Time Series Forecasting by Using Hybrid. Models for Monthly Streamflow Data

Transcription

1 Appled Mathematcal Sceces, Vol. 9, 215, o. 57, HIKARI Ltd, Tme Seres Forecastg by Usg Hybrd Models for Mothly Streamflow Data Sraj Muhammed Padha Departmet of Mathematcs Faculty of Scece, UverstTekolog Malaysa 8131, UTM Johor Bahru, Johor DarulTa azm, Malaysa A B Shabr Departmet of Mathematcs, Faculty of Scece, UverstTekolog Malaysa 8131, UTM Johor Bahru, Johor DarulTa azm, Malaysa Copyrght 215 Sraj Muhammed Padha ad A B Shabr. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract I ths study, ew hybrd models are developed by tegratg two etworks the dscrete wavelet trasform wth a artfcal eural etwork (WANN) model ad dscrete wavelet trasform wth least square support vector mache (WLSSVM) model. These model are used to measure for mothly stream flow forecastg. The mothly streamflow forecastg results are obtaed by applyg these models dvdually to forecast the rver flow data of the Idus Rver ad Neelum Rver Paksta. The root mea square error (RMSE), mea absolute error (MAE) ad the correlato (R) statstcs are used for evaluatg the accuracy of the WANN ad WLSSVM models. These results are frst compared wth each other ad the compared agast the results obtaed through the dvdual models called ANN ad LSSVM. The outcome of such comparso shows that WANN ad WLSSVM are more accurate ad effcet tha LSSVM, ANN. Ths study s dvded to three parts. I the frst part of the study, the mothly streamflow of the sad rver has bee forecasted usg these two models separately. The the secod part, the data obtaed s compared ad statstcally aalyzed. Later the thrd part, the results of ths aalytcal comparso show that WANN ad WLSSVM models are more precse tha the ANN ad LSSVM models the mothly streamflow forecastg.

2 281 Sraj Muhammed Padha ad A B Shabr Keywords: Artfcal Neural Network; Modelg; Least Square Support System; Dscrete Wavelet Trasform 1. Itroducto The accuracy of streamflow forecastg s a key factor for reservor operato ad water resource maagemet. However, streamflow s oe of the most complex ad dffcult elemets of the hydrologcal cycle due to the complexty of the atmospherc process. Paksta s maly a agrcultural coutry wth the world s largest cotguous rrgato system. Therefore, the ecoomy of Paksta heavly depeds o agrculture, hece rafall to keep ts rvers flowg. The Idus Rver ad Neelum Rver are the largest source of water dfferet provces of Paksta. It forms the backboe of agrculture ad food producto Paksta. Recet developmets tellget maches techology ad hgh speed data aalyss algorthms washed out covetoal methods of forecastg. Frst tellget model developed for stream flow predcto was based o artfcal eural etworks (ANN). Ulke mathematcal models that requre precse kowledge of all the cotrbutg varables, a traed artfcal eural etwork ca estmate process behavor eve wth complete formato. It s a prove fact that eural ets have a strog geeralzato ablty, whch meas that oce they have bee properly traed, they are able to provde accurate results eve for cases they have ever see before [13] [14]. Durg a mathematcal modelg process, two sets of formato are essetal for correct ad accurate results. Frst, there s the umber of parameters (varables) volved the process; secod, there s accurate kowledge of the terrelatoshps (o matter how complex ad olear) betwee dfferet parameters. Whe eural etworks are used to buld a fucto that estmates the process behavor, a complete kowledge of these two factors (all the parameters volved ad ther terrelatoshps) s ot a absolute ecessty. Usg ts massve coectvty, the eural etwork s able to costruct a hgh dmesoal space durg the trag. Ths s a dyamc ad adaptve process where a accurate approxmato of the fucto represetg the process behavor s made. Ths s doe by a cotuous chage ad adaptato of coecto stregth respose to put-output pars preseted to the etwork durg the trag. The developed eural etwork s put parameters ad desred output, where t wll raed usg some of the expermetal data such as teratvely adjust the weghts o the coectos betwee ts euros. Oce the trag s completed ad the etwork s stablzed, the complex tercoectos betwee the put,

