Two Hybrid Artificial Intelligence Approaches for Modeling Rainfall–Runoff Process

Journal of Hydrology 402 (2011) 41–59

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Journal of Hydrology

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Two hybrid Artificial Intelligence approaches for modeling rainfall–runoff process

Vahid Nourani a,c,⇑, Özgür Kisi b, Mehdi Komasi a

a Faculty of Civil Eng., Univ. of Tabriz , 29 Bahman Ave., Tabriz, Iranb Faculty of Engineering, Civil Eng. Dept., Univ. of Erciyes, 38039, Kayseri, Turkeyc National Center for Earth-surface Dynamics (NCED), St. Anthony Falls Lab., Dept. of Civil Eng., Univ. of Minnesota , Minneapolis, MN 55414, USA

a r t i c l e i n f o s u m m a r y

Article history:Received 12 February 2010Received in revised form 4 February 2011Accepted 3 March 2011Available online 9 March 2011This manuscript was handled by AndrasBardossy, Editor-in-Chief, with theassistance of Purna Chandra Nayak,Associate Editor

Keywords:Artificial Neural NetworkWavelet transformSARIMAXLighvanchai and Aghchai watersheds

0022-1694/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jhydrol.2011.03.002

⇑ Corresponding author at: Faculty of Civil Eng., Univ403 0332; fax: +98 411 334 4287.

E-mail addresses: [email protected], nourani@edu (V. Nourani), [email protected] (Ö. Kisi),(M. Komasi).

The need for accurate modeling of the rainfall–runoff process has grown rapidly in the past decades.However, considering the high stochastic property of the process, many models are still being developedin order to define such a complex phenomenon. Recently, Artificial Intelligence (AI) techniques such asthe Artificial Neural Network (ANN) and the Adaptive Neural-Fuzzy Inference System (ANFIS) have beenextensively used by hydrologists for rainfall–runoff modeling as well as for other fields of hydrology.

In this paper, two hybrid AI-based models which are reliable in capturing the periodicity features of theprocess are introduced for watershed rainfall–runoff modeling. In the first model, the SARIMAX (SeasonalAuto Regressive Integrated Moving Average with exogenous input)-ANN model, an ANN is used to findthe non-linear relationship among the residuals of the fitted linear SARIMAX model. In the second model,the wavelet-ANFIS model, wavelet transform is linked to the ANFIS concept and the main time series oftwo variables (rainfall and runoff) are decomposed into some multi-frequency time series by wavelettransform. Afterwards, these time series are imposed as input data to the ANFIS to predict the runoff dis-charge one time step ahead. The obtained results of the models applications for the rainfall–runoff mod-eling of two watersheds (located in Azerbaijan, Iran) show that, although the proposed models canpredict both short and long terms runoff discharges by considering seasonality effects, the second modelis relatively more appropriate because it uses the multi-scale time series of rainfall and runoff data in theANFIS input layer.

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1. Introduction

Accurate modeling of hydrological processes such as rainfall–runoff can provide important information for the urban andenvironmental planning, land use, flood and water resourcesmanagement of a watershed. It also plays an important role inmitigating the impact of drought on water resources systems.Therefore, numerous hydrological models have been developedin order to simulate this complex process and a comprehensiveclassification of these models was presented by Nourani et al.(2007). Due to the large number of obscure parameters involvedin the rainfall and runoff physical relationship in a watershed,black box lumped modeling may have some advantages over fullydistributed modeling (Nourani and Mano, 2007).

Conventional black box time series models such as Auto Regres-sive Integrated Moving Average (ARIMA) or Seasonal ARIMA witheXogenous input (SARIMAX) are widely used for hydrological time

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series forecasting (Salas et al., 1980; Tankersley et al., 1993).However, they are basically linear models assuming that data arestationary, and have limited ability in capturing that which isnon-stationary and non-linearity in hydrologic data.

Recently, Artificial Intelligence (AI) techniques have showngreat ability in modeling and forecasting non-linear hydrologicaltime series. AI techniques offer an effective approach for handlinglarge amounts of dynamic, non-linear and noisy data, especiallywhen the underlying physical relationships are not fully under-stood. This makes them well suited to time series modeling prob-lems of a data-driven nature. In general, the application of AItechnique does not require a prior knowledge of the process.Numerous papers have already been presented on the successfulapplication of ANNs for modeling the rainfall–runoff process as anon-linear complex phenomenon (e.g., Hsu et al., 1995; Dawsonand Wilby, 1998; Tokar and Johnson, 1999; Sajikumara andThandaveswara, 1999; Sudheer et al., 2000; Lallahem and Maina,2003; Senthil Kumar et al., 2004; Jain et al., 2004; Antar et al.,2006).

The model and data used for simulation of the rainfall–runoffprocess usually contain uncertainties. For example, an averagedvalue of the pointy measured rainfalls by the rain gauges over awatershed is usually assigned to the whole of the watershed. Using

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42 V. Nourani et al. / Journal of Hydrology 402 (2011) 41–59

this constant real number as the watershed rainfall in the ANN in-put layer can be a source of uncertainty. In such uncertain situa-tions, fuzzy numbers and the fuzzy system may be employed inthe estimation of uncertainties in real world problems. The hybridANN and fuzzy system is a research focus, which can make use ofthe advantages of both ANN and fuzzy system namely ANFIS(Adaptive Neuro-Fuzzy Inference System) (Rajaee et al., 2009).ANFIS is capable of combining the benefits of both these fields in asingle framework. There are a few studies on the application ofANFIS in rainfall–runoff modeling (e.g., Gautam and Holz, 2001;Nayak et al., 2004; Tayfur and Singh, 2006; Jothiprakash et al., 2009).

In spite of the suitable flexibility of ANN and ANFIS in modelinga hydrologic process such as rainfall–runoff, sometimes there is ashortage when signal fluctuations are highly non-stationary andthe physical hydrologic process operates under a large range ofscales varying from 1 day to several decades. In such a situation,ANN and ANFIS models may not be able to cope with non-station-ary data if pre-processing of the input and/or output data is notperformed (Cannas et al., 2006).

To overcome the above-mentioned shortage, the combinationof ANN and ANFIS with other approaches as hybrid models maybe an appropriate choice. The basic idea of model combinationin forecasting is to use each model’s unique features to capturedifferent patterns in the data. Both theoretical and empirical find-ings suggest that combining different methods can be an efficientway to improve forecasting (Zhang, 2003). One of such hybridmodels is the ARIMA-ANN model which was developed in the lastyears and is still being used in engineering and financial time ser-ies forecasting. This model was first proposed and evaluated byZhang (2003) in order to forecast a univariate time series withoutconsidering seasonal effect. This technique has enjoyed a fewapplications in hydrological time series modeling which usuallyshow highly non-stationary seasonal behavior (e.g., Mishra et al.,2007).

Furthermore, the wavelet-ANN is another reliable hybrid modelused in time series forecasting problems. Recently, wavelet trans-form analysis has become a popular analysis tool due to its abilityto elucidate simultaneously both spectral and temporal informa-tion within the signal. This overcomes the basic shortcoming ofFourier analysis, which is that the Fourier spectrum contains onlyglobally averaged information. Therefore, a data pre-processingcan be performed by time series decomposition into its subcompo-nents using wavelet transform analysis. Wavelet transforms pro-vide useful decompositions of the main time series, so thatwavelet-transformed data improve the ability of a forecastingmodel by capturing useful information on various resolution levels.The wavelet decomposition of a non-stationary time series into dif-ferent scales provides an interpretation of the series structure andextracts significant information about its history, using few coeffi-cients. For these reasons, this technique is largely applied to timeseries analysis of non-stationary signals (Nason and Von Sachs,1999; Adamowski, 2008a,b). Hence, a hybrid wavelet-ANN modelwhich uses multi-scale signals as input data may present moreprobable forecasting than a single pattern input.

The wavelet-ANN conjunction model was first presented byAussem et al. (1998) for financial time series forecasting. Zhangand Dong (2001) proposed a short-term load forecast model basedon ANN and the multi-resolution wavelet decomposition. Inhydrology, Wang and Ding (2003) applied a wavelet-network mod-el to forecast shallow groundwater level and daily discharge. Kimand Valdes (2003) proposed a conjunction model based on dyadicwavelet transform and ANNs to forecast droughts for the Conchesriver basin in Mexico; they used ANN to forecast sub-signals fromwavelet decomposition and also to reconstruct the main signalfrom the forecasted sub-signals. In both cited studies, ‘‘a trous’’algorithm accompanied by three-layered feed forward neural net-

works was used in order to predict the hydrological time series.Nourani et al. (2009a) predicted the monthly precipitation timeseries of a watershed by a combined wavelet-ANN model. Partaland Cigizoglu (2008) and Kisi (2008) used the neuro-wavelet tech-nique for forecasting daily suspended sediment and monthly riverflow, respectively. Partal and Cigizoglu (2008) decomposed a dailysediment time series into many components (sub-signals) usingwavelet transform and then composed a new time series by addingthe dominant sub-signals and used this time series in the ANN.They used the linear correlation coefficient between the main timeseries and sub-signals in order to determine the dominant compo-nents. However in a non-linear process, two time series may have aweak linear correlation but strong non-linear relation. Cannas et al.(2006) investigated the effects of data pre-processing on ANNmodel performance using continuous and discrete wavelettransforms; the results showed that networks trained with pre-processed data performed better than networks trained on unde-composed, noisy raw signals. Anctil and Tape (2004) decomposeda rainfall time series by wavelet into three sub-series depictingthe rainfall–runoff processes: short, intermediate and long waveletperiods, then three multi-layer networks were trained for thewavelet sub-series in order to estimate the runoff values. Their re-sults showed that short wavelet period fluctuations are thus thekey to any further improvement in ANN rainfall–runoff forecastingmodels. Since this model uses more than one ANN network, simul-taneous optimization of the networks could be a difficult and timeconsuming procedure.

