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- 1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND (Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME TECHNOLOGY (IJCIET)ISSN 0976 – 6308 (Print)ISSN 0976 – 6316(Online)Volume 3, Issue 2, July- December (2012), pp. 415-425 IJCIET© IAEME: www.iaeme.com/ijciet.aspJournal Impact Factor (2012): 3.1861 (Calculated by GISI) © IAEMEwww.jifactor.com WAVELET-NEURAL NETWORK CONJUNCTION MODEL IN FLOW FORECASTING OF SUBHIMALAYAN RIVER BRAHMAPUTRA Khandekar Sachin Dadu1 and Paresh Chandra Deka2 1 Research Scholar, 2Associate Professor, Department of Applied Mechanics and Hydraulics, National Institute of Technology Karnataka, Surathkal, Mangalore- 575025, India E-mail: paresh_deka@sify.com, khandekarsd@yahoo.com ABSTRACT In this current study, the Discrete Wavelet transform was hybridized with ANN naming Wavelet Neural Network (WLNN) for river flow forecasting at selective stations such as Pandu and Pancharatna of international river Brahmaputra within India, upto 4 time steps lead time. The main time series of daily, weekly and monthly discharge data were decomposed to multiresolution time series using discrete wavelet transformations and were used as input of the ANN to forecast the river flow at different multistep lead time. It was shown how the proposed model, WLNN, that makes use of multiresolution time series as input, allows for more accurate and consistent predictions with respect to classical ANN models. The proposed wavelet model (WLNN) results shows that it is better forecasted and consistent than single ANN model because of using multiresolution time series data as inputs. Keywords: Artificial neural network; Discrete Wavelet transform; Subtropical; Time series forecasting; Brahmaputra River; Hybridization 1. INTRODUCTION In the last decade, wavelet transform has become a useful technique for analysing variations, periodicities, and trends in time series (Lu 2002, Xingang et.al, 2003; Coulibaly and Burn, 2004;Partal and Kucuk,2006).Smith et al(1998) applied discrete wavelet transform(DWT) to quantify streamflow variability and suggested that streamflow could be effectively classified into distinct hydroclimatic categories using DWT.The dynamical link between streamflow and dominant modes of climatic variability in the Northern Hemisphere was explored by Coulibaly and Burn(2004) using wavelet analysis.Labat (2005)reviewed the most recent wavelet applications in the earth science field and explained new wavelet analysis methods in the field of hydrology. 415
- 2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME Combination of wavelet transformation with neural network models in time seriesforecasting has been reported since few years in the different fields such as oceanengg,Earthquake,hydrology( Cannas et al.,2005; Rao et al.,2009; Shiri and Kisi ,2010;Nourani et al., 2011 ;Deka and Prahlada,2012; Wang et al., 2011). Wang and Ding (2003)used Wavelet-ANN combination in hydrology to predict hydrological time series. A hybridwavelet predictor-corrector model was developed by Zhou et.al, (2008) for prediction ofmonthly discharge time series and showed that the model has higher prediction accuracy thanARIMA and seasonal ARIMA.All these studies revealed that Wavelet Transform is apromising tool for precisely locating irregularly distributed multiscale features of climaticelements in space and times Wavelet analysis is multiresolution analysis in time and frequency domain and is theimportant derivative of the Fourier transform.The original signal is represented by differentresolution intervals using Discrete Wavelet Transform(DWT).In other words, the complexdischarge time series are decomposed into several simple time series using aDWT.Thus,some features of the subseries can be seen more clearly than the original signalseries.These decomposed time series may be given as inputs to ANN which can handle non-linearity efficiently,higher forecasting accuracy may be obtained. Forecasts are more accuratethan that obtained by original signals due to the fact that the features of the subseries areobvious. This is why the hybridization of wavelet transformation and neural network canperforms better than single ANN model. In this work, an attempt has been made to investigate the potential and applicability ofHybrid Model by combining Wavelet and ANN with objectives to address the abovementioned scenarios using time series data of different frequencies for multistep lead timeforecasting. It is expected that this approach can improve the low level model accuracies inlong range (>1 time steps) flow forecasting. For this purpose, wavelet neural network(WLNN) algorithm was introduced and employed to develop a river flow forecasting modelwhich has an ability to