3 Tme seres forecastg by usg hybrd models 2811 hdde, ad output euros represet a approxmato of the complex fucto betwee put ad output parameters. Three-layer back propagato eural etworks (NNs) was used to predct mothly flow ad compared wth other models. Most referred source for the bascs of ANN s the text wrtte by [34]. There are some dsadvatages of ANN. Although, ANN has advatages of accurate forecastg, ther performace s some specfc stuato s cosstet [2]. ANN also ofte suffers from local mma ad over fttg, ad the etwork structure of ths model s dffcult to determe ad t s usually determe by usg a tral ad error approach [23]. I addto, ANN model has a lmtato wth statoary data ad may ot be able to hadle o-statoary data f preprocessg of the put data s ot doe [1]. Recetly, the support vector mache (SVM) method, whch was suggested by [37], has used a rage of applcato, cludg hydrologcal modelg ad water resources process [2] [41]. Several studes have bee carred out usg SVM hydrologcal modelg such as streamflow forecastg [4] [2] [27], rafall ruoff modelg [25] [37] ad flood stage forecastg [41]. I the hydrology cotext, SVM has bee successfully appled to forecast the flood stage [31] [41] ad to forecast dscharge [27, 4]. Prevous studes have dcated that SVM s a effectve method for streamflow forecastg [2] [27] [4]. As a smplfcato of SVM [36], have proposed the use of the least squares support vector maches (LSSVM). LSSVM has bee used successfully varous areas of patter recogto ad regresso problems [12] [18]. I order to make t computatoally expesve wthout compromsg over relablty ad accuracy LSSVM were troduced [36], ths approach depeds o computg least square error by cosderg the put vectors ad the obtaed vectors(results). The cluso of ths step esures a hgher level of accuracy comparso to SVM. Besdes, LSSVM s foud sutable for solvg lear equatos whch s a much eeded characterstcs. Ulke SVM, LSSVM works uder the fluece of equalty costrats. Equalty costrats are strumetal reducg computatoal speed. Regardg covergece effcecy LSSVM provdes a apprecatve level of precso ad carres good covergece [4] [6] [16] [3]. LSSVM s stll cosdered developg stages ad s rarely used addressg the problems of hydrologcal modelg [18]. However, the model was successfully appled for solvg regresso, patter recogto problems [12] [18] ad for modelg ecologcal ad evrometal systems [41]. Techcally both of these methods,.e. SVM ad LSSVM are equally effectve. I terms of mplemetato LSSVM s comparatvely easer tha SVM. I terms of ther geeralzed performace both of them are comparable [39] ad are foud relable. Though LSSVM s foud good terms of stablty ad accuracy ad gee-

4 2812 Sraj Muhammed Padha ad A B Shabr rally t seems to be a good choce for trag data. The oly cocer s that for gettg geeralzed results, lke ay other AI based techque LSSVM also requres huge amout of data for trag purposes. Wth ths dea.e. trag t o large database would make t able to deal wth most of the varatos whch are lkely to be there the collected dataset. So there s strog eed to utlze the capablty of LSSVM by optmzg the put data. The preset research addresses ths ssue ad successfully optmzes the trag ad testg data. Recetly, wavelet theory has bee troduced the feld of hydrology, [24] [37] [35]. Wavelet aalyss has recetly bee detfed as a useful tool for descrbg both rafall ad ruoff tme seres [24] [25]. I ths, regard there has bee a sustaed exploso of terest wavelet may dverse felds of study such as scece ad egeerg. Durg the last couple of decades, wavelet trasform (WT) aalyss has become a deal tool studyg of a measured o-statoary tmes seres, through the hydrologcal process. A tal terest the study of wavelets was developed by [11] [5] [29]. Daubeches [5] employed the wavelets techque for sgal trasmsso applcatos the electrocs egeerg. Foufoula Georgou ad Kumar [1] used geophyscal applcatos. Subsequetly, [29] attempted to apply wavelet trasformato to daly rver dscharge records to quatfy stream flow varablty. The wavelet aalyss, whch s aalogous to Fourer aalyss s used to decomposes a sgal by lear flterg to compoets of varous frequeces ad the to recostruct t to varous frequecy resolutos. Rao ad Bopardkar [33] descrbed the decomposto of a sgal usg a Haar wavelet techque, whch s a very smple wavelet. The ma cotrbuto of ths paper s to propose a hybrd model of least square support vector mache wth wavelet trasform ad ovel hybrd tegratg wavelet trasform ad ANN model streamflow rver data. I order to acheve ths target, the daly streamflow data of Idus Rver ad Neelum Rver were decomposed to subseres at dfferet scale by Mallat algorthm. The, effectve subseres were summed together ad the used as puts to the ANN model for streamflow forecastg. Fally to evaluate the model ablty, the proposed model was compared wth dvdual model LSSVM ad ANN. 2 Methods ad Materals 2.1 Artfcal Neural Network (ANN) The applcato of ANNs has bee the topc of a large umber of papers that have appeared the recet lterature. Therefore, to avod duplcato, ths secto wll be lmted to oly ma cocepts. A three layer, feed forward ANN s show (Fgure 2.1). It has put, output, ad hdde mddle layers. Each euro a layer s coected to all the euros of the ext layer, ad the euros oe layer

