Wavelet Network Model and Its Application to the Prediction of Hydrology

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1 Wvelet Network Model nd Its Appliction to the rediction of Hydrology Wensheng Wng, Jing Ding Deprtment of Hydrology nd Wter esources, Hydrulic School of Sichun University, Chengdu, Sichun 665, Chin, Abstrct: Bsed on the multi-time scle nd the nonliner chrcteristics of the time series, new hybrid model between wvelet nlysis nd rtificil neurl network (ANN): wvelet network model, hs been suggested. The present model bsorbs some merits of wvelet trnsform nd rtificil neurl network. Cse studies, the short nd long term prediction of hydrologicl time series, hve been reserched. The comprison results reveled tht the suggested model could increse the ccurcy nd prolong the length time of prediction. The wvelet network model is stisfied. [Nture nd Science 23;():67-7]. Keywords: wvelet nlysis; rtificil neurl network; wvelet network model; prediction of hydrology. Introduction The ccurcy prediction of hydrology nd wter resource cn give importnt informtion for the city plnning, lnd use, the design of civil project nd wter resource mngement. Hydrology system is influenced by mny fctors, such s wether, lnd with vegetl cover, infiltrtion, evportion nd trnspirtion, so it includes the good del of stochstic dependent component, multi-time scle nd highly nonliner chrcteristics. In generl, the hydrology system is predicted with regressive nlysis, stochstic theory models (Ding, 988) nd Grey model method (Deng, 992). In recent yers, rtificil neurl network (ANN), fuzzy theory nd chos theory hve been widely pplied in the sphere of hydrology nd wter resource. The studies hve demonstrted these pproches re not very stisfied in precision becuse of only considering some spects of its property. In order to rise the precision nd lengthen the time, the hybrid model bsed on some methods should be probed. In this pper, new hybrid model: wvelet network model, which is combined with wvelet nlysis nd ANN, hs been proposed. A wvelet network model mkes use of the merits of wvelet nlysis nd ANN, so it hs excellent performnce in simultion nd forecst. Some cse studies re presented (in section 5) in which wvelet network model hs been developed using the suggested methodology (in section 4) to forecst hydrologicl time series in Chin. The results show the technique nd the model re fesible. The conclusions of the study re given in section Study eview Wvelet nlysis hs become reserch hot point. Wvelet nlysis hs good time nd frequency multi-resolution, nd cn effectively dignose signl s min frequency component nd bstrct locl informtion of the time series. It hs huge dvnces in signl processing, imge compress nd encoding, tongue encoding, mode identifiction nd nonliner science fields. The reserches nd pplictions of wvelet nlysis hve lredy begun in hydrology nd wter resources. The document (Li, 997) points out the potentil pplictions of wvelet nlysis to hydrology nd wter resources. Li et l (999) probe long time intervl forecst of hydrologicl time series with combing neurl network models bsed on wvelet trnsform. Wng et l (2) hve proposed wvelet trnsform stochstic simultion model, which generte synthetic stremflow sequences tht re sttisticlly similr to stremflow sequences. The multi-time scle chrcteristics of hydrologicl vrible hve been studied by Wng et l (22). Wvelet nlysis will mke new reserch pproch for the system of hydrology nd wter resources nd broden the content of hydrology gretly. ANN is highly flexible function pproximtor tht hs self-lerning nd self-dptive feture. Mny studies ttempted to model runoff by ANN. For exmple, Hlf et l (993) designed three-lyer feed-forwrd ANN using the rinfll hyetogrphs s inputs nd hydrogrphs s output to predict runoff from wtershed. Tokr (999) reported tht their ANN model hd better prediction ccurcy nd flexibility thn sttisticl regression nd simple conceptul models. Applictions of ANN re widely reported in the hydrologicl literture (French, 992; mn, 995; Hu, 2; Qing, 22). ANN models hve shown their utility in brod rng of wter resources ppliction nd re powerful tool for forecsting nd prediction. 3. The Theory of Wvelet Anlysis 3. Wvelet Trnsform 67