make forecasts up to 4timesteps leadtime using flow time seriesobserved data. The results of WLNN model were compared with the results obtained fromsingle ANN model.Also,the proposed WLNN model performance were evaluated to assessthe model efficiency in the higher lead times alongwith different decomposition levels.2. WAVELET THEORY A Wavelet transformation is a signal processing tool like Fourier transformation withthe ability of analysing both stationary as well as non stationary data, and to produce bothtime and frequency information with a higher(more than one) resolution, which is notavailable from the traditional transformation.Wavelet means small wave,whereas by contrast,sinus and cosinus are big waves(Percival andWalden,2000).A function Ψ(.) is defined as wavelet if fulfill (1), (2)Commonly, wavelet is functions that have characteristic in equation(1),which is if it isintegrated on (-∞,∞) the result is zero, and the integration of the quadrate of function Ψ (.)equals to 1 as written in equation (2).There are two functions in wavelet transform,i.e.scalefunction(father wavelet) and mother wavelet. These two functions give a function family thatcan be used for reconstructing a signal. Some wavelet families are Haar wavelet, which is the 416
- 3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEMEoldest and simplest wavelet, besides that there are Meyer wavelet, Daubechies wavelet,Mexican Hat wavelet, Coiflet wavelet, and Last Asymmetric(Daubechies,1992). Ψa,b(t) canbe obtained by translating and expanding Ψ (t): Ψ a,b(t)= , a ϵ R, b ϵ R, a ≠ 0 (3)Where ߰a,b(t) is successive wavelet; a is scale or frequency factor, b is time factor; R is thedomain of real number. The successive wavelet transform of finite energy signal or timeseries f(t) ϵ L2(R)is defined as Wf(a,b)= dt (4)Where (t) is the complex conjugate function of (t); Wf(a,b) is the wavelet coefficient underdifferent resolution levels(scale) and different time.In general, the time series are discrete, so the discrete form of eq.4 can be written as: Wf(a,b) = (5) Where N is the number of discrete time step.∆t is the sample time interval.(Zhou etal,2008)Here, Wf(a,b) is the output of the time series f(t) or f(k ∆t) through the unit impulse responsefilter.The characteristics of the original time series in time(b) and frequency domain (a) at thesame time can be reflected through this output.Frequency resolution of wavelet transform islow for small value of ‘a’ with high time resolution.Also,for large ‘a’ value,it is just opposite.There are many discrete wavelet transform algorithms, but Mallat algorithm (1989) has beenadopted due to simplicity and most efficient case for practical purposes for decompositionand reconstruction of time series..It can be expressed as follows: cj+1 = Qcj for j=0,1,2……J and dj+1 = Gcj for j =0,1,2…..,J (6)where Q and G are low pass filter and high pass filter respectively.If co represents the originaltime series X,then X can be decomposed to d1,d2,….dj and cj through equation(6) where J isthe scale number.cj and dj are the approximated signal and the detail signal of original timeseries under the resolution level 2-j,respectively.In flow time series,cj represents thedeterministic components like tendency,period and approximate period,etc.dj represents thestochastic components and the noise.The basic idea of multiscale decomposition is trendinfluences Low frequency(L) components that tend to be deterministic.Whereas,Highfrequency(H) component is still stochastic.The time series is decomposed into one comprising its trend(approximation) and onecomprising the high frequencies and the fast events(details)(Kisi,2009).The decompositionsignals can be reconstructed as follows:( Zhou et al,2008) Cj = (Q*)j cj and Dj = (Q*)j-1 G*.dj for j = 0,2,…..J (7) 417
- 4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEMEWhere Q* and G* are dual operators of Q and G respectively.The reconstruction ofdecomposition signals will increase the number of signals.The wavelet transform series{D1,D2,…DJ,CJ},obtained by reconstructing d1,d2,…dj and cj,has the same length with theoriginal time series X and X = D1+D2+…DJ+CJ.3. STUDY AREA The study area is located in the international river Brahmaputra main stream withinIndia. The two consecutive discharge gauging sites namely Pandu(u/s) and Pancharatna(d/s)are selected for the study as these two stations recorded heavy discharge flows in the main-stream of the Brahmaputra river causing frequent floods to the downstream area. The Brahmaputra originates in Tibet region in China is the fourth largest river in theworld in terms of average discharge at mouth, with a flow of 19,830 cumecs(Goswami,1985). The hydrologic regime of the river responds to the seasonal rhythm of themonsoons and to the freeze-thaw