5 Tme seres forecastg by usg hybrd models 2813 are ot coected amog themselves. All the odes wth a layer act sychroously. The data passg through the coectos from oe euro to aother are multpled by weghts that cotrol the stregth of a passg sgal. Whe these weghts are modfed, the data trasferred through the etwork chages; cosequetly, the etwork output also chages. The sgal emaatg from the output ode(s) s the etwork s soluto to the put problem. Fgure 2.1: Three-layer feed-forward artfcal eural etwork archtecture used for flow predcto Each euro multples every put by ts tercoecto weght, sums the product, ad the passes the sum through a trasfer fucto to produce ts result. Ths trasfer fucto s usually a steadly creasg S-shaped curve, called a sgmod fucto. The sgmod fucto s cotuous, dfferetable everywhere, ad mootocally creasg. The output yj (, 1), ad the put to the fucto ca vary. Uder ths threshold fucto, the output yj from the jth euro a layer s 1 y j f w j x w (1) j x 1 e Where wj = weght of the coecto jog the jth euro a layer wth the th euro the prevous layer; ad x = value of the th euro the prevous layer. After trag s complete, the ANN performace s valdated. Depedg o the outcome, ether the ANN has to be retraed or t ca be mplemeted for ts teded use. A ANN s better traed as more put data are used. The umber of put, output, ad hdde layer odes deped upo the problem beg studed. If the umber of odes the hdde layer s small, the etwork may ot have suffcet degrees of freedom to lear the process correctly If the umber s too hgh, the trag wll take a log tme ad the etwork may sometmes over ft the data [19] [22] [34]. A ew geerato learg system based o recet advaces statstcal learg theory. SVMs delver state-of-the-art performace real-world applcatos such as text categorzato, had-wrtte character recogto, mage

6 2814 Sraj Muhammed Padha ad A B Shabr classfcato, bo-sequeces aalyss, rver flow ad traffc flow forecast etc., ad are ow establshed as oe of the stadard tools for mache learg ad data mg. Support vector maches (SVMs) appeared the early etes as optmal marg classfers the cotext of Vapk s statstcal learg theory. Sce the SVMs have bee successfully appled to real-world data aalyss problems, ofte provdg mproved results compared wth other techques. 2.2 Least Square Support Vector Maches (LSSVM) Model LSSVM optmzes SVM by replacg complex quadratc programmg. It acheves ths by usg least squares loss fucto ad equalty costrats. For the purpose of uderstadg about the costructo of the model cosder a trag sample set represeted by ( x, y ) where x represets the put trag vector. Suppose that ths trag vector belogs to dmesoal space.e. R, so we ca wrte x R. Smlarly, suppose that y represets the output ad ths output ca be descrbed as, y R. SVM ca be descrbed wth the help of Equato (2) T y( x) w ( x) b (2) Where (x) s a fucto that esures the mappg of olear values to hgher dmesoal space. LSSVM formulates the regresso problem accordg to Equato (3) m R( w, e) 1 2 T w w 2 The regresso model show Equato (2) works uder the fluece of equalty costrats y( x) b e, 1, 2,..., (4) T w ( x ) It troduces Lagrage multpler for: 1 e 2 (3) 1 T 2 T L( w, b, e, ) w w e { w ( x ) b e y} (5) Where represets Lagrage multplers. Sce Equato (5) volves more tha oe varable so, for studyg the rate of chage partal dfferetato of Equato (6-9) s requred. Therefore dfferetatg Equato (5) wth respect to w, b, e ad ad equatg them equal to zero yelds the followg set of Equatos. L w ( x ) (6) w 1

7 Tme seres forecastg by usg hybrd models 2815 L b 1 (7) L e e (8) L T w ( x) be y, 1, 2,..., (9) Substtutg Equato (6-8) Equato (5) we get the value of w. Ths w s descrbed accordg to Equato (19) w 1 Puttg Equato (1) Equato (4) ( x ) e ( x ) (1) 1 T y( x) ( x ) ( x ) b K( x, x) b 1 1 Where, K(x, x), represets a kerel such that: T K( x, x) ( x) ( x) (12) The vector whch s a Lagrage Multpler ad the Based ca be computed by solvg a set of lear equatos show Equato (13) 1 ( x ) T T 1 b 1 ( x I j ) y Where, y y...;, 1 1;...; 1,...; (11) (13) 1 ; y 1;. Ths evetually costtutes LSSVM model whch s descrbed accordg to Equato (14). y( x) K( x, x) b (14) 1 The model show Equato (14) deals wth the lear system ad soluto ths lear system s provded by, b. The hgh dmesoal feature space s defed by a fucto. Ths fucto s geerally kow as kerel fucto ad s represeted by K( x, x) There are varous choces avalable for pckg up ths fucto. 2.3 Dscrete Wavelet Trasform (DWT) Wavelets are becomg a creasgly mportat tool tme seres forecastg. The basc objectve of wavelet trasformato s to aalyze the tme seres data both the tme ad frequecy doma by decomposg the orgal tme seres dfferet frequecy bads usg wavelet fuctos. Ulke the Fourer

