that the forecast perfor-mance of the ANFIS model is very good. The lowestvalue of the RMSE of forecasting models is 1.234 (inANFIS_SUB). In addition, the lowest value of MAE is0.955 also (in ANFIS_SUB). The prediction accuracyfor the regression model was lower when compared toANFIS models for all the training, validation, andtesting phases. The results of regression model forDO at the top station were found to be lower than thosefor the corresponding model identified for the bottomstation (CC values of 0.505 for top station compared to0.868 for bottom station, in the testing phase; Table 4).

In order to determine the relative significance ofeach of the input water quality variables on thedissolved oxygen estimation, using the three devel-oped approaches, we have tested and compared variousinput combinations as shown in Table 5 (ten models).The ten combinations were considered and evaluatedin terms of the RMSE, MAE, and CC as main perfor-mance criteria. The trained ANFIS models establishthe relationship between the presented inputs (waterquality data) and the output (dissolved oxygen). Thetwo ANFIS models were developed using identical

inputs. For generation of the membership functionsassociated with each input variable, the grid partitionand subtractive clustering methods were employed forANFIS_GRID and ANFIS_SUB, respectively. In bothmodels, the Gaussian membership function wasassigned. In designing ANFIS models, the number ofmembership functions, the number of fuzzy rules, andthe number of training epochs are important factors tobe considered. The total number of parameters for eachmodel is summarized in Table 6. It appears that thesubtractive fuzzy clustering does significantly reducethe number of rules and numbers of parameters, whereANFIS_SUB model only need a few numbers of rules.In this section, general information about the resultsobtained by applying the two models is explained.

Station KRS12a [bottom] results

The results of the MLR models for DO modeling aresummarized in Table 7. As shown, the M10 model withinputs of TE, pH, SC, and SD had the best perfor-mance, although its performance was not significantlybetter than those of the M9 and M8 models. As seen inTable 7, the ten MLR models showed significant var-iations based on the three performance criteria. In thetraining phase, the lowest value of the RMSE of

Fig. 6 Scatterplots of calculated versus observed values of dissolved oxygen concentration (DO), for ANFIS_GRID model, at siteKRS12a [bottom], for a training, b validation, and c testing

Fig. 7 Comparisons betweenobserved and calculated data ofdissolved oxygen concentra-tion (DO) for the training, val-idation, and the testing sets ofthe ANFIS_ GRID model, atsite KRS12a [bottom]

612 Environ Monit Assess (2014) 186:597–619

forecasting models is 1.207 (in MLR M10), and thehighest value of the CC is 0.933 (in MLR M10). Inaddition, the lowest value of MAE is also 1.027 (inMLR M10). Table 7 indicates that the MLR (M10) has

the smallest MAE (1.014) and RMSE (1.210), and thehighest CC (0.934) in the validation phase. In the testingphase the MLR M10 has the smallest MAE (0.990) andRMSE (1.204), and the highest CC (0.935).

Table 12 Performances of the MLR models in different phases for KRS12a [top] station

Model Training Validation Testing

MAE RMSE CC MAE RMSE CC MAE RMSE CC

M10 1.005 1.430 0.887 1.027 1.499 0.876 1.027 1.465 0.888

M9 1.010 1.431 0.886 1.031 1.503 0.876 1.062 1.473 0.887

M8 1.011 1.433 0.886 1.034 1.500 0.876 1.059 1.472 0.887

M7 1.969 2.446 0.613 1.996 2.482 0.602 2.000 2.470 0.629

M6 1.911 2.537 0.573 1.864 2.531 0.581 1.868 2.476 0.581

M5 1.008 1.433 0.886 1.032 1.503 0.876 1.060 1.476 0.886

M4 1.927 2.563 0.562 1.869 2.551 0.572 1.869 2.506 0.616

M3 2.151 2.667 0.508 2.164 2.707 0.492 2.156 2.645 0.555

M2 1.970 2.446 0.613 1.997 2.482 0.602 2.00 2.469 0.629

M1 2.238 2.744 0.464 2.291 2.820 0.423 2.295 2.840 0.449

Table 13 Performances of the two ANFIS models in different phases for KRS12a [top] station