cycle of the Himalayan snow. The discharge is highlyfluctuating in nature. Discharge per unit drainage area in the Brahmaputra Basin River isamong the highest of major rivers of the world. The basin lies between latitudes 24013´ and31030´9 North and longitudes 820 and 96049´ East. The catchment area upto pandustation(u/s) is 500,000 sqkm and upto Pancharatna(d/s) is 532,000 sqkm. The location of thetwo discharge gauging stations namely Pandu and Pancharatna are shown in the figure 1below. Figure.1: Location of the study area4. METHODOLOGY Here, considering the dominance of persistence in the flow time series, future riverflow to be forecasted from the past/previous flow data. River flow upto previous four timesteps were taken as predictor variables.The input scenarios formed by various predictorconfigurations are;[ 1] Q(t) [2] Q(t),Q(t-1) [3] Q(t),Q(t-1),Q(t-2) [4] Q(t),Q(t-1),Q(t-2),Q(t-3).Where Q(t)—current discharge .Also Q(t-1),Q(t-2),Q(t-3) are onetime step,two time step andthree time step past discharge respectively. The predictand is Q (t+n) where n is the lead time.These input and output scenarios are same for both ANN and WLNN model.Here, wavelet 418
- 5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEMEand ANN techniques are used together as a combined method.While wavelet transform is employed todecompose discharge time series into their spectral bands, ANN is used as a predictive tool thatrelates predictand (output) and predictors (inputs). The original non-stationary time series were decomposed into a certain number of stationarytime series through discrete wavelet transform such as, periodic properties,non-linearity anddependence relationship of the original time series were separated.Hence, each wavelet transformseries has obvious regularities. Then the ANN model was used to simulate the wavelet transformseries in the form of approximations and details coefficients. Therefore, the prediction accuracy wasexpected to improve.4.1. Artificial neural network ANN is a flexible mathematical structure having an interconnected assembly of simpleprocessing elements or nodes, which emulates the function of neurons in the human brain. It possessesthe capability of representing the arbitrary complex non-linear relationship between the input andoutput of any system. Mathematically, an ANN can be treated as universal approximators having anability to learn from examples without the need of explicit physics. In this study, A single layer feed forward network with a back propagation learning algorithmhas been selected for the ANN model .Here, TRAIN LM (Levenberg-Marquardt) learning function,Tangent Sigmoid as transfer function has been chosen and the analysis was carried out for differentinput scenarios of previous time steps discharge data. The optimal structure of the ANN is selectedbased on mean square error during training. The ANN model implementation was carried out usingMATLAB routines.4.2 Wavelet Neural Network (WLNN) In the proposed (WLNN) model, the Discrete Wavelet Transformation discretizes theinput data (Q) in to number of sub signals in the form of approximations and details and henceforth,these sub signals has been used as input to ANN. The schematic diagram of proposed WLNN modelis shown in figure 2. The proposed Hybrid model which uses multiscale signals as input data maypresent more probable forecasting rather than a single pattern input. The objectives of WLNN model is to forecast multitime steps ahead discharge from previoustime steps discharge.Here,future discharge are taken as predictand and past discharges aspredictor.After decomposing the time series into several resolution levels,each level subseriespredictand data is estimated from its corresponding separated predictor level. The proposed WLNNmodel focused on improving the precision and prolonging the forecasting time period. Here, ANN part was constructed with appropriate sub-series belongs to different scales asgenerated by DWT.These new series consists of details and approximations were used as input toANN .In the proposed WLNN model,only input signals were decomposed into wavelet coefficients sothat ANN was exposed to large number of weights attached with higher input nodes during training.Hence,the higher adaptability can be achieved for input- output mapping. Figure.2: Schematic diagram of the proposed WLNN model 419