As a preliminary study to the current research, the authors usedthe wavelet-ANN (WANN) combined approach for modeling Ligh-vanchai watershed rainfall–runoff process on a daily time scale(Nourani et al., 2009b).

Considering the ability of ANFIS in modeling hydrologicalprocesses which usually involve some degree of uncertainty, thehybrid wavelet-ANFIS (WANFIS) approach is proposed for multi-variate hydrological modeling in the current paper. The univariatedaily precipitation forecasting model proposed by Partal and Kisi(2007) is the sole application of a wavelet and nero-fuzzy conjunc-tion model in the hydrological literature but it differs from the pro-posed multivariate WANFIS model in employing wavelet analysis.

In this paper, two new multivariate black box models based onAI techniques are proposed for the rainfall–runoff modeling of twowatersheds located in Azerbaijan, Iran which have different clima-tologic characteristics. In the first model, considering the existenceof seasonality in the time series, a seasonal ARIMA model withexogenous inputs (i.e., rainfall and runoff values), SARIMAX, iscombined with the ANN approach to construct the hybrid SARI-MAX-ANN model. A multivariate wavelet-ANFIS model is intro-duced as the second model. In this model the daily and monthlyrainfall and runoff time series of the watersheds are decomposedinto sub-signals in various resolution levels and periodicity scales;then these sub-signals are entered into the ANFIS model to recon-struct the forecasted runoff time series. One of the main aims ofthis study is to investigate the effect of simulation time scale,and also the watershed’s climatologic conditions, on the model’sperformance. Finally, in order to evaluate the models’ abilities,the results are compared with those of individual ANN, ANFISand SARIMAX models as well as the previously presented hybridWANN model.

The remainder of this paper is organized as follows. In the nextthree sections, the concepts of wavelet transform, ANNs and ANFISare briefly reviewed, respectively. Section 5 presents the formula-tions and structures of the proposed hybrid models. In Sections 6and 7, the efficiency criteria and study area are introduced and inSection 8 the models performances are evaluated, discussed andcompared with some other conventional methods. Concluding re-marks are in the final section of the paper.

V. Nourani et al. / Journal of Hydrology 402 (2011) 41–59 43

2. Wavelet transform

The wavelet transform has increased in usage and popularity inrecent years since its inception in the early 1980s, yet still does notenjoy the wide spread usage of the Fourier transform. Fourier anal-ysis has a serious drawback. In transforming to the frequency do-main, time information is lost. When looking at a Fouriertransform of a signal, it is impossible to tell when a particular eventtook place but wavelet analysis allows the use of long time inter-vals where we want more precise low-frequency information,and shorter regions where we want high-frequency information.Fig. 1 compares Fourier and wavelet transforms.

In the field of earth sciences, Grossmann and Morlet (1984),who worked especially on geophysical seismic signals, introducedthe wavelet transform application. A comprehensive literature sur-vey of wavelet in geosciences can be found in Foufoula-Georgiouand Kumar (1995) and the most recent contributions are cited byLabat (2005). As there are many good books and articles introduc-ing the wavelet transform, this paper will not delve into the theorybehind wavelets and only the main concepts of the transform arebriefly presented; recommended literature for the wavelet noviceincludes Mallat (1998) or Labat et al. (2000).

The time-scale wavelet transform of a continuous time signal,x(t), is defined as (Mallat, 1998):

Tða; bÞ ¼ 1ffiffiffiap

Z þ1

�1g�

t � ba

� �xðtÞ � dt ð1Þ

where � corresponds to the complex conjugate and g(t) is calledwavelet function or mother wavelet. The parameter a acts as a dila-tion factor, while b corresponds to a temporal translation of thefunction g(t), which allows the study of the signal around b. Themain property of wavelet transform is to provide a time-scale local-ization of processes, which derives from the compact support of itsbasic function. This is opposed to the classical trigonometric func-tion of Fourier analysis. The wavelet transform searches for correla-tions between the signal and wavelet function. This calculation isdone at different scales of a and locally around the time of b. Theresult is a wavelet coefficient (T(a, b)) contour map known as a sca-logram. In order to be classified as a wavelet, a function must havefinite energy, and it must satisfy the following ‘‘admissibility condi-tions’’ (Mallat, 1998):Z þ1

�1gðtÞdt ¼ 0; Cg ¼

Z þ1

�1

jgðwÞj2

jwj dw <1 ð2Þ

where gðwÞ is Fourier transform of g(t); i.e., the wavelet must haveno zero frequency component.

Fig. 1. Comparison of Fourier

In order to obtain a reconstruction formula for the studied sig-nal, it is necessary to add ‘‘regularity conditions’’ to the previousones (Mallat, 1998):Z þ1

�1tkgðtÞdt ¼ 0 where k ¼ 1;2; . . . ;n� 1 ð3Þ

So the original signal may be reconstructed using the inverse wave-let transform as (Mallat, 1998):

xðtÞ ¼ 1cg

Z þ1

�1

Z 1

0

1ffiffiffiap g

t � ba

� �Tða; bÞda � db

a2 ð4Þ

For practical applications, the hydrologist does not have at hisor her disposal a continuous – time signal process but rather a dis-crete – time signal. A discretization of Eq. (1) based on the trape-zoidal rule maybe is the simplest discretization of the continuouswavelet transform. This transform produces N2 coefficients froma data set of length N; hence redundant information is locked upwithin the coefficients, which may or may not be a desirable prop-erty (Addison et al., 2001).

To overcome the mentioned redundancy, logarithmic uniformspacing can be used for the a scale discretization with correspond-ingly coarser resolution of the b locations, which allows for Ntransform coefficients to completely describe a signal of length N.Such a discrete wavelet has the form (Mallat, 1998):

gm;nðtÞ ¼1ffiffiffiffiffiffiam

0

p gt � nb0am

0

am0

� �ð5Þ

where m and n are integers that control the wavelet dilation andtranslation respectively; a0 is a specified fined dilation step greaterthan 1; and b0 is the location parameter and must be greater thanzero. The most common and simplest choice for parameters area0 = 2 and b0 = 1.

This power-of-two logarithmic scaling of the translation anddilation is known as the dyadic grid arrangement. The dyadicwavelet can be written in more compact notation as (Mallat,1998):

gm;nðtÞ ¼ 2�m=2gð2�mt � nÞ ð6Þ

Discrete dyadic wavelets of this form are commonly chosen tobe orthonormal; i.e. (Mallat, 1998):

Z þ1

�1gm;nðtÞgm0 ;n0 ðtÞdt ¼ dm;m0dn;n0 ð7Þ

which d is Kronecker delta.

and Wavelet transforms.


This allows for the complete regeneration of the original signalas an expansion of a linear combination of translates and dilatesorthonormal wavelets.

For a discrete time series, xi, the dyadic wavelet transform be-comes (Mallat, 1998):

Tm;n ¼ 2�m=2XN�1

i¼0

gð2�mi� nÞxi ð8Þ

where Tm,n is wavelet coefficient for the discrete wavelet of scalea = 2m and location b = 2mn. Eq. (8) considers a finite time series,xi, i = 0, 1, 2, . . . , N � 1; and N is an integer power of 2: N = 2M. Thisgives the ranges of m and n as, respectively, 0 < n < 2M�m � 1 and1 < m < M. At the largest wavelet scale (i.e., 2m where m = M) onlyone wavelet is required to cover the time interval, and only onecoefficient is produced. At the next scale (2m�1), two wavelets coverthe time interval, hence two coefficients are produced, and so ondown to m = 1. At m = 1, the a scale is 21, i.e., 2M�1 or N/2 coefficientsare required to describe the signal at this scale. The total number ofwavelet coefficients for a discrete time series of length N = 2M isthen 1 + 2 + 4 + 8 + � � � + 2M�1 = N � 1.

In addition to this, a signal smoothed component, �T is left,which is the signal mean. Thus, a time series of length N is brokeninto N components, i.e., with zero redundancy. The inverse discretetransform is given by (Mallat, 1998):

xi ¼ �T þXM

m¼1

X2M�m�1

n¼0

Tm;n2�m=2gð2�mi� nÞ ð9Þ

or in a simple format as (Mallat, 1998):

xi ¼ �TðtÞ þXM

m¼1

WmðtÞ ð10Þ

Which �TðtÞ is called approximation sub-signal at level M and Wm(t)are details sub-signals at levels m = 1, 2, . . . , M.

The wavelet coefficients, Wm(t) (m = 1, 2, . . . , M), provide the de-tail signals, which can capture small features of interpretationalvalue in the data; the residual term, �TðtÞ, represents the back-ground information of data.

Because of simplicity of W1(t), W2(t), . . . , WM(t), �TðtÞ, someinteresting characteristics, such as period, hidden period,dependence and jump can be diagnosed easily through waveletcomponents.