2 Wvelet nlysis is multi-resolution nlysis in time nd frequency domin, nd is the importnt milestone of the Fourier Trnsform. Wvelet function (t) is clled mother wvelet, which hs shock properties nd cn reduce zero rpidly. It cn be defined s + dt = mthemticlly. ( ) cn be cquired through, b t compressing nd expnding (t) : 2 t b, b = ( ) b,, () Where,b (t) is successive wvelet; is scle or frequency fctor, b is time fctor; is the domin of rel number. If,b (t) stisfies eqution (), for the energy finite signl or time series f (t) L 2 (), successive wvelet trnsform of f (t) is defined s: t b W f (, b) =< f, b 2, >= f ( t ) ( ) dt (2) Where (t) is complex conjugte functions of (t). Eqution (2) describes tht wvelet trnsform is the decomposition of f (t) under different resolution level (scle). In other words, the essence of wvelet trnsform is to filter wve for f (t) with different filter. In rel ppliction successive wvelet is often j j discrete. Let =, = kb, >, b b, k, j re integer number. Discrete wvelet trnsform of f(t) is written s: j / 2 j W f j, = f ( t kb ) dt (3) ( When =2, b =, eqution (3) becomes binry wvelet trnsform: j / 2 j W f ( j, = 2 f (2 t dt (4) f (, b) W f ( j, W or cn reflect the chrcteristics of originl time series in frequency ( or j) nd time domin (b or t the sme time. When or j is smll, the frequency resolution of wvelet trnsform is low, but the time domin resolution is high. When or j becomes lrge, the frequency resolution of wvelet trnsform is high, but the time domin resolution is low. Tht is, wvelet nlysis is mthemtic microscope. 3.2 The Algorithm of Wvelet Trnsform In rel world time series re discrete, such s rinstorm process, flood process, monthly stremflow process, nd dily runoff sequence. So discrete wvelet trnsform must be selected for decomposition nd reconstruction of time series. There re mny discrete wvelet trnsform lgorithm, such s Mllt lgorithm (Mllt, 989; Mllt, 989) nd A Trous lgorithm (Shens, 992; Aussum, 997). A Trous lgorithm hs been dopted in the pper. Let Z(t) (or C(t)) denote the originl discrete time series. A Trous decomposition lgorithm s following: i C t) h( l ) C ( t + 2 l) (i =,2, L) (5) = + i( i l= W i = Ci ( t) Ci (i =,2, L) (6) Where h(l) is the discrete low-pss filter; C i (t), W i (t) (i=,2, ) re scle coefficient (bckground informtion) nd wvelet coefficient (detil informtion) t the resolution level i respectively. W re clled ( t ), W 2 ( t ), L, W ( t ) nd C ( t ) discrete wvelet trnsform with the resolution level. In eqution (5), extending of boundries my be crried out in different wys. We took n intuitively cceptble pproch by tking C(n+=C(n-. The wvelet coefficients, W i (t) (i=,2, ), provide the detil signl, which cn cpture smll fetures of interprettionl vlue in the dt; the residul term C(t) represents the dt s bckground informtion. Becuse of simplicity of W, W2, W, C (we cn see from section 4), some interesting chrcteristics, such s period, hidden period, dependence, jump, cn be dignosed esily through wvelet components W, W2, W, C. It is possible to reconstruct the originl hydrologicl time series from wvelet components { W, W2, W, C }. The wvelet reconstruction of the originl time series, in term of wvelet coefficients, is given by Z( t) = C p p + W i= i (7) Eqution (7) provides reconstruction formul for originl time series. Tht is A Trous reconstructing lgorithm. A Trous decomposition nd reconstructing lgorithm re simple nd rpid. It s key is to determine the discrete low-pss filter. 4. Wvelet Network Model 4. Min Ide First, originl time series cn be decomposed into certin number of sub-time series {W,W 2,,W,C } by wvelet trnsform lgorithm. W,W 2,,W re detil time series, nd C is bckground time series. These ply different role in the originl time series nd the behvior of ech sub-time series is distinct. So the contribution to originl time series vries from ech other. Then, ANN is constructed in which the sub-time series t t time re input of ANN nd the originl time series t t+t time re output of ANN, where T is the time length of forecst. Lst, the wvelet network model (WNM) is formed in which the weighs re lerned with some lgorithm. The key of wvelet network model is wvelet decomposition of time series nd the 68