- 6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME4.3 Performance Indices The conventional performance evaluation such as correlation coefficient is seems tobe unsuitable for model evaluation (Legates and McCabe, 1999). However, Mean AbsolutePercentage Error and Mean Square Error can be used for better evaluation of modelperformance. In this study, following performance indices based on goodness of fit are used. 2 MSE = ∑ (X − Y ) MAPE = 1 N ∑ X −Y × 100 N , N i =1 XWhere, X=observed values, Y=predicted values, N = total number of values,4.4 Data division Daily, weekly average and monthly average discharge data for 20 years has beencollected for both the gauging stations of Pandu and Pancharatna are divided in training andtesting sets. Initial 70% of time series flowdata were used for training and remaining 30%time series data were used for testing. The analysis is carried out adopting Artificial NeuralNetwork using varying input scenarios. Later, hybrid model combination of Wavelet-ANNwas proposed to minimise the errors obtained in multistep lead time forecasting. The softwareused for analysis is MATLAB (2009) using ANN and Wavelet Toolboxes.5. RESULTS AND DISCUSSSION In this study, a number of ANN models has been developed and the best model(optimised structure) out of various input combinations were selected.The best ANN modeltesting results obtained for input three (3rd scenarios) with seven (7) neurons in the hiddenlayer based on various performances indices were presented in Table 1 and Table 2 for Pandustation and Pancharatna station respectively.It can be seen from the Table 1 and Table 2 thatMSE and MAPE values for ANN model are more than WLNN model using daily flowdata.Although the predictive performance of ANN model within acceptable accuracy such asMAPE is 4.34% for Pandu and 3.81% for Pancharatna but well below than the WLNNperformance.The mean squared error (MSE) also followed the similar trend with high errorfor ANN model. Table 1 Testing results at station Pandu(1day ahead)Decom WLNN ANNpositio DB-4 COIFLET-2 SYMHLET-4n Level MSE MAPE MSE MAPE MSE MAPE MSE MAPE 6 6 6 (x10 ) (%) (x10 ) (%) (x10 ) (%) (x106) (%) (cumec)2 (cumec)2 (cumec)2 (cumec)2 1 0.96 1.22 8.11 4.01 15.19 4.05 23.40 4.34 2 7.20 3.96 7.29 3.97 13.64 4.72 3 2.77 1.38 7.42 3.92 13.72 4.74 4 3.03 1.95 8.21 4.03 8.73 2.97 5 2.10 1.52 7.59 4.06 13.63 4.73This may be due to significant fluctuations of the data around mean value such as skewnessand standard deviation are high,where short term regression between data is minimised. 420
- 7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME.From the Table 3, it is clear that the ANN model doesnot perform well in longtermstreamflow forecasting.The results of weekly forecast are relatively weaker than those ofdaily forecasts and the monthly forecast are quite weak with higher MAPE and MSE. Table 2 Testing results at Pancharatna station (1day ahead) Decomp WLNN ANN osition DB-4 COIFLET-2 SYMHLET-4 Level MSE MAPE MSE MAPE MSE MAPE MSE MAPE (x106) (%) (x106) (%) (x106) (%) (x106) (%) (cumec) (cumec) (cumec) (cumec) 2 2 2 2 1 3.52 1.72 3.94 2.03 4.24 2.34 9.21 3.81 2 4.42 2.93 4.83 2.97 4.98 3.52 3 8.91 2.73 9.23 3.12 10.02 3.73 4 10.81 3.49 11.45 3.56 11.56 3.64 5 6.30 2.48 7.12 2.75 7.85 2.85 Also the best input combinations are not same for all prediction intervals.Inputcombination 3 is coming best for daily forecast and combination 2 is best for weekly andmonthly forecast.For multistep leadtime forecasting, again optimal combinations weredifferent which consists of more lagged discharge. In the second stage, for hybrid wavelet neural network (WLNN) model, pre-processeddischarge time series data were given to ANN model to improve the model accuracy byadopting proper selection of wavelet type and decomposition levels. For these objectives,Discrete Wavelet Transformation (DWT) was used and various type of wavelets such asDaubechies wavelet order-4 (DB-4), COIFLET-2, SYMHLET-4(Daubechies, 1992; Mallat,1998) were selected as a mother wavelet considering the shape similarity with time seriessignal. The selected mother wavelets are of exact reconstruction possibilities and arecompactly supported and Asymmetric in shape. Similar to ANN models, here also a number of WLNN models were developed usingdifferent input combinations (mentioned earlier) with different ANN architecture. The bestresults in terms of performance indices were obtained for third input scenarios (three inputs)for various decomposition levels and results are presented for all three type of mother waveletsuch as DB-4,COIFLET-2 and SYMHLET-4 in Table 1 and Table 2 for one day leadtime forboth the stations. In this work, the effects of various decomposition levels on model efficiency havealso investigated to optimize the result. The output result from the discrete wavelettransformation in the form of ‘approximations’ and ‘details’ sub signals at different levels arepresented in figure 3 for db-4.The mechanism inside the network was somewhat transparentin WLNN. When coefficients are used as inputs, as the number of input layers increasesaccordingly number of weights also increases. The analysis has been done for differentdecomposition levels from level 1 to 5 to obtain optimal results. In each case, as thedecomposition level increases, the number of input layers also increases and the network wastrained and tested accordingly. 421