Fig. 2. A three-layered feed-forward neura

3. Artificial Neural Networks (ANNs)

ANN is widely applied in hydrology and water resource studiesas a forecasting tool. In ANN, feed forward back-propagation (BP)network models are common to engineers. It has proved that BPnetwork model with three-layer is satisfied for the forecastingand simulating in any engineering problem (Hornik, 1988; ASCE,2000; Nourani et al., 2008).

As shown in Fig. 2, three-layered feed forward neural networks(FFNNs), which have been usually used in forecasting hydrologictime series, provide a general framework for representing non-linear functional mapping between a set of input and outputvariables. Three-layered FFNNs are based on a linear combinationof the input variables, which are transformed by a non-linear acti-vation function.

In the Fig. 2 i, j and k denote input layer, hidden layer and out-put layer neurons, respectively and w is the applied weight by theneuron. The term ‘‘feed forward’’ means that a neuron connectiononly exists from a neuron in the input layer to other neurons in thehidden layer or from a neuron in the hidden layer to neurons in theoutput layer and the neurons within a layer are not interconnectedto each other. The explicit expression for an output value of athree-layered FFNN is given by (Kim and Valdes, 2003):

yk ¼ fo

XMN

j¼1

wkj � fh

XNN

i¼1

wjixi þwjo

!þwko

" #ð11Þ

where wji is a weight in the hidden layer connecting the ith neuronin the input layer and the jth neuron in the hidden layer, wjo is thebias for the jth hidden neuron, fh is the activation function of thehidden neuron, wkj is a weight in the output layer connecting thejth neuron in the hidden layer and the kth neuron in the outputlayer, wko is the bias for the kth output neuron, fo is the activationfunction for the output neuron, xi is ith input variable for input layerand yk, y are computed and observed output variables, respectively.NN and MN are the number of the neurons in the input and hiddenlayers, respectively. The weights are different in the hidden and out-put layers, and their values can be changed during the process ofthe network training.

4. The Adaptive Neuro-Fuzzy Inference System (ANFIS)

Each fuzzy system contains three main parts, fuzzifier, fuzzydata base and defuzzifier. Fuzzy data base contains two main parts,fuzzy rule base, and inference engine. In fuzzy rule base, rules

l network with BP training algorithm.

Fig. 3. (a) Sugeno’s fuzzy if–then rule and fuzzy reasoning mechanism. (b) Equivalent ANFIS structure.


related to fuzzy propositions are described (Jang et al., 1997).Thereafter, analysis operation is applied by fuzzy inference engine.There are several fuzzy inference engines which can be employedfor this goal, which Sugeno and Mamdani are the two of well-known ones (Lin et al., 2005). Neuro-fuzzy simulation refers tothe algorithm of applying different learning techniques producedin the neural network literature to fuzzy modeling or a fuzzy infer-ence system (FIS) (Brown and Harris, 1994). This is done by fuzzifi-cation of the input through membership functions, where a curvedrelationship maps the input value within the interval of [0 1]. Theparameters associated with input as well as output membershipfunctions are trained using a technique like back propagationand/or least squares. Therefore, unlike the Multi-Layer Perceptron(MLP), where weights are tuned, in ANFIS, fuzzy language rules orconditional (if–then) statements, are determined in order to trainthe model (Rajaee et al., 2009).

The ANFIS is a universal approximator and as such is capable ofapproximating any real continuous function on a compact set toany degree of accuracy (Jang et al., 1997). The ANFIS is functionallyequivalent to fuzzy inference systems (Jang et al., 1997). Specifi-cally the ANFIS system of interest here is functionally equivalentto the Sugeno first-order fuzzy model (Jang et al., 1997). The gen-eral construction of the ANFIS is presented in Fig. 3.

Fig. 3a shows the fuzzy reasoning mechanism for the Sugenomodel to derive an output function f from a given input vector[x, y]. The corresponding equivalent ANFIS construction is shownin Fig. 3b. According to this figure, it is assumed that the FIS hastwo inputs x and y and one output f. For the first order Sugenofuzzy model, a typical rule set with two fuzzy if–then rules canbe expressed as (Aqil et al., 2007):

Rule ð1Þ : If lðxÞ is A1 and lðyÞ is B1; then f 1 ¼ p1xþ q1yþ r1

Rule ð2Þ : If lðxÞ is A2 and lðyÞ is B2; then f 2 ¼ p2xþ q2yþ r2

where A1, A2 and B1, B2 are the membership functions for inputs xand y, respectively; p1, q 1, r1 and p2, q2, r2 are the parameters ofthe output function. The functioning of the ANFIS is as follows:

Layer 1: Each node in this layer produces membership grades ofan input variable. The output of the ith node in layer k isdenoted as Q k

i . Assuming a generalized bell function as themembership function, the output Q 1

i can be computed as (Jangand Sun, 1995):

Q1i ¼ lAi

ðxÞ ¼ 1

1þ ððx� ciÞ=aiÞ2bið12Þ

where {ai, bi, ci} are adaptable variables known as premiseparameters.Layer 2: Every node in this layer multiplies the incoming signals(Jang and Sun, 1995):

Q2i ¼ wi ¼ lAi

ðxÞ � lBiðyÞ i ¼ 1;2 ð13Þ

Layer 3: The ith node of this layer calculates the normalized fir-ing strengths as (Jang and Sun, 1995):

Q3i ¼ �wi ¼

wi

w1 þw2i ¼ 1;2 ð14Þ

Layer 4: Node i in this layer calculates the contribution of the ithrule towards the model output, with the following node func-tion (Jang and Sun, 1995):

Q4i ¼ �wiðpixþ qiyþ riÞ ¼ �wifi ð15Þ

where �w is the output of layer 3 and {pi, qi, ri} is the parameterset.Layer 5: The single node in this layer calculates the overall out-put of the ANFIS as (Jang and Sun, 1995):

Q5i ¼

Xi

�wifi ¼

Pi

wifiPi

wið16Þ

The learning algorithm for ANFIS is a hybrid algorithm, which isa combination of the gradient descent and least-squares method(Aqil et al., 2007). The parameters for optimization are the premise


parameters {ai, bi, ci} and the consequent parameters {pi, qi, ri}. Inthe forward pass of the hybrid learning approach, node outputsgo forward until layer (4) and the consequent parameters are iden-tified by the least-squares technique. In the backward pass, the er-ror signals propagate backward and the premise parameters areupdated by gradient descent. More information for ANFIS and hy-brid algorithm can be found in related literature (Jang and Sun,1995; Jang et al., 1997).

5. Proposed hybrid models

5.1. Hybrid SARIMAX-ANN (SANN) Model

For construction of the SANN model, the proposed methodologyby Zhang (2003) in developing the ARIMA-ANN model is followedin this paper but with considering two other components. Firstly,because of the seasonality characteristics of the hydrological timeseries and secondly in order to employ two variables (i.e., rainfalland runoff) in the modeling, SARIMAX-ANN is proposed and usedfor rainfall–runoff modeling instead of ARIMA-ANN model whichconsiders just autoregressive property of the time series in themodeling and also its formulation contains only one variable.

According to the complex behavior of the hydrological time ser-ies (e.g., rainfall and runoff time series), they may include both lin-ear and non-linear components. Therefore, application of anindividual linear model (such as ARIMA or SARIMA) for such sig-nals may not be adequate. On the other hand, with all advantagesof the ANNs, these models are not universal model suitable for allcircumstances and may yield mixed results in modeling linearproblems. Hence, it is not wise to apply ANNs blindly to any typeof data without any data pre or post-processing. For example in aprocess which includes seasonal and non-stationary characteris-tics, a temporal data pre-processing may improve the efficiencyof the modeling (Zhang, 2003; Cannas et al., 2006); but if the pro-cess shows trend in space, a spatial data pre-processing such asspatial clustering should be employed prior to the main modeling(Nourani and Kalantari, 2010). Therefore, since it is difficult toknow the characteristics of the data in a real problem completely,hybrid modeling can be a good methodology for practical use andby combing different models; different aspects of the underlyingpatterns may be captured.

The proposed hybrid SARIMAX-ANN model consists of twosteps. In the first step, a SARIMAX model is initially fitted to modelthe linear component of the runoff time series so that the rainfall isalso used as external (regressor) variable in this exogenous model-ing. The result of this stage is the estimation of actual value of run-off time series one time ahead (i.e., Qðt þ 1Þ) via the used SARIMAX(p, d, q)(P, D, ~Q)sI(t) model in the general form of (Box and Jenkins,1976):

/pðBÞUPðBÞ½rdrDs ðQðtÞ � cIðtÞÞ � �QðtÞ� ¼ hqðBÞH~Q ðBÞeðtÞ ð17Þ

Fig. 4. The schematic flowchart

In this equation s is periodicity; /pðBÞ ¼ 1� /1B� /2B2 � . . .