3 construction of ANN. 4.2 Decomposition of Observed Time Series The low-pss filter h, which is B 3 spline, defined s 3 (,,,, ), is used. This is of compct support nd point-symmetric. First the resolution level must be determined. In generl there is INT ( lg n) resolution scle number, where n is the length of time series nd INT stnds for integer number, the lg n is common logrithm. The wvelet coefficients nd scle coefficients of the monthly groundwter level time series derived from A Trous decomposition lgorithm re shown in Figure. In Figure W ( t ) nd W 2 denote wvelet coefficients t the resolution level nd 2 respectively; C 2 denotes scle coefficients t resolution level 2. Figure. Wvelet Decomposed rocess of Monthly Groundwter Level Time Series 4.3 The Structure of ANN ANN is widely pplied in the forecsting of hydrology nd wter resource. In ANN, B network models re common to engineer. So clled B network models, tht is the feed-forwrd rtificil neurl network structure nd bck-propgtion lgorithm (B). It hs proved tht B network model with three-lyer is stisfied for the forecsting nd simulting in the science of wter. The input of B network is X=[W (t),w 2 (t),,w (t),c (t)] T,nd the nodes of input lyer re +. The output is Y=[L(t+T)], nd the node is. The nodes of hidden lyer re determined by tril nd error. The network weights re lerned by stndrd B lgorithm or self-dpted B lgorithm nd so on. The detils on B lgorithm re vilble in the references, hence these re not repeted here. 5. Cses Studies 5. Cse One: Shllow Groundwter Level Forecst There is 2-yer ( ) record of shllow monthly groundwter level in Beijing of Chin from the litertures (Lu, 997), tht is {Z(t),t=,2,,44}. The first nine yers time series re used for clibrtion /trining of the model, nd the remining three yers dt re used for verifiction or testing purposes. In this reserch n = 9 2, then the scle number =2. Through A Trous lgorithm, the groundwter level time series re decomposed into the sub-time series: {W (t),w 2 (t),c 2 (t)}, nd re listed in Figure. Here three-lyer network: input lyer, hidden lyer nd output lyer, is dopted. The number of nodes in hidden lyer is equl to 3. So the structure of WNM is The weight prmeters of network re estimted by self-dpted B lgorithm. The number of trining of WNM is 5. Given four forecsting periods (T= month, 2 month, 3 month, nd 4 month), the fitting nd forecsting results of groundwter level re shown in Tble. The results of groundwter level of 992~994 re shown in Figure 2 (T= month) nd Figure 3 (T=3 month). In order to compre, the results of clibrtion nd verifiction of AMA model (Lu, 997) nd threshold utoregressive model (TA) bsed on genetic lgorithm (Jing, 2) re listed in Tble. From Tble it cn be seen tht, for T= month, the clibrtion precision of WNM is lmost good s AMA nd TA, but the verifiction precision is better thn the ltter. At the sme time, when the forecsting period (Tble, Figure 3, Figure 4) is become long, the fitting nd testing precision of WNM is lso very higher thn the other models (not listed). level(m) month Figure 2. Comprison of the Originl Groundwter Level Time Series nd Forecsted Time Series of WNM (Where T= Month) 5.2 Cse Two: Dily Dischrge Forecst Dily dischrge dt for Yngtze iver bsin of Chin t Cuntn Sttion re used. The yers re selected, the first 8 yers dt re used for build the wvelet network model, nd the remining two yers dt re used for verifiction of WNM. Here n = 8 365, then scle number =3. 69

4 Tble. Error Anlysis of Vlidtion nd Verifiction of WNM Model ercent of clibrtion bsolute error flling into the demrction ercent of verifiction bsolute error flling into the demrction [,.] [,.2] [,.3] [,.4] [,.] [,.2] [,.3] [,.4] AMA (T=) TA (T=) WNM (T=) WNM (T=2) WNM (T=3) WNM (T=4) Tble 2. The Clibrted nd Vlidted esults of Wvelet Network Model (%) Model ercent of vlidtion ercent of verifiction <% <2% <3% <% <2% <3% TA(T=d) TA(T=2d) TA(T=3d) TA(T=4d) TA(T=54d) WNM(T=d) WNM(T=2d) WNM(T=3d) WNM(T=4d) WNM(T=5d) Through A Trous lgorithm, the dily dischrge time series re decomposed into the sub-time series: {W (t),w 2 (t),w 3 (t),c 3 (t)}. Three lyers network is dopted too. The number of nodes in hidden lyer is equl to 4. So the structure of WNM is The weight prmeters of network re estimted by modified B lgorithm. The number of trining of WNM is 2. Given five forecsting periods (T= dy, 2 dy, 3 dy, 4 dy nd 5 dy), the fitting nd forecsting results of dily dischrge re shown in Tble 2. The results of dily dischrge of 2 yer re shown in Figure 4 (T= dy) nd Figure 5 (T=3 dy). In order to compre, the results of vlidtion nd verifiction of TA model re listed in Tble 2. It ws noticed tht WNM is better thn TA model. 