8 2816 Sraj Muhammed Padha ad A B Shabr trasform, whch tme seres are aalyzed usg se ad cose fuctos, wavelet trasformatos provde useful decomposto of orgal tme seres by capturg useful formato o varous decomposto levels. Assumg a cotuous tme seres x (t), t [, ], a wavelet fucto ca be wrtte as 1 t ( t, s) (15) s s where t stads for tme, for the tme step whch the wdow fucto s terated, ad s [, ] for the wavelet scale. (t) called the mother wavelet ca be defed as ( t) dt. The cotuous wavelet trasform (CWT) s gve by 1 t W (, s) x( t) dt (16) s s where (t) stads for the complex cojugato of (t). W(, s) presets the sum over all tme of the tme seres multpled by scale ad shfted verso of wavelet fucto (t). The use of cotuous wavelet trasform for forecastg s ot practcally possble because calculatg wavelet coeffcet at every possble scale s tme cosumg ad t geerates a lot of data. Therefore, Dscrete Wavelet Trasformato (DWT) s preferred most of the forecastg problems because of ts smplcty ad ablty to compute wth less tme. The DWT volves choosg scales ad posto o powers of 2, so-called dyadc scales ad traslatos, the the aalyss wll be much more effcet as well as more accurate. The ma advatage of usg the DWT s ts robustess as t does ot clude ay potetally erroeous assumpto or parametrc testg procedure [24][9][31]. The DWT ca be defed as m t 1 t s m, m m s / 2 (17) s s Where, m ad are tegers that cotrol the scale ad tme, respectvely; s s a specfed, fxed dlato step greater tha 1; ad s the locato parameter, whch must be greater tha zero. The most commo choces for the parameters s = 2 ad = 1. For a dscrete tme seres x (t) where x (t) occurs at dscrete tme t, the DWT becomes Where, W m, N 1 m / 2 m m, 2 (2 t ) x( t) t W (18) s the wavelet coeffcet for the dscrete wavelet at scale m s 2 ad 2 m. I Eq. (18), x (t) s tme seres (t = 1, 2,, N-1), ad N s

9 Tme seres forecastg by usg hybrd models 2817 a teger to the power of 2 (N= 2 M ); s the tme traslato parameter, whch chages the rages < < 2M m, where 1 < m < M. Accordg to Mallat s theory [1], the orgal dscrete tme seres x(t) ca be decomposed to a seres of learty depedet approxmato ad detal sgals by usg the verse DWT. The verse DWT s gve by [11] [24] [31]. or a smple format as x M m1 M 2 m / 2 m ( t) T Wm, 2 (2 t ) (19) m1 t M x( t) A ( t) D ( t) (2) M m1 whch A M (t) s called approxmato sub-seres or resdual term at levels M ad D m (t) ( m = 1, 2,..., M) are detal sub-seres whch ca capture small features of terpretatoal value the data. 3 Applcato I ths study, the tme seres of mothly streamflow data of the Neelum ad Idus rver of Paksta are used. The Neelum Rver catchmet covers a area of km2 ad the Idus Rver catchmet covers 1165 km2. The frst set of data comprses of mothly streamflow data of Neelum Rver from Jauary 1983 to February 212 ad the secod data of set of streamflow data of Idus Rver Jauary 1983 to March 213. I the applcato, the frst 75% of the whole data set were used for trag the etwork to obta the parameters model. Aother 2% of the whole dataset was used for testg. The comprehesve assessmets of model performace atleast mea absolute error (MAE) measures, root mea square error (RMSE) ad correlato coeffcet (R). Evaluated the results of tme seres forecastg data to check the performace of all models for forecastg data ad trag data. 1 MAE y t yˆ t (21) RMSE t1 1 m y t yˆ t t1 (22) R 1 1 t1 t1 y y y yˆ 2 1 y y yˆ yˆ t t t t1 t t 2 (23) Where s the umber of observato, the observed rafall at all-tme t. y t stads for the forecastg rafall, yt s

10 2818 Sraj Muhammed Padha ad A B Shabr 4 Results ad Dscusso 4.1 Fttg ANN Model ad LSSVM Model to the Data I the frst part of the study, the ANN models are obtaed, for the tme seres forecastg. The archtecture of the ANN cossts of a umber of hdde layers ad the umber of euros the put layer, hdde layers ad output layer. The three layer ANN s chose for the curret study, whch comprse the put layer wth m odes, where m s daly rafall, the hdde layer wth h odes (euros) ad the output layer wth oe ode. The hyperbolc taget sgmod trasfer fucto the hdde layer ad the output layer. The lear fuctos from the hdde layer to a output layer are used for forecastg mothly rver flow tme seres. The hdde layer plays mportat roles may successful applcatos of ANN. It has bee prove that oly oe hdde layer s suffcet for ANN to approxmate ay complex olear fucto wth ay desred accuracy. I the case of the popular oe hdde layer etworks, several practcal umbers of euros the hdde layer were detfed for better forecastg accuracy. The optmal umber of odes the hdde layer was detfed usg several practcal gudeles. Berry ad Loff [3] clamed that the umber of hdde odes should ever be more tha 2I, where I s the umber of puts. Hecht-Nelse [14] clamed that the umber of hdde euro s equal to 2I + 1. However, as far as the umber of hdde euros s cocered, there s curretly o theory to determe how may odes the hdde layer are optmal. I the preset study, the umber of hdde odes was progressvely creased from 1 to 2I + 1. A program code cludg the wavelet toolbox was wrtte MATLAB laguage for the developmet of the ANN model. The optmal complexty of ANN model, that s, the umber of put ad hdde odes, was determed by a tral-ad-error approach. (Table 2) shows the best performace results of dfferet ANN wth dfferet umbers of hdde euros from 1 to 2I + 1 for both data seres. The trag set s used to estmate parameters for ay specfc ANN archtecture. The testg set s the used to select the best ANN amog all umbers of hdde euros cosdered. For trag ad forecastg perod, obta the best results for MAE, RMSE ad R. Sx models (M1 M6) havg varous put structures are traed ad test by ANN models. The etwork was traed for 5 epochs usg the back-propagato algorthm wth a learg rate of.1 ad a mometum coeffcet of.9. (Table 4.1) lsts model performace evaluato results of the M1 M6 models.