- 8. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME Figure 3: DB-4 Sub signals after data decomposition through DWT at level-5 The results from the above model (WLNN) for different decomposition levels clearlyrevealed the better performance of the proposed model (WLNN) both in low as well as higherlead time compared to ANN ( Table 3 ), considering various performance indices. The basicWLNN model of decomposition level 1 (L-1) with DB-4 mother wavelet was performingbetter than best ANN model and other type of wavelets considering coefficient of efficiencyand least error criteria. Also, other WLNN improved models based on differentdecomposition levels (L-1,L-2,L-3, L-4, and L-5) performed better than ANN model.Also,theother type of wavelets are performing similar to DB-4 but well above ANN performance. For shorter lead times, performances of WLNN models were almost similar to ANNand observed no significant variations. But in the higher lead time forecast, significantvariations were observed among the performance of WLNN models. For low lead time withlow decomposition levels, the model is performing in a better way than in higher lead times. Again from the time series plot in figure 4 for one day leadtime prediction, it wasobserved that the ANN and WLNN model results were closely following the observed data 422
- 9. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEMEboth in low and high flow.The scatter diagram shows close agreements between WLNNmodel results and observed flow as shown in figure 5. As lead time increases, theperformances of ANN decreases drastically but, WLNN performance decreases gradually asthe variation of MAPE for different lead time forecasting was presented for ANN and WLNNin Table 3. Similarly for the Pancharatna station, WLNN model performance were similar toPandu station as the statistical behaviour were different but capturing the hidden knowledgeas presented in Table 1 and Table 2 in flow forecasting. Table 3 Testing results at both the stations (weekly and monthly data) StationModel 1 week lead 4 week lead 1 week lead 4 week lead type MSE MAPE MSE MAPE MSE MAPE MSE MAPE (x106) (%) (x106) (%) (x106) (%) (x106) (%) PANDU WLNN 5.24 3.66 6.78 4.22 5.76 7.14 7.11 9.27 ANN 6.76 3.93 11.23 4.97 8.21 14.21 12.34 21.79 PANCHA WLNN 5.42 3.57 6.87 4.12 5.67 5.45 6.17 8.27 ANN 7.67 3.89 11.32 5.79 8.12 7.21 12.53 22.97 Figure.4: Model performance in testing for 1day leadtime at Pandu Figure.5: Scatter plot of Observed and WLNN for 1 day leadtime at Pandu The main reason for this improvement is that the WLNN model can extract thebehaviour of discharge variation processes through decomposing the nonstationary timeseries of daily, weekly and monthly discharge into several stationary time series.These 423
- 10. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEMEstationary time series can exhibit the finer details of the discharge time series and alsoreducing the interference between the deterministic components and the stochasticcomponents. Hence the stability of the data variation has increased which results in improvedprediction accuracy.5.1 Effects of decomposition level and type of wavelet In the WLNN, the results obtained for different lead times has undergone differentdecomposition levels starting from 1 to 5 for various type of mother wavelets. In each leadtime analysis, there was an increasing trend in performance error from low decompositionlevels towards higher decomposition levels as presented results shown in Table 1and Table2.This may be because of higher decomposition levels lead to a large number of parameterswith more complex nonlinear relationships in the ANN . This follows the net errors resultsreducing the model performance. In this study,level-1 may be considered as best andappropriate decomposition level alongwith DB-4 mother wavelet for the given data sets and itwas considered as the best model among the WLNN models. Based on the results, it is noticed that the number of decomposition levels has littleimpact on the predictive performance of WLNN models. Since the random parts of originaltime series were mainly in the first resolution level, the prediction errors were also mainly inthe first resolution level. Thus the errors were not increased proportionately with theresolution number. Again for higher lead time forecast, higher model efficiency was obtainedat selected decomposition levels. These may be due to the effect of correlation of moresmoothened signals with flattened variability between the inputs and output.6. CONCLUSIONS In this study, a