Model Number of rules Training Validation Testing

MAE RMSE CC MAE RMSE CC MAE RMSE CC

ANFIS_GRID M10 81 0.255 0.370 0.992 0.451 0.622 0.982 0.528 0.686 0.981

M9 27 0.419 0.568 0.983 0.520 0.676 0.977 0.633 0.812 0.972

M8 27 0.331 0.456 0.989 0.452 0.623 0.982 0.579 0.737 0.978

M7 27 2.000 2.383 0.638 1.986 2.404 0.641 2.006 2.422 0.652

M6 27 1.045 1.533 0.868 1.169 1.655 0.855 1.277 1.745 0.855

M5 9 0.488 0.635 0.978 0.576 0.744 0.971 0.664 0.856 0.967

M4 9 1.477 1.985 0.767 1.499 1.961 0.779 1.558 2.070 0.764

M3 9 1.416 1.911 0.787 1.476 1.937 0.786 1.633 2.075 0.762

M2 9 1.570 2.137 0.724 1.582 2.136 0.727 1.669 2.215 0.719

M1 9 1.523 2.010 0.761 1.629 2.105 0.737 1.682 2.210 0.723

ANFIS_SUB M10 13 0.371 0.507 0.986 0.470 0.632 0.979 0.583 0.759 0.976

M9 7 0.501 0.676 0.976 0.611 0.801 0.969 0.724 0.978 0.957

M8 15 0.367 0.494 0.987 0.498 0.661 0.978 0.498 0.737 0.977

M7 14 1.021 1.481 0.878 1.098 1.553 0.867 1.125 1.562 0.873

M6 15 0.951 1.410 0.890 1.064 1.529 0.875 1.247 1.660 0.863

M5 6 0.529 0.684 0.975 0.612 0.790 0.968 0.717 0.933 0.968

M4 6 1.334 1.835 0.806 1.348 1.845 0.808 1.411 1.941 0.797

M3 12 1.176 1.692 0.838 1.238 1.735 0.832 1.344 1.831 0.820

M2 6 1.569 2.121 0.729 1.550 2.100 0.738 1.668 2.190 0.727

M1 11 1.393 1.868 0.798 1.471 1.948 0.780 1.546 2.021 0.776

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The proposed ANFIS_SUB model is evaluated here interms of its performance in modeling dissolved oxygenconcentration and the results are compared to the predic-tion models obtained using ANFIS_GRID. Results aretabulated in Table 8 for abovementioned station (bottom).It is worth noting that the presented results are briefed foronly Gaussian membership function as it had the bestresults than others membership functions.

According to Table 8, in the training phase, thevalues of CC, RMSE, and MAE, ranged from 0.827 to0.995, 0.300 to 1.888, and 0.216 to 1.261, respectively.In addition, in the validation phase, the values of CC,RMSE, and MAE, ranged from 0.812 to 0.995, 0.317 to1.989, and 0.240 to 1.403, respectively. Finally, in thetesting phase, the values of CC, RMSE, and MAE,ranged from 0.800 to 0.995, 0.320 to 2.066, and 0.231to 1.398, respectively. It may be seen from Table 8, theCC values for all the ten models are reasonably good,being smallest (0.800) for M2 model and greatest(0.995) for M10 model. The values of other modelperformances such as RMSE, and MAE indicate thatthe forecast performance of the ANFIS_SUB model isvery good. As can be observed from Table 8, the

ANFIS_SUB (M10) model performed better than theother models in the training, validation, and testingphases. During training, the ANFIS_SUB (M10) per-forms slightly better than the others. Also, in the valida-tion and testing phases, the ANFIS_SUB (M10) outper-forms all others models in terms of various performancecriteria. As shown in Table 8, the ANFIS_SUB (M10)predictions for the DO concentration yield a mean ab-solute error of 0.216, a root mean square error of 0.300,and a correlation coefficient of 0.995 in the trainingphase. Table 8 indicates that the ANFIS_SUB (M10)has the smallest MAE (0.240) and RMSE (0.317), andthe highest CC (0.995) in the validation phase; in thetesting phase, the ANFIS_SUB (M10) has the smallestMAE (0.231), RMSE (0.320), and the highest CC(0.995). These values show that the ANFIS_SUB pre-dicts DO very well. The ANFIS_SUB (M10) predic-tions for the dissolved oxygen concentration are indicat-ed in Fig. 4 in the training, validation, and testing phase,respectively. The statistical performances of these pre-dictions are almost as good as those observed value.