11 Tme seres forecastg by usg hybrd models 2819 Table 4.1: The model structures for forecastg streamflow Iput Orgal streamflow data 1 2 y t 1 y y DWT of streamflow data DW t 1 DW t DW, t 1 t 2 1, t 2 3 y t 1, yt 2, yt 3 1, 2, 3 4 y t 1, yt 2, yt 3, yt 4 1, 2, 3, 4 5 y t 1, yt 2, yt 3, yt 4, yt 5 1, 2, 3, 4, 5 6 y t 1, yt 2, yt 3, yt 4, yt 5, yt 6 1, 2, 3, 4, 5, 6 As show as (Table 4.2) the performace of ANN varyg accordace wth the umber of euros the hdde layer. It ca be see that the trag data of Neelum Rver, the Iput Model 5 wth the hdde euro 8 obtaed the best results for RMSE ad R statstcs of.747 ad.9233, respectvely for forecastg data the put model 5 wth 1 umber of hdde euro had the best results.1467 ad For Idus Rver, the put model 2 wth the hdde euro 4 obtaed the best results for RMSE ad R statstcs of.213 ad.8769, respectvely for forecastg data the put mode 2 wth 4 umber of hdde euro had the best results.922 ad Table 4.2: Trag ad Testg Performace of dfferet etwork archtecture of ANN model Data Neelum Idus Network Archtecture Trag Testg Iput Hdde MSE MAE R MSE MAE R I the secod part, the LSSVM model s tested. I ths study the same puts structures of the datasets whch s M1 to M6 were used. I order to obta the optmal model parameters of the LSSVM, a grd search algorthm ad cross-valdato method were employed. May works o the use of the LSSVM

12 282 Sraj Muhammed Padha ad A B Shabr tme seres modelg ad forecastg have demostrated favorable performaces of the RBF [41][28]. Therefore, RBF s used as the kerel fucto for streamflow forecastg ths study. The LSSVM model used here has two parameters (, 2 ) to be determed. The grd search method s a commo method whch was appled to calbrate these parameters more effectvely ad systematcally to overcome the potetal shortcomgs of the trals ad error method. It s a straghtforward ad exhaustve method to search parameters. I ths study, a grd search of (, 2 ) wth the rage 1 to 1 ad 2 the rage.1 to 1. was coducted to fd the optmal parameters. I order to avod the dager of over fttg, the cross-valdato scheme s used to calbrate the parameters. For each hyper parameter par (, 2 ) the search space, 1-fold cross valdato o the trag set was performed to predct the predcto error. The best ft model structure for each model s determed accordg to the crtera of the performace evaluato. As show (Table 4.3) the performace results obtaed the trag ad testg perod of the regular LSSVM approach (.e. those usg orgal data). For the trag ad testg phase Neelum Rver, the best values of the MSE (.21), MAE (.281) ad R (.7912) were obtaed usg put model 5. I the put model 6 had the smallest MSE (.279) ad MAE (.111) whereas t had the hghest value of the R (.847). For Idus Rver trag ad testg phase, the best value of MSE (.4) ad MAE (.95) ad R (.971) were obtaed usg put model 2, whereas the for the testg phase the best value of MSE (.85), MAE (.267) ad R (.8968) were obtaed usg put model 3. Table 4.3: Trag ad testg performace dcates of LSSVM Model Data Neelum Idus Trag Testg Iput MSE MAE R MSE MAE R Fttg Hybrd Models Wavelets-ANN Model ad Wavelet-LSSVM Model to the Data Two hybrd models Wavelet-ANN (WANN) model ad Wavelet-LSSVM