�/pBp, UPðBÞ ¼ 1�U1Bs �U2B2s � � � � �UPBPs, hqðBÞ ¼ 1� h1B�h2B2 � � � � � hqBq and H~Q ðBÞ ¼ 1�H1Bs �H2B2s � � � � �H~Q B

~Qs arethe autoregressive and moving average polynomials in B of degreep, q, P, ~Q; B is the backward shift operator (e.g., BI(t) = I(t � 1)),r ¼ 1� B;rs ¼ 1� Bs; d, D are the numbers of regular and sea-sonal differences respectively; c is the regressor’s coefficient; Q(t)and I(t) are observed runoff and rainfall time series with the meansof �QðtÞ;�IðtÞ, respectively; e(t) is the estimated residual at time twhich should be linearly independently and identically distributedas normal random variables with mean � 0 and variance r2. TheSARIMAX model involves the following three iterative cycles (Boxand Jenkins, 1976):

(a) Identification of the model structure in which ACF (AutoCorrelation Function) and PACF (Partial Auto CorrelationFunction) plots can be good guides in this step.

(b) Estimation of the parameters. The maximum likelihoodmethod may be used for this purpose.

(c) Goodness-of-fit test on the estimated residuals and diagnos-tic checking of the model adequacy. The AIC (Akaike Infor-mation Criterion) is a reliable choice for checking themodel performance as well as model structure sufficiency.

Now, considering the actual time series to be composed of a lin-ear structure and a non-linear component:

Qðt þ 1Þ ¼ Qðt þ 1Þ þ eðt þ 1Þ ð18ÞThe SARIMAX model just yields the linear estimation for runoff

(Qðt þ 1Þ) and cannot handle the non-linear patterns; hence, theresidual time series of the SARIMAX (i.e., e(t + 1)) may containthe non-linear relationship which this relation can be detectedby an ANN via the second step of the SANN modeling. With NN in-put neurons in the first layer of the ANN where include residualvalues of NN previous days and according Eq. (11), the ANN modelfor the residuals will be as follows:

eðt þ 1Þ ¼ f ðeðtÞ . . . eðt � NNÞÞ þ etþ1

¼ f0

XMN

j¼1

wkjfhðXNN

i¼1

wjjeðt þ 1� iÞ þwjoÞ þwko

" #þ eðt þ 1Þ

ð19Þwhere f(t) is a non-linear function determined by the ANN ande(t + 1) is the random error. Consequently, considering Eqs. (18)and (19), the forecasted value for the runoff at time t + 1 will becomputed by Eq. (20). This procedure is illustrated in Fig. 4.

Qðt þ 1Þ ¼ Qðt þ 1Þ þ f ðtÞ þ etþ1 ð20Þ

5.2. Hybrid Wavelet-ANFIS (WANFIS) model

The proposed WANFIS model consists of a five-layer Sugenorule based structure so that the first layer is the wavelet neurons

for SARIMAX-ANN model.

Fig. 5. The schematic flowchart for WANFIS model.


unit with the fuzzified inputs of rainfall and runoff sub-signals ob-tained via a wavelet transform. The schematic diagram of thedeveloped model is shown in Fig. 5. For non-linear calculations, atrail–error back propagation process can be employed to obtainthe weighting through training the WANFIS.

In the proposed approach, the rainfall and runoff signals arefirstly decomposed into sub-signals with different scales, i.e., alarge scale sub-signal and several small scale sub-signals in orderto obtain temporal characteristics of the input time series. For agiven time series, the time series corresponding to a(t) (i.e., Ia(t)or Qa(t)) is approximation sub-signal (large scale) of the original

Fig. 6. (a) Haar wavelet, (b) db4 wavelet

signal and ith or jth detailed sub-signal (small scale) is identifiedby i or j (i.e., Ii(t) or Qj(t)) where i and j are decomposition levelsof the rainfall (I(t)) and runoff (Q(t)) time series respectively. An-nual or seasonal data are decomposed into large scale sub-signaland daily, weekly and monthly data in the small periods, aredecomposed into detailed sub-signals.

The number of neurons in the input layer is determined asi + j + 2 because the model uses two variables and each time seriesis decomposed into i + 1 or j + 1 sub-signals. In this study someirregular mother wavelets such as Haar, db4 (Daubechies waveletof order 4), sym3 and coif1 were used which are illustrated in

(c) sym3 wavelet (d) coif1 wavelet.


Fig. 6. Mallat (1998) can be referred for more information aboutthese wavelets. Since both rainfall and runoff time series are mea-sured pointy and in a discrete form, in this study dyadic discretewavelet transform was used rather than a continuous wavelet.Also, ‘‘a trous’’ is another alternative of decomposition and recon-struction algorithm and can be considered for the modeling. It is asimple and fast algorithm but it is a redundant algorithm and itskey is to determine an appropriate low-pass filter (usually a Splineis selected) which its determination is not usually convenient foran inexpert.

6. Efficiency criteria

The model that yields the best results in terms of determinationcoefficient and root mean squared error on the training and verify-ing steps can be determined through trial and error process. Forthis purpose the following measures of evaluation have been used

Fig. 7. Location of t

Table 1Statistic characteristics of rainfall and runoff data for case studies.

Case study Time scale Rainfall time series (mm)

Max Min Mean

Calibration data setLighvanchi Monthly 0.61 0.05 0.19

Daily 19 0 0.76Aghchai Monthly 1 0.05 0.20

Daily 29 0 0.69

Verification data setLighvanchi Monthly 0.45 0.05 0.20

Daily 22 0 0.76Aghchai Monthly 0.88 0.05 0.25

Daily 30 0 0.91

to compare the performance of the different models (Nourani,2009):

R2 ¼ 1�PN

i¼1ðOobsi� Ocomi

Þ2PNi¼1ðOobsi

� �OobsÞ2ð21Þ

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1ðOobsi

� OcomiÞ2

N

sð22Þ

where R2, RMSE, N, Oobsi, Ocomi

and �Oobs are determination coefficient,Root Mean Squared Error, number of observations, observed data,computed values and mean of observed data, respectively. Also,the Eq. (23) can be used to compare the ability of different modelsin capturing the peak values in runoff time series as similar as Eq.(21) for the total data.

R2Peak ¼ 1�

Pni¼1ðQ PCi

� Q POiÞ2Pn

i¼1ðQ POi� �Q POÞ2

ð23Þ

he study areas.

Runoff time series (m3/s)

Variance Max Min Mean Variance

0.02 0.89 0.05 0.17 0.034.21 9 0 0.63 0.910.02 1 0.05 0.19 0.036.75 85.40 0.12 4.03 33.81

0.01 1 0.05 0.23 0.065.02 6 0 0.99 1.810.03 0.81 0.07 0.22 0.038.96 68.30 0.10 4.72 33.54


where R2peak determination coefficient for peak values, n number of

peak values, QPOi, QPCi

and �QPO are observed data, computed valuesand mean of observed data for peak values, respectively.

The RMSE is used to measure forecast accuracy, which producesa positive value by squaring the errors. The RMSE increases fromzero for perfect forecasts through large positive values as the dis-crepancies between forecasts and observations become increas-ingly large. Obviously high value for R2 (up to one) and smallvalue for RMSE indicate high efficiency of the model. Legates andMcCabe (1999) indicated that a hydrological model can be suffi-ciently evaluated by these two statistics. Also in modeling process,the total data were split into training (calibration) and testing (ver-ification) sets. For this purpose the data set is divided into twoparts: the first 75% of total data were used as training set andthe second 25% are used for verifying the models. The time seriesdata before going through the network are usually normalized be-tween 0 and 1.

7. Study area

The data used in this paper are from Lighvanchi and Aghchaiwatersheds, located in northwest Iran at Azerbaijan province.These watersheds are main sub-branches of the Ajichai and ArasRivers which discharge to Urmieh and Caspian Lake, respectively(Fig. 7).

The time series data for 12 years (from 1995 to 2007) were usedin the modeling process (the first 8 years for training and the rest4 years for verification). The statistic characteristics of rainfalland runoff for both watersheds in daily and averaged monthly timescales are tabulated in Table 1. The other characteristics of thewatersheds are introduced in cases (1) and (2) comprehensively.

7.1. Case (1): Lighvanchai watershed

Lighvanchai watershed is located between 37�450 and 37�500

North latitude and 46�250 and 46�260 East longitude. The wa-tershed area is 75 km2 (Fig. 8). Watershed elevation is varying be-tween 2140 m and 3620 m above sea level and its longestwaterway has 17 km length. The geology formation of the wa-tershed is hard volcanic and surface layer is constructed 15 cmapproximately thick, dark brown in color with a varying texture

Fig. 8. Lighvanchai watershed

of sandy to silt and clay. Watershed contains moderate to mediumvegetative cover as a rural region. The topography is steep withaverage slope 11%. Hence the soils are susceptible to erosion tosome extended. The prevailing climate of the study area is rainyand sub-humid having four well defined seasons viz. spring, sum-mer, autumn and winter. During the wet season, the area is underthe influence of middle latitude westerlies, and most of the rainthat occurs over the region during this period is caused by depres-sions moving over the area, after forming in the Mediterranean Seaon a branch of the polar jet stream in the upper troposphere. Themean daily temperatures vary from �22 �C in January up to 40 �Cin July with a yearly average of 9 �C. The dominant winds overthe area blow from the northeast and the southwest. The meanand maximum value of precipitation and runoff are presented inTable 1 in daily and monthly time scales.

7.2. Case (2): Aghchai watershed

Aghchai watershed is located between 38�400 and 39�300 Northlatitude and 44�100 and 44�570 East longitude. The watershed areais 1440 km2 (Fig. 9). Watershed elevation is varying between1168 m and 3280 m above sea level and its longest waterwayhas 64.88 km length. The topography is steep with average slope25%. Due to adjacency watersheds, the other climatologic and geo-logical characteristics of the Aghchi watershed are similar to Ligh-vanchai watershed.