For ANFIS_SUB model when fuzzy systems aredesigned by using fuzzy clustering, each cluster

Fig. 8 Scatterplots of calculated versus observed values of dissolved oxygen concentration (DO), for ANFIS_GRID model, at siteKRS12a [top], for a training, b validation, and c testing

Fig. 9 Scatterplots of calculated versus observed values of dissolved oxygen concentration (DO), for ANFIS_SUB model, at siteKRS12a [top], for a training, b validation, and c testing

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corresponds to a fuzzy rule. Hence, the number ofclusters determines the number of rules. We have deter-mined the number of clusters experimentally, by devel-oping various models and studying the rules and theirconsequent parameters (Table 9). If the cluster radius wasspecified as a small number, then there will be many smallclusters in the data that results in many rules. In contrast,specifying a large cluster radius will yield a few largeclusters in the data resulting in fewer rules. By trial anderror, the cluster radius was determined as 0.35 (Table 9).The appropriate number of clusters resulted to be 15,which were labeled MF1 to MF15 for each input variable.The parameters of these membership functions were givenin Table 10. The ANFIS_SUB (M10) model and relatedfuzzy rule base obtained was given in Table 11. Theperformance of the actual output with that of the modeloutput is shown in Fig. 5. Figure 5 shows that the modeloutput is well comparable with the actual output (solidcurve) with acceptable mean square error.

According to the grid method, for each input vari-able, three Gaussian membership function (MFs) wasassigned. Although ANFIS_GRID gives accurate re-sults as that of the proposed approach, theANFIS_GRID model is less transparent (81 rulesagainst only 15) in comparison with ANFIS_SUB andconsumes more computational time. The ANFIS_GRID predictions for the dissolved oxygen concentra-tion are shown in Table 8. These predictions result in aMAE of 0.144, a RMSE of 0.204, and a CC of 0.998.These results reveal that the ANFIS_ GRID predictionsfor dissolved oxygen concentration are even better thanthe ANFIS_SUB in the training phase. Similarly, the

ANFIS_SUB yields slightly better MAE, RMSE, andCC values in validation and testing phase, respectivelyas shown in Table 8. The ANFIS_GRID predictions forthe dissolved oxygen concentration are indicated inFig. 6 in the training, validation, and testing phase,respectively. The performance of the actual output withthat of the model output is shown in Fig. 7. Figure 7shows that the model output is well comparable with theactual output (solid curve) with acceptable mean squareerror. For ANFIS_GRID model where each rule hasonly a single possible output (dissolved oxygen concen-tration), a complete rule base requires that a rule bedefined for every combination of antecedent conditions.A process is defined by four input variables, each de-scribed by three linguistic terms (three Gaussian mem-bership function), then 34 (or 81) rules would constitutethe complete rule base.

Station KRS12a [top] results

Table 12 presents the performance of different MLRmodels in estimating hourly dissolved oxygen concen-tration for the case study in terms of RMSE, MAE, andCC statistics, respectively. As seen from Table 12, theten MLR models have shown significant variationsbased on the three performance criteria. The lowestvalue of the RMSE of forecasting models is 1.430 (inMLRM10) and the highest value of the CC is 0.888 (inMLR M10). In addition, the lowest value of MAE is1.005 also (in MLRM10). From the results of training,validation, and testing all the ten models developed inthis study are evaluated all together, and the M10, M9,

Fig. 10 Comparisons be-tween observed and calcu-lated data of dissolved ox-ygen concentration (DO)for the training, validation,and the testing sets of theANFIS_ GRID model, atsite KRS12a [top]

Fig. 11 Comparisons be-tween observed and calcu-lated data of dissolved oxy-gen concentration (DO) forthe training, validation, andthe testing sets of theANFIS_ SUB model, at siteKRS12a [top]

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M8, and M5 models are conspicuous. Among these, theM10 andM9models have quite lowMAE and high CC,and the M10 model is very successful on testing phase.All these four models were examined comparing theirability on predicting hourly dissolved oxygen concen-tration. During training, the MLR (M10) performsslightly better than the others. Also, in the validationand testing phases, the MLR (M10) outperforms allother models in terms of various performance criteria.

As seen fromTable 13, the ten ANFIS_GRIDmodelshave shown significant variations based on the threeperformance criteria. The lowest value of the RMSE offorecasting models is 0.370 (in ANFIS_GRID M10)and the highest value of the CC is 0.992 (inANFIS_GRID M10). In addition, the lowest value of

MAE is 0.255 also (in ANFIS_GRIDM10). Also, it canbe seen from Table 13, the ten ANFIS_SUB modelshave shown significant variations based on the threeperformance criteria. The lowest value of the RMSE offorecasting models is 0.507 (in ANFIS_SUB M10) andthe highest value of the CC is 0.986 (in ANFIS_SUBM10). In addition, the lowest value ofMAE is 0.371 also(in ANFIS_SUB M10). It can be observed fromTable 13, the ANFIS_GRID models performed betterthan the ANFIS_SUB estimates for the training, valida-tion, and testing phases. During training, theANFIS_GRID (M10) performs much better than theothers. Also, in the validation and testing phases, theANFIS_GRID (M10) outperforms all other models interms of various performance criteria. Table 13 indicates