13 Tme seres forecastg by usg hybrd models 2821 (WLSSVM) model are obtaed by combg two methods, dscrete trasform (DWT) ad ANN model ad dscrete trasform (DWT) ad LSSVM. I WANN ad WLSSVM, the orgal tme seres was decomposed to a certa umber of sub-tme seres compoets whch were etered ANN ad LSSVM order to mprove the model accuracy. I ths study, the Deubeches wavelet, oe of the most wdely used wavelet famles, s chose as the wavelet fucto to decompose the orgal seres [11] [24] [31] [32]. The observed seres was decomposed to a umber of wavelet compoets, depedg o the selected decomposto levels. Decdg the optmal decomposto level of the tme seres data wavelet aalyss plays a mportat role preservg the formato ad reducg the dstorto of the datasets. However, there s o exstg theory to tell how may decomposto levels are eeded for ay tme seres. To select the umber of decomposto levels, the followg formula s used to determe the decomposto level [24] [32] M = log (). Where, s legth of the tme seres ad M s decomposto level. I ths study, = 35 ad = 483 mothly data are used for Neelum ad Idus, respectvely, whch approxmately gves M = 3 decomposto levels. Three decomposto levels are employed ths study, the same as studes employed by [3]. The observed tme seres of dscharge flow data was decomposed at 3 decomposto levels (2 4 8 moths). The effectveess of wavelet compoets s determed usg the correlato betwee the observed streamflow data ad the wavelet coeffcets of dfferet decomposto levels. (Table 4.4) shows the correlatos betwee each wavelet compoet tme seres ad orgal mothly stream flow data. It s observed that the D1 compoet shows low correlatos. The correlato betwee the wavelet compoet D2 ad D3 of the mothly stream flow ad the observed mothly stream flow data show sgfcatly hgher correlatos compared to the D1 compoets. Afterward, the sgfcat wavelet compoets D2, D3 ad approxmato (A3) compoet were added to each other to costtute the ew seres. For the WANN model ad WLSSVM model, the ew seres s used as puts to the ANN ad LSSVM model. (Fgure 4.1) ad (Fgure 4.2) shows the orgal streamflow data tme ad ther Ds, that s the tme seres of 2-moth mode (D1), 4-moth mode (D2), 8-moth mode (D3), approxmate mode (A3), ad the combatos of effectve detals ad approxmato compoets mode ( A2 + D2 + D3). Sx dfferet combatos of the ew seres put data (Table 4.1) s used for forecastg as the prevous applcato. A program code cludg wavelet toolbox was wrtte MATLAB laguage for the developmet of ANN ad LSSVM. The forecastg performaces of the Wavelet-ANN (WANN) ad Wavelet-LSSVM (WLSSVM)

14 2822 Sraj Muhammed Padha ad A B Shabr models are preseted (Table 4.5), ad (Table 4.6) respectvely, terms of MSE, MAE ad R trag ad testg perods. Table 4.5 shows that WANN model has a sgfcat effect o streamflow forecast. As see from (Table 4.5), for the Neelum rver data, the put model 5 s has the smallest MSE (.7,.11) ad MAE (.172,.247) ad the hghest R (.988,.9735) the trag ad testg phase. The best archtecture accordg MSE, MAE ad R crtero for trag data ad testg data has 6 put layer euros, 4 hdde layer euros, ad 1 output layer euro (6-4-1). For the Idus rver data the best archtecture accordg to MSE value are (.7,.15), MAE value are (.152,.29) ad R value are (.9871,.9597) crtero for trag data ad testg data s (5-1-1). As see the (Table 4.6), the WLSSVM models are evaluated based o ther performaces the trag data ad testg data. For the trag phase of Neelum rver, the best value of the MSE (.6), MAE (.168) ad R (.9919) statstcs for put data model 6. However, for the testg phase, the best MSE (.53), MAE (.52) ad R (.975) were obtaed for the put combato model 5. I the other had, for the Idus rver, the put model 6 obtaed lowest value of the MSE (.) ad MAE (.32) ad the hghest R (.9926) the trag phase. However, for the testg phase, the best MSE (.2) ad MAE (.178) ad R (.9398) was obtaed for the put combato model 2. Table 4.4: Correlato coeffcets betwee each of sub-tme seres for streamflow data Data Dscrete Wavelet Compoets D t-1/q t D t-2/q t D t-3/q t D t-4/q t D t-5/q t D t-6/q t Mea Absolute Correlato D Neelum D D A D Idus D D A

15 Tme seres forecastg by usg hybrd models 2823 A3-approxmato Moths D Moths D Moths D Moths Fgure 4.1: Decomposed wavelets sub-seres compoets (Ds) of streamflow data of Neelum Rver.1 A3-approxmato Moths D Moths D Moths D Moths Fgure 4.2: Decomposed wavelets sub-seres compoets (Ds) of streamflow data of Idus Rver

16 2824 Sraj Muhammed Padha ad A B Shabr Table 4.5: Trag ad testg performace of dfferet etwork archtecture of WANN model Data Neelum Idus Network Archtecture Trag Testg Iput Hdde MSE MAE R MSE MAE R Table 4.6: Trag ad testg performace dcates of WLSSVM model Data Neelum Idus Trag Testg Iput MSE MAE R MSE MAE R Comparso of forecastg Models Fally, order to evaluate the effcecy of the proposed hybrd models, the obtaed results were also compared wth the results of ANN, LSSVM, WANN ad WLSSVM models usg the same data. The compresso has bee summarzed the (Table 5.1). (Table 5.1) shows that the hybrd models WANN ad WLSSVM has good performace durg the testg phase, ad the outperform sgle model ANN ad LSSVM term of all the stadard statstcal measures. It s observed that the proposed model yelds better result tha the other models for both streamflow data. Ths result shows that the ew put seres from dscrete wavelet trasforms have sgfcat extremely postve effect o ANN ad LSSVM model results.