As whole, by comparison the both watershed characteristicssuch as runoff, slope, and area, it is concluded that the Aghchai wa-tershed can be categorized as a wild watershed with respect to theLighvanchai watershed. For instance, the average slope ratio(25:11) for Aghchai watershed is twice bigger than Lighvanchaiwatershed. This character causes the Aghchai watershed to havemore quick response toward the Lighvanchai watershed for anevent of precipitation. Furthermore, according to the statistics ofthe watersheds presented in Table 1, it is obvious that the variancevalues and also the difference between the minimum and maxi-mum values of the data for the Aghchai are more considerable thanthe values for the Lighvanchai. Also it has been seen that there aremore instantaneously jumps in the Aghchai time series in compar-ison with the Lighvanchai time series.

(location and DEM map).

Table 2Results and structures of ANN model.

Case study Modeling time scale Network structurea Input variablesb Determination coefficient (R2)

Calibration Verification

Lighvanchai Daily (4, 6, 1) Qt, Qt�1 0.75 0.64Monthly (4, 5, 1) 0.72 0.61

Aghchai Daily (4, 7, 1) It, It�1 0.67 0.53Monthly (4, 4, 1) 0.72 0.63

Lighvanchai Daily (6, 8, 1) Qt, Qt�1, Qt�2 0.78 0.71Monthly (6, 8, 1) 0.69 0.60

Aghchai Daily (6, 10, 1) It, It�1, It�2 0.76 0.70Monthly (6, 7, 1) 0.69 0.57

Lighvanchai Daily (8, 12, 1) Qt, Qt�1, Qt�2, Qt�3 0.82 0.74Monthly (8, 5, 1) 0.77 0.69

Aghchai Daily (8, 10, 1) It, It�1, It�2, It�3 0.80 0.71Monthly (8, 10, 1) 0.78 0.70

Lighvanchai Daily (10, 15, 1) Qt, Qt�1, Qt�2, Qt�3, Qt�4 0.81 0.73Monthly (10, 8, 1) 0.79 0.68

Aghchai Daily (10, 17, 1) It, It�1, It�2, It�3, It�4 0. 80 0.69Monthly (10, 8, 1) 0.77 0.67

a The result has been presented for the best structure.b Qt+1: Output Variable.

Fig. 9. Aghchai watershed (location and DEM map).


8. Results and discussion

8.1. The ANN model

At first, a Multi-Layer Perceptron (MLP) feed forward ANNmodel without any data pre-processing was used to model thewatersheds’ rainfall–runoff process. This kind of ANN modelaccompanied by back propagation training algorithm is widelyused in hydrologic modeling (ASCE, 2000). In this study four struc-tures were examined for the ANN model and the results have beenshown in Table 2. Each MLP was trained with 3–20 hidden neuronsin a single hidden layer using the Levenberg–Marquardt trainingalgorithm and 150 epochs with 10�4 as goal performance. No greatimprovement in model performance was found when the numberof hidden neurons was increased from a threshold, which is similarto the experience reported by Abrahart and See (2000). The train-

ing was terminated at the point where the error in the validationdata set begins to rise. This ensures that the network does not overfit the training data and then fails to generalize the unseen testdata set. At this stage, the model efficiency criterion (Determina-tion coefficient, R2) showed the low performance of the model evenwhen a window format of the previous (at time stepst, t � 1, t � 2, . . .) days (or months) rainfall–runoff data was usedas input neurons. This was probably because of significant fluctua-tions of the data around the mean value, so that the short-termregression between data was reduced.

8.2. The SANN model

Afterwards, the first proposed hybrid model (i.e., SANN) wasexamined for the rainfall–runoff modeling of the watersheds. Theresults of the model for four selected types are presented in Table

Table 3Structures and results of SANN model.

Case study Modeling time scale SARIMAX structure Input variables to ANNa Determination coefficient (R2)


Lighvanchai Daily (2, 1, 1)(1, 2, 1)3I(t) et, et�1 0.75 0.65Monthly 0.68 0.61

Aghchai Daily 0.73 0.63Monthly 0.68 0.59

Lighvanchai Daily (2, 1, 1)(1, 1, 1)12I(t) et, et�1, et�2 0.90 0.86Monthly 0.88 0.81


Lighvanchai Daily (2, 1, 1)(2, 1, 0)12I(t) et, et�1 0.83 0.75Monthly 0.73 0.64


Lighvanchai Daily (2, 1, 0)(0, 1, 0)12I(t) et, et�1, et�2 0.81 0.72Monthly 0.78 0.66


a Output variable is et+1.


3 as examples. The best SANN structure in daily modeling belongsto the SARIMAX(2, 1, 1)(1, 1, 1)12 I(t) model in which the obtainedresiduals were entered into the ANN with a window format of 3successive days’ residuals (i.e., e(t),e(t � 1),e(t � 2)) in order to findthe one time step ahead residual (i.e., e(t + 1)). In this model, at firsta SARIMAX model was fitted to the calibration runoff data set sothat the everyday rainfall value was also considered as theexogenous input of the model. The fitted model gave the estimatedvalues for the runoff ðQðtÞÞ, then the residual time series of thecalibration data was computed by subtracting the actual observeddata (Q(t)) from the estimated values ðQðtÞÞ (i.e., eðtÞ ¼QðtÞ � QðtÞ). This residual time series of the calibration data setwas then used in order to train an ANN model to detect the non-linear relationship among the residuals. The best architecture ofthe ANN was (3, 6, 1), obtained via a trial–error process using afeed forward network trained by the back propagation Leven-berg–Marquardt algorithm. The trained network was then usedin order to find the residual values of the verification data setday by day. On the other hand, SARIMAX(2, 1, 1)(1, 1, 1)12 I(t)was separately used in order to find the linear estimation of therunoff values of the verification data set. Finally, the values ofthe computed runoff for the verification data set were obtainedby Eq. (20) and could be compared with the observed runoff interms of the determination coefficient. The same methodologyaccompanied by a trial–error process was also used for monthlymodeling. In addition to the results presented in Table 3, the scat-ter plot of the verification stage is shown in Fig. 10 for dailymodeling.

Fig. 10. SANN scatter plot for verification data (daily modeling).

8.3. The WANFIS model

The second hybrid model, WANFIS, was designed to catch thecapability of the ANFIS and wavelet in non-linear rainfall–runoffmodeling simultaneously. In this modeling, the pre-processed data(by wavelet transform) were entered into the ANFIS model in orderto improve the model’s accuracy. For this purpose, the discretewavelet transform was used. Wavelet algorithms process data atdifferent temporal scales (levels) thereby the permitting grossand small features of a signal to be separated.

The WANFIS model, which contains both wavelet and ANFISconcepts, has numerous calibration parameters such as numberof membership functions (MF), type of membership function, num-ber of decomposition levels and mother wavelet type. Therefore,

the simultaneous calibration of all parameters in the unique frame-work of the model may be a time consuming process. Hence as apreface to the WANFIS model, an ANFIS model was used and itsstructure and parameters were calibrated by the available data.Thereafter, most of the parameters of the identified ANFIS modelwere also utilized in the hybrid WANFIS model where the wavelettransform is combined with the ANFIS model.

In ANFIS modeling two points are important and particularattention must be paid to them: firstly the ANFIS architecture(e.g., type and number of MF) and secondly training iteration num-ber (epoch), the appropriate selection of which can improve themodel’s efficiency in both steps of calibration (training) and verifi-cation; it also prevents the model from being over trained. In thisstudy, sensitivity analysis was performed to determine an efficientANFIS architecture. For this purpose the following input combina-tions (combs (1)–(5)), which include different numbers of inputvalues (Q and I), were considered in the input layer to predict theone time step ahead runoff (Qt+1).

Comb. (1): Qt, It

Comb. (2): Qt, Qt�1, It, It�1

Fig. 11. The daily and monthly ANFIS structures versus determination coefficients for calibration step.

Table 4Results of daily and monthly time series modeling by ANFIS.