6. Conclusions This pper hs reported new hybrid model wvelet network model. It plys n importnt role in improving the precision nd prolonging the forecsting time period 7

5 or hydrology nd wter resource time series. level(m) month Figure 3. Comprison of the Originl Groundwter Level Time Series nd Forecsted Time Series of WNM (Where T=3 Month) dischrge(m 3 /s) dy Figure 4. Comprison of the Originl Dily Dischrge Time Series nd Forecsted Time Series of WNM (Where T= Dy) dischrge(m 3 /s) dy Figure 5. Comprison of the Originl Dily Dischrge Time Series nd Forecsted Time Series of WNM (Where T=3 dy) Clibrtion nd verifiction of wvelet network model for prediction of hydrology nd wter resource in cse studies hve shown tht the method is functionl. The suggested strtegy is suit to ny other wter resource time series. Elementry ttempt t developing the hybrid model is success. Future studies will be opened up from the mnner of recombined with wvelet nlysis nd ANN to pplictions of wvelet network model. We would like to thnk the Ntionl Science Foundtion Committee of Chin to support this reserch project (No: ). Correspondence to: Wensheng Wng Hydrulic School of Sichun University Chengdu, Sichun 665, Chin Telephone: E-mil: wngws7@sin.com eferences Aussum A, Cmpbell J, Murfgh F. Wvelet-bsed feture extrction nd decomposition strtegies for finncil forecsting. Journl of Computtionl Intelligence in Frnce 997:-7. Chui CK, ed. Wvelet A Tutoril in Theory nd Applictions. Xin Jiotong University ress, Xin, Chin, 995: (in Deng J, ed. Grey rediction nd Grey Decision. Huzhong Ligong University ress, Wuhn, Chin, 992:74-92 (in Ding J, Deng Y, ed. Stochstic Hydrology. Chengdu University of science nd technology ress, Chengdu, Chin, 988:9-23 (in French MN, Krjewski WF, Cuykendll. infll forecsting in spce nd time using neurl network. J Hydrol 992;37:-3. Hlf AH, Hlff HM, Azmoodeh M. redicting runoff from rinfll using rtificil neurl networks. roc Engng Hydrol 993; ASCE:76-5. Hu T, Lm KC, Ng ST. iver flow time series prediction with rnge dependent neurl network. Hydrol Sci J 2;46(5): Jing J, Ding J, ed. Genetic Algorithm nd Its Appliction to Wter Science. Sichun University ress, Chengdu, Chin, 2:25-6 (in Li X, Ding J, Li H. Combing neurl network models bsed on Wvelet trnsform. Journl of Hydrulic 999; 2:-4 (in Li X, Ding J, Li H. Wvelet nlysis nd its potentil ppliction to hydrology nd wter resources. Journl of Sichun Union University (Engineering Science) 997;(4): (in Liu G, Ding J. Groundwter level forecst with Bp network model. Journl of Xin Geology College, 997;2:45-5 (in Lu H. Forecst of shllow groundwter level in Beijing. Geotechnicl Investigtion & Surveying, 997;:67-7. Mllt S G. Multifrequency chnnel decomposition of imges nd wvelet models. IEEE Trns on ASSp 989;37(2):29-. Mllt SG. A theory for multiresolution signl decomposition: the wvelet representtion. IEEE Trns on AMI 989;(7): Qing G, Ding J, Liu G. Self-dpted B lgorithm nd its ppliction to flood forecst of river. Advnce in Wter Science 22;3():37-4(in mn H, Sunilkumr N. Multivrite modeling of wter resource time series using rtificil neurl network [J]. Hydrol Sci J 995;4(2): Shens MJ. Discrete wvelet trnsform: wedding the A Trous nd Mllt lgorithm [J]. IEEE Trnsctions on Signl rocessing 992;4: Tokr AS, Johnson A. in-runoff modeling using rtificil neurl network. J Hydrol Engng ASCE 999;4(3): Wng W, Ding J, Xing H. The multi-time scle nlysis of hydrologicl time series with wvelet trnsform. Journl of Sichun University 22;35(4):4-7 (in Wng W, Yun, Ding J. Wvelet nlysis nd its ppliction to the stochstic simultion of dily dischrge process. Journl of Hydrulics 2;:43-8 (in Zhng ed. Artificil neurl network models nd its ppliction. Fudn University ress, Shnghi, Chin. 994:-65 (in 7