17 Tme seres forecastg by usg hybrd models 2825 Table 5.1: The performace results ANN, LSSVM, WANN ad WLSSVM Approach durg testg perod Data Model MSE MAE R Neelum Idus ANN LSSVM WANN WLSSVM ANN LSSVM WANN WLSSVM Cocluso I ths study, two ew methods based o the WANN ad WLSSVM was developed by combg the dscrete wavelet trasforms (DWT) ad ANN model ad DWT ad LSSVM model for forecastg streamflows. The mothly streamflow tme seres was decomposed at dfferet decomposto levels by DWT. Each of the decompostos carred most of the formato ad plays dstct role orgal tme seres. The correlato coeffcets betwee each of sub-seres ad orgal streamflow seres were used for the selecto of the ANN model ad LSSVM model puts ad for the determato of the effectve wavelet compoets o streamflow. The mothly streamflow tme seres data was decomposed at 3 decomposto levels (2 4 8 moths). The sum of effectve detals ad the approxmato compoet were used as puts to the LSSVM model. The WANN ad WLSSVM models were traed ad tested by applyg dfferet put combatos of mothly streamflow data of Neelum Rver ad Idus Rver of Paksta. The, ANN ad LSSVM models are costructed wth ew seres as puts ad orgal streamflow tme seres as output. The performace of the proposed WANN model ad WLSSVM model was compared to the regular ANN ad LSSVM model for mothly streamflow forecastg. Comparso results dcated that the WANN model ad WLSSVM model were substatally more accurate tha ANN ad LSSVM models. The study cocludes that the forecastg abltes of the ANN model ad LSSVM model are foud to be mproved whe the wavelet trasformato techque s adopted for the data pre-processg. The decomposed perodc compoets obtaed from the DWT techque are foud to be most effectve yeldg accurate forecast whe used as puts the ANN model ad LSSVM model. Refereces [1] J. Adamowsk ad K. Su, Developmet of a coupled wavelet trasform ad eural etwork method for flow forecastg of o-pereal rvers sem-ard watersheds, Joural of Hydrology, vol. 39 (21), o. 1-2, pp

18 2826 Sraj Muhammed Padha ad A B Shabr [2] Asefa, T., Kemblowsk, M., McKee, M., ad Khall, A., Mult-tme scale stream flow predctos: The support vector maches approach, J. Hydrol., 318 (1 4), (26) [3] M. J. A. Berry ad G. S. Loff, Data Mg Techques: For Marketg, Sales, ad Customer Support, Joh Wley & Sos, New York NY, USA, (1997). [4] Bose, G. E. P., ad Jeks, G. M. Tme seres aalyss forecastg ad cotrol, Holde Day, Sa Fracsco, (197). [5] I. Daubeches, Orthogoal bases of compactly supported wavelets, Commu. Pure Appled Math., 41 (1988): [6] Delleur, J. W., Tao, P. C., ad Kavvas, M., A evaluato of the practcalty ad complexty of some rafall ad ruoff tme seres model. Water resour. res., 12(5) (1976), [7] Dbke, Y. B., Velckov, S., Solomate, D. P., ad Abbott, M. B., Model ducto wth support vector maches: troducto ad applcatos, ASCE J. Comput. Cvl Eg., 15(3) (21), [8] Elshorbagy, A., Corzo, G., Srvasulu, S., ad Solomate, D. P. Expermetal vestgato of the predctve capabltes of data drve modelg techques hydrology Part 2: Applcato, Hydrol. Earth Syst. Sc. Dscuss., 6 (29), [9] M. Frat, Comparso of Artfcal Itellgece Techques for rver flow forecastg, Hydrol. Earth Syst. Sc., 12, (28). [1] Foufoula-Georgou & P. E Kumar, Wavelets Geophyscs, ed. E. Foufoula-Georgou & P. Kumar, (Sa Dego ad Lodo: Academc), (1994). [11] A. Grossma, ad J. Morlet, Decomposto of Harley fuctos to square tegral wavelets of costat shape, SIAM J. Math.Aal. 15, (1984) pp [12] Habay, D. A expert system based o least square support vector maches for dagoss of valvular heart dsease, Expert Syst. Appl., 36(4) (29),