hybrid model of wavelet and ANN (WLNN) has been developed toforecast discharge for higher lead times such as daily, weekly and monthly at two gaugingstations of India. The accuracy of WLNN model has been investigated for forecasting riverdischarge in the present study by adopting various decomposition levels with respect todifferent type of mother wavelets. The WLNN models were developed by combining twotechniques such as ANN and DWT.The WLNN model results were also compared withsingle ANN model in the study. The WLNN and ANN model performance were tested byapplying to different input scenarios of past discharge data at the two gauging stations of theriver Brahmaputra in Assam within India. The accuracy of WLNN models was found to bemuch better than ANN model in modeling for all time steps flow discharge value.Theirregular and asymmetric shaped DB-4 wavelet provides better results than other models forall the decomposition levels showing superiority at the first level. The proposed hybridWLNN model plays an important role in improving the precision and prolonging theforecasting time period of hydrological time series.The appropriate selection of motherwavelet and decomposition levels also remains as partially conclusive as further analysis arerequired with more lengthy and more stations data.The suggested strategy can be adopted toother selective hydrological time series of similar statistical behaviour.ACKNOWLEDGEMENT The authors greatly acknowledged the support provided by Water Resourcesdepartment, Govt. of Assam, India for providing the necessary data for the analysis. 424
- 11. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308(Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEMEREFERENCES[1] Coulibaly,P.,and Burn,H.D.2004.“Wavelet analysis of variability in annual Canadianstreamflows.Water Resources Research, 40(3), 1-14.[2] Cannas, B., Fanni, A, Sias, G, Tronchi, S, Zedda, M.K. 2005. “River flow forecastingusing Neural Networks and Wavelet Analysis.” EUG (2005), European Geosciences Union,Vienna, Ausrtia, vol.7, 24-29.[3] Deka,P.C. and R.Prahlada.2012. “Discrete Wavelet Neural Network approach insignificant wave height forecasting for multistep lead time.”Ocean Engineeringjournal(ISSN-0029-8018)April, Vol-43pp..32-42[4] Daubechies,I.,1992. “Ten lectures on wavelets.Society for Industrial and AppliedMathematics”,Philadelphia,PA.[5] Goswami, D.C.1985.“Brahmaputra river, Assam, India; Physiographic, basin denudationand channel aggradation”.Water Resources Research, 21,959-978.[6] Lu,R.Y.2002. “Decomposition of interdecadal and interannual components for NorthChina rainfall in rainy season”,Chin.J.Atmosp.,26,611-624.[7] Labat,D.2005.“Recent advances in wavelet analysis:PartI.A review of concepts”,J.ofHydrology,314(1-4),289-311.[8] Legates, D. R.; McCabe, Jr., 1999. “Evaluating the use of goodness-of-fit measures inhydrologic and hydroclimatic model validation”. Water Resour. Res., 35 (1), 233-241.[9] Mallat S.G. 1989. “A theory for multiresolution signal decomposition: The waveletrepresentation.”IEEE Trans. Pattern Anal. Mach. Intell., 11(7), 674-693. [10] Mallat S G.1998. “A wavelet tour of signal processing”Academic,San Diego [11] Nourani, V.,Kisi,O.,Komasi.M. 2011. “Two hybrid artificial intelligence approaches formodeling rainfall – runoff process.” Journal of Hydrology, 402, 41-49. [12] Partal,T., and Kucuk,M.2006 “Flow forecasting for a Hawaii stream using rating curvesand neural networks”.J.Hydrology,317,63-80. [13] Percival,D.B and Walden,A.T.2000. “Wavelet methods for time series analysis”.Cambridge University Press,Cambridge. [14] Rao,Y.R. Satyaji and B. Krishna 2009, “Modelling Hydrological Time Series data usingWavelet Neural Network Analysis”, IAHS Publication -333,101-110. [15] Smith,L.C.,Turcotte,D.L.,Isacks,B.1998 “Streamflow characterization and featuredetection using a Discrete wavelet transform”.Hydrological Process.,12,233-249. [16] Shiri, J., and Kisi, O. 2010. “Short term and long term stream flow forecasting using awavelet and neuro-fuzzy conjunction model.” Journal of Hydrology, 394, 486-493. [17] Wang.W.,Hu.S.,Li. Y.2011. “Wavelet transform method for synthetic generation ofdaily streamflow”.Water Resources Management, 25:41-57. [18] Wang . W and Ding.J, 2003. “Wavelet network model and its application to theprediction of hydrology.” Nature and Science, 1(1) ,67-71. [19] Xingang,D.,Ping,W.,and Jifan,C.2003. “Multiscale characteristics of the rainy seasonrainfall and interdecadal decaying of summer monsoon in North China”. Chin.Sci.Bull.,48,2730-2734. [20] Zhou H.C,Peng Y,Liang G.H.,2008 “.The research of monthly discharge predictor-corrector model based on wavelet decomposition.”Water Resources Management 22:217-227. 425

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