Case study Lighvanchai Aghchai

Modeling step Calibration Verification Calibration Verification

Combination Time scale R2 RMSE R2 RMSE R2 RMSE R2 RMSE

Comb (2) Daily 0.88 0.027 0.85 0.037 0.89 0.026 0.86 0.036Monthly 0.89 0.022 0.84 0.039 0.80 0.043 0.70 0.049

Comb (4) Daily 0.87 0.030 0.81 0.038 0.88 0.027 0.84 0.037Monthly 0.79 0.043 0.73 0.047 0.89 0.022 0.86 0.029

Note: The result has been presented for the best ANFIS structure (i.e., gbellmf-2)


Comb. (3): Qt, Qt�1, Qt�2, It, It�1, It�2

Comb. (4): Qt, Qt�1, Qt�2, Qt�3, It, It�1, It�2, It�3

Comb. (5): Qt, Qt�1, Qt�2, Qt�3, Qt�4, It, It�1, It�2, It�3, It�4

The results of the sensitivity analysis for daily and monthlymodeling are presented in Fig. 11. According to the figure, it canbe concluded that comb (1) and comb (5) are not very accuratefor daily modeling in the Aghchai and Lighvanchai watersheds,respectively. It is clear that the efficiency of comb (5) for the Ligh-vanchai watershed (black column) is relatively low in comparisonwith combs (1), (2) and (4). Furthermore, comb (5) includes twoand six more parameters than combs (4) and (2), respectively,and has a more complex structure without any considerableimprovement in efficiency. Also, comb (3) has the worst perfor-mance of all the combinations. Hence combs (2) and (4) with thebell function as the membership function and two MFs (gbellmf-

2) are satisfied for the ANFIS model in the modeling of daily andmonthly time scales. In addition, by training different ANFIS struc-tures via different training epoch numbers, we found that 200epochs are sufficient to train the network with 10�4 as goal perfor-mance. Hence the above-mentioned ANFIS structures were utilizedfor developing the WANFIS model for both watersheds. The resultsobtained by the ANFIS model are presented in Table 4. The resultspresented in Fig. 11 indicate the improved performance of thecomb (3) data set in the monthly modeling; however, it was thelowest in daily modeling. In the authors’ opinion this behaviormay be related to the seasonal and periodic patterns of the process.In monthly modeling using the comb (3) data set, all data of eachseason (3 months) which can represent a complete pattern of aseason are taken into account. However, regarding daily modeling,and according to the hydrological regime of the study region, theprecipitation over the watersheds usually takes one or 2 days


and rarely extends to the third day. Therefore in modeling usingthe comb (3) data set, the third neuron of the rainfall at the inputlayer usually takes a zero value so that these zero values act as thesignal’s noises and decrease the efficiency of the model. However,in combs (4) and (5) when the number of such zero values is in-creased at the input layer, the network can learn and adapt itselfto the situation.

In the next step when wavelet analysis is combined with theANFIS model, we also aimed to examine the effects of the motherwavelet type used, as well as the decomposition level, on the

Fig. 13. Approximation and details sub-sign

Fig. 12. The calibration determination coefficients versus

model’s efficiency (Fig. 12). Hence, time series were decomposedinto four, five, six, seven, eight and nine levels by different kindsof wavelet transforms, i.e., 1-the Haar wavelet, a simple wavelet;2-the Daubechies wavelet (db), a most popular wavelet; 3-thesym wavelet, with three sharp peaks; 4-the coif1 wavelet (Mallat,1998). Fig. 12 shows that the db4 and Haar mother wavelets havebetter performance and the subsequent results of the WANFISmodel are presented for these mother wavelets.

In decomposition level 5 we have 5 details (21-day mode,22-day mode, 23-day mode which is nearly weekly mode, 24-day

als of Lighvanchai rainfall time series.

different mother wavelets and decomposition levels.


and 25-day mode which is nearly monthly mode) and one approx-imation signal. In decomposition level 7, we also have two moredetails (i.e., 26-day mode and 27-day mode) and in level 8 we alsohave another sub-signal which is the 28-day mode (nearly annualmode) and finally in level 9, the other detail is the 29-day mode.For instance, the approximation and detail sub-signals of theLighvanchai rainfall time series, decomposed by the db4 waveletat level 8, are presented in Fig. 13.

To continue, the calibration data set of rainfall and runoff sub-signals extracted by the selected mother wavelets (i.e., Haar anddb4) at different decomposition levels were then considered as in-put neurons to predict the runoff 1 day ahead (as output layer neu-ron) via an ANFIS model. Then the trained model was validated bythe verification data set. The results obtained for the daily timescale have been added up in Table 5.

Table 6Results of WANFIS model in monthly time scale.

Motherwavelettype

Decompositionlevel (i = j)

ANFISstructure


RMSE(m3/s)

R2 RMSE(m3/s)

R2

Case study (1): LighvanchaiHaar 2 gbellmf-2 0.028 0.88 0.036 0.86Haar 3 gbellmf-2 0.022 0.89 0.029 0.86Haar 4 gbellmf-2 0.024 0.91 0.029 0.89db4 2 gbellmf-2 0.025 0.90 0.038 0.86db4 3 gbellmf-2 0.023 0.92 0.026 0.88db4 4 gbellmf-2 0.018 0.93 0.020 0.92

Case study (2): AghchaiHaar 2 gbellmf-2 0.026 0.89 0.036 0.86Haar 3 gbellmf-2 0.024 0.86 0.031 0.84Haar 4 gbellmf-2 0.019 0.94 0.019 0.91db4 2 gbellmf-2 0.021 0.92 0.023 0.89db4 3 gbellmf-2 0.019 0.92 0.019 0.90db4 4 gbellmf-2 0.018 0.93 0.017 0.92

Note: Input variables are I(t) and Q(t) (i.e., Rainfall(t) and Runoff(t)) and Q(t + 1) isoutput.

Table 5Results of WANFIS model in daily time scale.

Motherwavelettype

Decompositionlevel (i = j)

ANFISstructure


RMSE(m3/s)

R2 RMSE(m3/s)

R2

Case study (1): LighvanchaiHaar 5 gbellmf-2 0.026 0.89 0.036 0.86Haar 6 gbellmf-2 0.021 0.91 0.027 0.88Haar 7 gbellmf-2 0.024 0.86 0.031 0.84Haar 8 gbellmf-2 0.020 0.93 0.023 0.91Haar 9 gbellmf-2 0.019 0.93 0.023 0.91db4 5 gbellmf-2 0.021 0.92 0.023 0.89db4 6 gbellmf-2 0.017 0.94 0.021 0.90db4 7 gbellmf-2 0.019 0.92 0.020 0.90db4 8 gbellmf-2 0.018 0.96 0.018 0.92db4 9 gbellmf-2 0.016 0.94 0.019 0.92

Case study (2): AghchaiHaar 5 gbellmf-2 0.027 0.88 0.037 0.86Haar 6 gbellmf-2 0.022 0.89 0.029 0.86Haar 7 gbellmf-2 0.029 0.84 0.036 0.82Haar 8 gbellmf-2 0.024 0.91 0.028 0.89Haar 9 gbellmf-2 0.024 0.92 0.027 0.89db4 5 gbellmf-2 0.025 0.90 0.028 0.87db4 6 gbellmf-2 0.023 0.92 0.025 0.88db4 7 gbellmf-2 0.021 0.92 0.023 0.90db4 8 gbellmf-2 0.020 0.95 0.019 0.92db4 9 gbellmf-2 0.018 0.94 0.020 0.91

Note: Input variables are I(t) and Q(t) (i.e., Rainfall(t) and Runoff(t)) and Q(t + 1) isoutput.

The presented hybrid wavelet-ANFIS methodology for dailyrainfall–runoff modeling was also used for monthly time seriesforecasting. However, according to the time scale resolution (i.e.,monthly), only decomposition levels 2, 3, and 4 were consideredin which the 24-monthly mode approximately represents the an-nual time scale and can be considered as the upper limit of thetime series decomposition level. The results of the monthly model-ing have been presented in Table 6.

It should be noted that in Tables 5 and 6 the rainfall and runoffdecomposition levels (i.e., i and j) can be substituted by differentvalues but considering the direct relation between rainfall and run-off amount, it is expected that both the rainfall and runoff time ser-ies will have the same seasonal levels. Hence, the decompositionlevels for the rainfall (i) and runoff (j) time series were consideredequal in the current study.

For example, in decomposition level of 8 (i = j = 8) 9 sub-signals(one approximation and eight details) will be at hand for each rain-fall and runoff time series. Therefore, 18 neurons will be requiredfor the input layer of the ANFIS in order to find the runoff value1 day ahead as the model output. By comparing the obtained re-sults (Table 5), it can be clearly seen that by increasing the decom-position level to 8, the model’s performance increases but thisincrease is not perceptible from level 8 to 9, therefore level 8 canbe considered as proper decomposition level for the data. The rea-son for this result may be related to the fact that decomposition le-vel 8 includes a 28-day mode which is nearly an annual mode andthis periodicity is a very important and dominant seasonal cycle ina hydrologic time series. However, in a comparison of decomposi-tion levels 8 and 9, although both levels contain a 28-day mode, thehigh decomposition level leads to a large number of parameters inthe complex non-linear relationship of the ANFIS model. Conse-quently, although this relationship may monitor and fit the calibra-tion data appropriately, each parameter produces an error in theforecasting data and net errors decrease the model’s efficiency atthe verification stage. Also the large amount of input data requiresthe amount of runtimes in the training network. Therefore, thedecomposition level must be determined according to the time ser-ies seasonality periods. This result is in contrast to some otherstudies which offer the following formula to determine the maxi-mum decomposition level (Wang and Ding, 2003):

L ¼ int½logðNÞ� ð24Þ

where L and N are the decomposition level and number of timeseries data respectively. This experimental equation was derivedfor fully autoregressive signals, only considering time series lengthwithout paying any attention to seasonal effects. The increase ofsub-signals may lead to essential deterioration in the networkwhich is usually reflected in network over fitting. Hence, to opti-mize the network architecture and training speed and reduce thenumber of input, it was tried to distinguish the dominant and mosteffective sub-signals in the decomposed rainfall and runoff timeseries. For this purpose, the different subsets of sub-signals wereselected as input to the WANFIS model. The results of this methodare presented in Tables 7 and 8 for daily and monthly modeling,respectively. The input structure column in the tables representsthe time series and sub-signals which were examined as input tothe ANFIS model. For instance, the term ‘‘I(t) and Q(t) in levels 1, 3and 5’’ indicate that the rainfall and runoff time series in decom-posed levels 1, 3 and 5 are considered as inputs. In addition tothe periodicity effect which is represented by the decomposedsub-signals, the autoregressive property of the time series can alsobe taken into account by considering the main time series as themodel input. Hence, for each specific input structure, a parallelstructure which also includes the main rainfall and runoff time ser-ies (I(t) and Q(t)) without any decomposition, in addition to thedecomposed sub-signals, were also examined as the model’s input.