Table 14 Performances of the ANFIS_SUB model for different values of the radius parameter for KRS12a [top] station

Radius Number of rules Number of MFs Training Validation Testing

MAE RMSE CC MAE RMSE CC MAE RMSE CC

0.90 02 02 0.786 1.043 0.941 0.841 1.104 0.935 0.846 1.081 0.943

0.80 04 04 0.571 0.801 0.966 0.644 0.875 0.960 0.702 0.928 0.960

0.70 05 05 0.457 0.603 0.980 0.524 0.709 0.973 0.594 0.782 0.973

0.60 07 07 0.400 0.534 0.985 0.469 0.623 0.980 0.576 0.745 0.976

0.35 13 13 0.371 0.507 0.986 0.470 0.632 0.979 0.583 0.759 0.976

The table entries in bold show superior results among other ANFIS_SUB structure

Table 15 Final Gaussian membership function parameters (MF) of the inputs data for ANFIS_SUB model for KRS12a [top] station

Membership functions Temperature pH Specific conductance Sensor depth

a b a b a b a b

MF1 0.420 −1.186 0.454 −0.801 0.907 0.528 0.593 −0.680MF2 0.420 −0.070 0.454 −0.960 0.907 −0.509 0.593 1.088

MF3 0.420 1.481 0.454 1.432 0.907 −0.698 0.593 −0.343MF4 0.420 −0.146 0.454 −0.641 0.907 −0.667 0.593 −0.090MF5 0.420 1.263 0.454 −0.801 0.907 −0.352 0.593 0.330

MF6 0.420 −1.365 0.454 −0.801 0.907 −0.163 0.593 1.004

MF7 0.420 −1.096 0.454 −0.960 0.907 0.34 0.593 −1.943MF8 0.420 −0.134 0.454 1.432 0.907 −0.447 0.593 0.667

MF9 0.420 −0.018 0.454 0.634 0.907 0.34 0.593 −1.185MF10 0.420 −0.288 0.454 1.272 0.907 1.694 0.593 −1.606MF11 0.420 1.648 0.454 −0.0036 0.907 −0.730 0.593 0.246

MF12 0.420 −0.570 0.454 0.793 0.907 3.362 0.593 −0.090MF13 0.420 −0.134 0.454 −0.641 0.907 −0.352 0.593 2.351

a represents Gaussian MFs center and b determines Gaussian MFs width

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that the ANFIS_GRID (M10) has the smallest MAE(0.451), RMSE (0.622), and the highest CC (0.982) inthe validation phase; and in the testing phase theANFIS_GRID (M10) has the smallest MAE (0.528),RMSE (0.686), and the highest CC (0.981). In order toshow the potential of the ANFIS_GRID models, theforecast results of the ANFIS_SUB models is alsopresented. Overall, the performance of the two ANFISis very good. The results demonstrate that theANFIS_GRID can be successfully applied to establishthe forecasting models that could provide accurate andreliable reference dissolved oxygen prediction.

Table 13 indicates that unlike the KRS12a [bottom]station, the ANFIS_GRIDmodel gives accurate estimates.The performance indices reveal that the ANFIS_GRIDmodel is superior in dissolved oxygen concentration esti-mation compared with the ANFIS_SUB. In the trainingphase, theANFIS_GRID improved theANFIS_SUB fore-cast of about 27.02 and 31.26 % reduction in RMSE andMAE values, respectively. In addition, improvements ofthe forecast results regarding the correlation coefficient

value during the training phase were approximately0.6 %. In addition, in the validation phase as seen inTable 13, the values with the ANFIS_GRID predictionwere able to produce a good forecast, as compared to thosewith ANFIS_SUB prediction. In the validation phase, theANFIS_GRID improved the ANFIS_SUB forecast ofabout 1.58 and 4.04 % reduction in RMSE and MAEvalues, respectively. In addition, improvements of the fore-cast results regarding the correlation coefficient value dur-ing the validation phase were approximately 0.3 %. In thetesting phase, the ANFIS_GRID improved theANFIS_SUB forecast of about 9.61 and 9.43 % reductionin RMSE and MAE values, respectively. In addition,improvements of the forecast results regarding the correla-tion coefficient (CC) value during the testing phase wereapproximately 0.5 %.