19 Tme seres forecastg by usg hybrd models 2827 [13] Hayk, S. Neural etwork, a comprehesve foudato, IEEE, New York, (1994). [14] Hecht-Nelse, R. Neurocomputg, Addso-Wesley, New York, (1991). [15] R. Hecht-Nelse, Neurocomputg, Addso-Wesley, Readg, Mass, USA, (199). [16] Hurst, H. E. Log term storage capacty of reservors. Tras. ASCE, 116 (1961), [17] S. Ismal, R. Samsudd, ad A. Shabr, A hybrd model of self orgazg maps ad least square support vector mache. (21), Hydrol. Earth Syst. Sc. Dscuss., Volume 7, [18] Kag, Y. W., L, J., Cao, G. Y., Tu, H. Y., L, J., ad Yag, J., Dyamc temperature modelg of a SOFC usg least square support vector maches, J. Power Sources, 179 (28), [19] Karuadh, N., Greey, J., Whtley, D., Bovee, K., Neural etworks for rver flow predcto. J. Comp. Cv. Egrg. 8 (2) (1994), [2] M. Khashe ad M. Bjar, A ovel hybrdzato of artfcal eural etworks ad ARIMA models for tme seres forecastg, Appled Soft Computg Joural, vol. 11, o. 2, pp. (211), [21] Khashe, M. ad Bjar, M. A artfcal eural etwork (p,d,q) model for tme seres forecastg, Expert Syst. Appl., 37 (21), [22] Ks, O., Rver flow modelg usg artfcal eural etworks, (24), J. Hydrol. Eg., 9(1), [23] Ks, O., Neural etwork ad wavelet cojucto model for modellg mothly level fluctuatos Turkey, Hydrologcal Processes, vol. 23, o. 14 (29), pp [24] Ks, O., Wavelet regresso model for short-term streamflow forecastg. J. Hydrol., 389 (21): [25] B. Krsha, Y. R. Satyaj Rao, P.C. Nayak, Tme Seres Modelg of Rver

20 2828 Sraj Muhammed Padha ad A B Shabr Flow Usg Wavelet Neutral Networks, Joural of Water Resources ad Protecto, 3 (211), [26] D. Labat, R. Ababou ad A. Mag, Rafall-Ruoff Relatos for Karastc Sprgs: Part II. Cotuous Wavelet ad Dscrete Orthogoal Multresoluto Aalyss, Joural of Hydrology, Vol. 238, No.3 4 (2), pp [27] L, J. Y., Cheg, C. T., ad Chau, K. W. Usg support vector maches for log-term dscharge predcto, Hydrology Sc. J., 51 (4) (26), [28] Lu, L. ad Wag, W., Exchage rates forecastg wth least squares support vector maches, Iteratoal Coferece o Computer Scece ad Software Egeerg, (28), [29] S. G. Mallat, A theory for mult resoluto sgal decomposto: The wavelet represetato, IEEE Trasactos o Patter Aalyss ad Mache Itellgece, 11(7) (1998): [3] Matalas, N. C., Mathematcal assessmet of symmetrc hydrology. Water Resour. Press, 3(4) (1967), [31] PY. Ma, A fresh egeerg approach for the forecast of facal dex volatlty ad hedgg strateges, PhD thess, Quebec Uversty, Motreal, Caada, (26). [32] S. Padha ad A. Shabr, Tme Seres Forecastg Usg Wavelet-Least Squares Support Vector Maches ad Wavelet Regresso Models for Mothly Stream Flow Data," Ope Joural of Statstcs, Vol. 3 No. 3 (213), pp [33] R. M Rao, ad A.S. Bopardkar, Wavelet Trasforms: Itroducto to Theory ad Applcatos, Addso Wesley Logma, Ic., (1998), 31 pp. [34] Rumelhart, D. E., Hto, G. E., ad Wllams, R. J., Learg teral represetato by error propagato, Parallel dstrbuted processg. Volume 1(1986): Foudatos, eds., MIT Press, Cambrdge, Mass. [35] LC. Smth, DL. Turcotte, B. Isacks, Streamflow characterzato ad feature detecto usg a dscrete wavelet trasform, Hydrologcal processes 12(1998):

21 Tme seres forecastg by usg hybrd models [36] Suykes, J. A. K., Vadewalle, J. Least squares support vector mache classfers. Neural Processg Letters, Volume 3 (1999), [37] Vapk, V. The ature of Statstcal Learg Theory, Sprger Verlag, Berl, (1995). [38] D. Wag ad J. Dg, Wavelet etwork Model ad Its Applcato to the Predcto of Hydrology, Nature ad Scece, Vol. 1 (23), pp [39] Wag, H. ad Hu, D. Comparso of SVM ad LS-SVM for Regresso, IEEE, (25), [4] Wag, W. C., Chau, K. W., Cheg, C. T., ad Qu, L., A Comparso of Performace of Several Artfcal Itellgece Methods for Forecastg Mothly Dscharge Tme Seres, J. Hydrol., 374 (29), [41] Yu, P. S., Che, S. T., ad Chag, I. F., Support vector regresso for real-tme flood stage forecastg, J. Hydrol., 328 (3 4) (26), Receved: March 9, 215; Publshed: Aprl 12, 215