Table 8Results of monthly time series modeling by WANFIS with different inputs (Mother wavelet is db4).



No. Input structure R2 RMSE R2 RMSE R2 RMSE R2 RMSE

I I(t) and Q(t) in levels 1 and 2 0.86 0.028 0.84 0.036 0.89 0.024 0.88 0.030

II I(t) and Q(t) in levels 1 and 2 0.90 0.023 0.88 0.029 0.90 0.021 0.89 0.025I(t) and Q(t) time series

III I(t) and Q(t) in levels 3 and 4 0.87 0.027 0.84 0.032 0.87 0.026 0.86 0.029

IV I(t) and Q(t) in levels 3 and 4 0.92 0.023 0.89 0.027 0.92 0.023 0.89 0.026I(t) and Q(t) time series

V I(t) and Q(t) in levels 1, 2, 3 and 4 0.90 0.023 0.88 0.025 0.87 0.027 0.84 0.032

VI I(t) and Q(t) in levels 1, 2, 3 and 4 0.96 0.021 0.93 0.023 0.94 0.021 0.93 0.023I(t) and Q(t) time series

Table 7Results of daily time series modeling by WANFIS with different inputs (Mother wavelet is db4).



No. Input structure R2 RMSE R2 RMSE R2 RMSE R2 RMSE

I I(t) and Q(t) in levels 1 0.89 0.026 0.86 0.036 0.88 0.027 0.86 0.037

II I(t) and Q(t) in levels 1 0.91 0.021 0.88 0.027 0.89 0.022 0.86 0.029I(t) and Q(t) time series

III I(t) and Q(t) in levels 1 and 3 0.86 0.024 0.84 0.297 0.84 0.029 0.82 0.036

IV I(t) and Q(t) in levels 1 and 3 0.93 0.020 0.91 0.023 0.91 0.024 0.89 0.028I(t) and Q(t) time series

V I(t) and Q(t) in levels 1, 3 and 5 0.92 0.021 0.89 0.023 0.90 0.025 0.87 0.028

VI I(t) and Q(t) in levels 1, 3 and 5 0.94 0.019 0.90 0.021 0.92 0.023 0.88 0.025I(t) and Q(t) time series

VII I(t) and Q(t) in levels 1, 3, 5 and 8 0.92 0.017 0.91 0.019 0.91 0.021 0.89 0.023

VIII I(t) and Q(t) in levels 1, 3, 5 and 8 0.94 0.017 0.92 0.019 0.93 0.020 0.92 0.020I(t) and Q(t) time series

XI I(t) and Q(t) in levels 1, 2, 3, 4, 5, 6, 7 and 8 0.92 0.019 0.91 0.019 0.90 0.023 0.89 0.023

X I(t) and Q(t) in levels 1, 2, 3, 4, 5, 6,7 and 8 0.94 0.017 0.91 0.020 0.92 0.021 0.89 0.024I(t) and Q(t) time series


As shown in Table 7, the optimal input structure in the dailyWANFIS model is the combination of model No. VIII. In this combi-nation, the rainfall and runoff time series in levels 1, 3, 5 and 8were imposed on the input layer. By considering the mathematicalconcepts of discrete wavelet transform, it can be concluded thatlevels 1, 3, 5 and 8 reflect the time series seasonality of 2 days(21-day mode), nearly a week (23-day mode), nearly a month (25-day mode) and nearly a year (28-day mode). Also, the optimal inputstructure in a monthly time scale (Table 8) is model no. VI whichthe monthly rainfall and runoff sub-time series in levels 1, 2, 3and 4 were considered. These levels can attribute the periodic pat-tern in the time series.

In the WANFIS model, when multi-level sub-signals are enteredinto the model as input neurons, the applied weights to them byANFIS will be different at different decomposition levels (timescales) (Fig. 5), so that high weights will be applied to the appropri-ate level of the signal. For example, in using level 5 as the decom-position level which yields six sub-signals for both rainfall andrunoff time series, Q(t+1) may be related more to Id5(t) than toId4(t) because Id5(t) (detail sub-signal of rainfall) is the short periodin the rainfall time series and has a significant role in predictingrunoff at t + 1 (Q(t+1)). Therefore, the network magnifies its weightas comparatively as the other sub-signals. Partal and Cigizoglu(2008), in a univariate model, only used those components which

had highly linear correlations with the main time series as inputsto the wavelet-ANN model. This may lead to missing some impor-tant components (details) in the modeling that have low linear cor-relation but may have a high non-linear contribution in the maintime series. However, in the presented WANFIS model, the ANFISclarifies the more dominant components by applying high weights.It was concluded that the selection of input components (sub-sig-nals) should be done according to the hydrological characteristicsof the process.

In addition to decomposition level which plays a significant rolein the modeling as formerly mentioned, the selection of motherwavelet type is another important task in WANFIS modeling. Thereare many jumps in the runoff time series because of the suddenstart and cessation of rainfall over the related watershed. There-fore, because of the structure of the Haar or db4 wavelets (Fig. 6)which are similar to the signal, they could capture the signal’s fea-tures, especially peak values, well and yielded comparatively highefficiency. Consequently it is strongly recommended that wavelettype be selected according to the main signal formation. The cali-bration and verification time series of level five, decomposed bythe db4 mother wavelet which was reconstructed via the trainedANFIS, are shown in Figs. 14 and 15 for the Lighvanchai andAghchai watersheds as examples. According to Fig. 14 in dailyhydrograph for the Lighvanchai, we can see some constant

Fig. 14. The results of WANFIS model for Lighvanchai watershed in daily time step (a) Computed and observed runoff (b) Detail (c) Scatter plot.

Fig. 15. The results of WANFIS model for Aghchai watershed in daily time step. (a) Computed and observed runoff (b) Detail (c) Scatter plot.


observed discharge values for some successive days in the timeseries in which some of them may be related to the measurementproblem and can be considered as the noise. Without a shadow of adoubt, if available, the utilization of a high quality data set in anydata driven method will lead to more reliable results.

8.4. Comparison of the models

Finally, in order to evaluate the efficiency of the proposed hy-brid models, the obtained results were also compared with the re-sults of SARIMAX, ANN, ANFIS and WANN (Nourani et al., 2009b)models. The comparison has been summarized in Table 9. It isnotable that the best structure of the SARIMAX model is SARIMAX(2, 1, 1)(1, 1, 1)12 I(t), obtained via a trial–error process.

The tabulated results in Table 9 indicate that, although the lin-ear SARIMAX model has the ability to capture the seasonal featuresof the process, it was unable to completely model the complexnon-linear process because of its linear inherence. However, whenANN is used to find the relationship among the residuals, its non-linear inherency helps the SANN model to detect and catch thenon-linear features of the phenomenon. The ad hoc ANN model,which only has autoregressive modeling ability (using the windowinput values of the previous time steps), performs much betterthan the SARIMAX model which demonstrates the existence andimportance of non-linearity and non-stationary behavior in therainfall–runoff process. However the ANN model is less efficientcompared to the hybrid models because it only considers short-term autoregressive features of the process without any attention

Table 9Comparison of different rainfall–runoff models.

Case study Lighvanchai Watershed Aghchai watershed

Modela Modeltype

Determinationcoefficient (R2)

Determinationcoefficient (R2)

Calibration Verification Calibration Verification

Daily modelingSARIMAX Linear 0.71 0.63 0.67 0.61ANN Non-

linear0.82 0.74 0.80 0.71

SANN Hybrid 0.90 0.86 0.82 0.76WANNb Hybrid 0.95 0.92 0.90 0.88ANFIS Hybrid 0.88 0.85 0.89 0.86WANFIS Hybrid 0.94 0.92 0.93 0.92

Monthly modelingSARIMAX Linear 0.67 0.56 0.63 0.54ANN Non-

linear0.77 0.69 0.78 0.70

SANN Hybrid 0.88 0.81 0.79 0.64WANNb Hybrid 0.90 0.87 0.91 0.88ANFIS Hybrid 0.89 0.84 0.89 0.86WANFIS Hybrid 0.96 0.93 0.94 0.93

a In this table the best result for each model has been presented.b The reference for WANN model is Nourani et al. (2009b).

Table 10The ability of different models in capturing peak values.

Case study Modeling timescale

Determination coefficient for peak values

(R2peak)

ANN ANFIS SANN WANN WANFIS

Lighvanchai Daily 0.41 0.60 0.64 0.82 0.85Monthly 0.53 0.79 0.79 0.91 0.96

Aghchai Daily 0.47 0.72 0.70 0.78 0.95Monthly 0.41 0.71 0.76 0.85 0.96

Note: In this table the best result for each model has been presented.

Table 11Comparison of different models in modeling peak values (in%).

Case Study Lighvanchai watershed Aghchai watershed

Model ANN WANN ANFIS WANFIS ANN WANN ANFIS WANFIS

Daily modelingANN – 50 32 52 – 40 34 50WANN – �27 4 – �9 18ANFIS – 30 – 24WANFIS – –

Monthly modelingANN – 42 32 45 – 52 42 57WANN – �20 5 – �17 10ANFIS – 18 – 25WANFIS – –


to long-term seasonality. Hence, by combining the ANN and wave-let concepts to the SARIMAX and ANN approaches, respectively,and constructing the hybrid models, the aforementioned weak-nesses are averted.