The scatterplots of the observed versus calculatedvalues of the dissolved oxygen concentration (DO) ofthe ANFIS_GRID and ANFIS_SUB analyzed hereinare shown in Figs. 8 and 9 for the training, validation,and testing phases, respectively. Figures 10 and 11 show

Table 16 Fuzzy rule base for the ANFIS_SUB model for KRS12a [top] station

Rule number Rule description

Premise parameters Consequents parameters

1 If TE is TE_MF1 and pH is pH_MF1and SC is SC_MF1 and SD is SD_MF1

THEN DO=0.08×TE +1.36×pH −0.11×SC −0.42×SD +1.47

2 If TE is TE_MF2 and pH is pH_MF2and SC is SC_MF2 and SD is SD_MF2

THEN DO=−1.73×TE +0.64×pH −0.23×SC +0.33×SD −1.15

3 If TE is TE_MF3 and pH is pH_MF3and SC is SC_MF3 and SD is SD_MF3

THEN DO=0.30×TE +1.75×pH +0.06×SC +0.77×SD −2.8

4 If TE is TE_MF4 and pH is pH_MF4and SC is SC_MF4 and SD is SD_MF4

THEN DO=−1.32×TE +0.65×pH −0.26×SC +0.07×SD −0.40

5 If TE is TE_MF5 and pH is pH_MF5and SC is SC_MF5 and SD is SD_MF5

THEN DO=1.60×TE +1.33×pH −0.23×SC +0.64×SD −3.14

6 If TE is TE_MF6 and pH is pH_MF6and SC is SC_MF6 and SD is SD_MF6

THEN DO=2.00×TE +0.68×pH +0.03×SC −0.04×SD +3.84

7 If TE is TE_MF7 and pH is pH_MF7and SC is SC_MF7 and SD is SD_MF7

THEN DO=−0.22×TE −0.07×pH +0.18×SC −0.85×SD −1.51

8 If TE is TE_MF8 and pH is pH_MF8and SC is SC_MF8 and SD is SD_MF8

THEN DO=0.04×TE +1.18×pH +0.12×SC −0.28×SD −0.13

9 If TE is TE_MF9 and pH is pH_MF9and SC is SC_MF9 and SD is SD_MF9

THEN DO=0.42×TE +0.98×pH −0.01×SC −0.22×SD +0.25

10 If TE is TE_MF10 and pH is pH_MF10and SC is SC_MF10 and SD is SD_MF10

THEN DO=−0.38×TE +1.14×pH +0.07×SC +0.14×SD −0.29

11 If TE is TE_MF11 and pH is pH_MF11and SC is SC_MF11 and SD is SD_MF11

THEN DO=3.65×TE +1.34×pH +0.46×SC +0.44×SD −6.87

12 If TE is TE_MF12 and pH is pH_MF12and SC is SC_MF12 and SD is SD_MF12

THEN DO=0.25×TE +0.76×pH +0.16×SC +0.31×SD +0.11

13 If TE is TE_MF13 and pH is pH_MF13and SC is SC_MF13 and SD is SD_MF13

THEN DO=−1.38×TE +1.30×pH −0.53×SC −0.23×SD +0.71

Environ Monit Assess (2014) 186:597–619 617

the plots between observed and model calculated valuesof DO in training, validation, and testing sets for the twomodels, ANFIS_GRID and ANFIS_SUB, respectively. Itcan be seen from these figures, for the training phase, theANFIS_GRID model is closer to the exact fit line than theANFIS_SUB. The figures nicely demonstrate that (1) themodels performances are, in general, accurate; (2) theANFIS_GRID model is consistently superior to theANFIS_SUB in all phases. By trial and error, the clusterradius was determined as 0.35 (Table 14). The appropriatenumber of clusters resulted to be 13, which were labeledMF1 to MF13 for each input variable. The parameters ofthese membership functions were given in Table 15. TheANFIS_SUB model and related fuzzy rule base obtainedwas given in Table 16.

Conclusion

In this work the development of an ANFIS model fordissolved oxygen concentration (DO) using water qualitydata as input was presented. Results show that the trainedmodel can be considered as very satisfactory. four waterquality variables, such as water pH, temperature, specificconductance, and sensor depth, were used as input param-eters and the ANFIS model results showed a very goodagreement with the measured values (CC=0.992–0.998).Two types of ANFIS networks were considered regardingto the structure identificationmethod. The applicability andcapability of the ANFIS models are investigated throughthe use of great number data sets collected at the USGSstation. However, it has been proven that both ANFISmodels (ANFIS_SUB and ANFIS_GRID) yield very reli-able results compared with measured value of DO. ANFISmodels were compared withMLRmodels and presented ahigher correlation coefficient and a better accuracy ofprediction.

Acknowledgments The author thanks the staffs of USGS webserver for providing the data that makes this research possible.

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