In a comparison of the two first hybrid models (i.e., WANN andSANN), the WANN is more efficient (Table 9). The reason for thismay be related to the wavelet used in the WANN model; thus, bydecomposing the main time series into multi-scale sub-signals,each sub-signal represents a separate seasonal scale and therefore,the multi-seasonality features of the time series can be handled.Whereas, in the SANN model only one seasonal scale can be takeninto account (for example annual, s = 12 or monthly s = 1); but ithas been shown that sometimes there is more than one seasonaland periodicity scale in a hydrological time series.

As whole, the results of ANFIS and WANN models are more sat-isfactory compared to the ANN model in terms of prediction accu-racy by considering uncertainty and multi-resolution, respectively(Table 9). This is consistent with the results of previous researches(e.g., Jothiprakash et al., 2009; Nayak et al., 2004; Nourani et al.,2009b) which showed that by employing fuzzy and wavelet con-cepts linked to the ANN framework, the uncertainty and seasonal-ity of the phenomena respectively, can be better handled. Theresults of Table 9 indicate that the WANFIS and WANN modelshave comparable accuracy. However, by comparing the results ofthe Lighvanchi and Aghchai watersheds, it is found out that the dif-ference between the efficiency criteria of the WANFIS and WANNmodels for the Aghchai watershed was more than for the Lighvan-chi watershed. Hence it can be concluded that the WANFIS modelperforms much better than the other models in a wild watershed(e.g., Aghchai) because of the unknown response and uncertainnature of the wild watershed. As discussed in Section 3 (study areasection), the evidence is sufficient to prove this claim that theAghchai watershed has a more uncertain and ambiguous hydrolog-ical behavior than the Lighvanchai watershed. Therefore, it was ex-pected that the WANFIS model, which contains the fuzzy theoryconcept might lead to more reliable results than the WANN modelfor a wild watershed such as Aghchai. This is in agreement with thepreviously obtained results of the authors in forecasting suspendedsediment load using ANFIS model (Rajaee et al., 2009).

Since the feasible estimation of the peak values is usually themost important factor in any flood mitigation program, another

key point when comparing different models is the capability ofthe models in estimating peak values. For this purpose, peak valueswere sampled by considering the threshold of the top 5% of thedata from the original runoff time series contractually. The perfor-mances of the various models for this modeling were evaluatedusing Eq. (23) and are presented in Table 10.

According to Table 10, it can be concluded that the seasonalmodels (i.e., SANN, WANN and WANFIS) are more efficient thanthe autoregressive models (i.e., ANN and ANFIS) in monitoringpeak values. It is evident that extreme or peak values in the rainfalland runoff time series, which occur in a periodic pattern, can be de-tected by the seasonal models accurately. This is obviously seen inthe Figs. 14 and 15 which show the appropriate estimation of thepeak runoff values by the WANFIS model on a daily time scale.Also, as depicted in Table 10, the determination coefficients forpeak values are more precise on a monthly time scale comparedto a daily time scale. Since in monthly scale modeling the autore-gressive characteristics of the time series are decreased by averag-ing the time series data over a month, therefore the seasonalpattern is highlighted as the main characteristic of the time serieswhich can be captured by wavelet analysis in term of sub-signals.

Among the seasonal models, the WANFIS model is more effi-cient in modeling extreme runoff values compared to the othermodels (Table 10). This result is reasonable because both theuncertainty and seasonality of the process could be taken into con-sideration by the WANFIS model, simultaneously. However, theWANFIS model has a costly structure but this is unavoidable whenyou consider it can catch all uncertain, autoregressive and multi-scale seasonality effects in a modeling process perfectly.

For the final evaluation, the two by two comparisons of themodels for daily and monthly modeling of extreme values havebeen presented in Table 11 percentile. For instance, by assemblingthe wavelet theory to the ANN framework, the efficiency ofmonthly modeling for the Lighvanchai watershed was improvedup to 42%. It is obvious that the performance of the WANFIS modelis far superior to the ANN, ANFIS and WANN models. Besides, it canbe deduced that although the data pre-processing process by thewavelet transform can improve the modeling performance in both


time scales, this improvement is more considerable in monthlymodeling. This result is reasonable because the seasonality patternin the monthly time scale is more highlighted in compared to thedaily time scale. In other words, the autoregressive feature is moresignificant in daily modeling, whereas the seasonality feature is thedominant factor in monthly modeling.

According to the obtained results for two distinct watershedsusing two different time scales data, it is clear that the proposedmodels are as case-sensitive and site-specific as any other blackbox model and, as highlighted in this study, it is recommended thatthe hybridization of the models should be done with completeknowledge of the hydrological behavior of the watershed at differ-ent spatio-temporal scales.

9. Concluding remarks

The data pre-processing technique warrants further investiga-tion. In fact it should be noted that in general, and in the Lighvanc-hai and Aghchai basins in particular, rainfall and runoff time seriesare characterized by high non-linearity, non-stationary and sea-sonality behavior. Neural network and SARIMAX models may sim-ply be unable to cope with these different features if pre-processing of the input and/or output data is not performed.

In this study the wavelet transform, ANN, SARIMAX and ANFISapproaches were combined in order to develop two hybrid blackbox models for the multivariate rainfall–runoff simulation of twowatersheds located in Iran.

In the first model, an ANN model was used to find the non-lin-ear relationship among the residuals of the fitted linear SARIMAXmodel on the runoff data. The model could capture both non-linearand seasonal features of the runoff time series using daily ormonthly rainfall values as exogenous input. However, only onetime scale seasonality can be handled in this hybrid model inwhich, in the current study, annual periodicity (s = 12) was thedominant seasonality. In the case of using noisy data in a non-lin-ear model, the data errors may be magnified and result in unreli-able output (Nourani et al., 2007). However, SANN uses non-linear ANN for modeling only the residuals which usually haveminor values and will not be magnified in passing through thenon-linear ANN model. Therefore, the SANN model may be a prom-ising choice for modeling a noisy time series.

In the second proposed hybrid model, the wavelet transform,which can capture the multi-scale features of a signal, was usedto decompose the Lighvanchai and Aghchai rainfall and runoff timeseries. The sub-signals were then used as input to the ANFIS modelto predict the runoff discharge one time step ahead. It is recom-mended that the decomposition level in modeling by WANFISshould be selected according to time series periodicity rather thansignal length.

Overall, these results provide evidence of the promising role ofcombining data clustering and discrete wavelet transforms in run-off forecasting, especially on a monthly time scale. Furthermore,the effect of wavelet transform type on the WANFIS model perfor-mance was investigated using different kinds of wavelet trans-forms. The model results showed the high merit of Haar and db4mother wavelets in comparison to the others (i.e., sym2, sym3,db2, db3 and coif1). In general, runoff time series peaks can beapproximated as a single-peaked event of varying duration. Forthis reason, single-peaked wavelets such as the Haar (as similaras a step function) may provide a good approximation of sharpevents contained in the runoff records.

These hybrid multi-variable models showed a great improve-ment in rainfall–runoff modeling and produced better forecaststhan either the SARIMAX or ANFIS model alone. By using theANN model in both hybrid models, the developed models had a

non-linear kernel so that they could simulate the non-linearbehavior of the phenomenon more accurately than other linearmodels. Moreover, the periodic and autoregressive patterns ofthe process could be simultaneously captured through multi-reso-lution wavelet analysis whereas ANN and ANFIS could only moni-tor the autoregressive pattern of the time series.

Generally, a fully autoregressive black box model such as ANN,ANFIS, and ARIMA, underestimates the peak values of a hydrolog-ical time series when these extreme values may have been createdbecause of the sudden imposition of extreme inputs on the hydro-logical system, such as in the case of a severe storm. Therefore,fully regressive models which use only current and some previoustime steps’ data as model inputs, cannot cope with the reasonabledetection of instantaneous jumps in the time series. However, sea-sonal models such as SANN, WANN and WANFIS which have long-term periodicity memory and utilize stored information of extremeevents which have occurred in the previous month, season, year oreven decade, can estimate peak values more accurately. It is worthnoting that employing wavelet analysis as a data pre-processingtool is more profitable in monthly modeling than in daily simula-tion because seasonal pattern is the most governed feature in arainfall–runoff time series, whereas the autoregressive pattern isthe main trait in the daily time series. Also, from the point of viewof uncertainty, the WANFIS model is more effective than other sea-sonal models in modeling wild watersheds and also in estimatingthe extreme values of the time series.

In order to complete the current study, it is recommended touse the presented methodology to forecast the runoff 2, 3, . . . daysor months ahead and also to model the rainfall–runoff process of awatershed by adding other hydrological time series and variables(e.g., temperature or/and evapotranspiration) to the input layerof the model. Furthermore, it may be instructive to apply the pro-posed models on other heterogeneous watersheds in order toinvestigate the overall effect of watershed climatic conditions onthe models’ performances.

Acknowledgements

The research was financially supported by a grant from Re-search Affairs of Univ. of Tabriz.

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Documents

Two Hybrid Artificial Intelligence Approaches for Modeling Rainfall–Runoff Process