Hybrid Wavelet and Chaos Theory for Runoff Forecasting

Transcription

1 Proceedgs of the 5th WSEAS/IASME It. Cof. o SYSTEMS THEORY ad SCIENTIFIC COMPUTATION, Malta, September 5-7, 5 (pp75-79) Hybrd Wavelet ad Chaos Theory for Ruoff Forecastg Che X, Jag Chuawe, Wag Yu, Zhou Ja epartmet of Electrcal Egeerg, Shagha Jaotog Uversty Shagha 3, P.R.Cha Abstract: -Ths paper troduced a method of decomposg o-statoary ruoff tme. By wavelet decomposg, the ruoff tme s decomposed to statoary tme ad stochastc tme, ad AR() model be mposed for forecastg statoary tme. By studyg chaos characterstc of stochastc tme, ths paper put forward a olear chaos dyamcs-forecastg model to dspose ruoff tme wth hgh-embedded dmeso. It ca effectvely decrease the Lyapuov expoetal sum added dmesos of recostructo set whe the dmesos of recostructed space are creased. Fally, the forecastg result s recostructed based o wavelet theory. The forecastg result of orgal ruoff tme s acheved. The method s hgh precso ad feasble through example test. Key Words:- Wavelet; Ruoff; Chaos; Forecastg Itroducto The olear theores of ruoff forecastg, such as the applcatos of eural etwork ad chaos, have recetly made cosderable progress []-[4]. However, t s stll ecessary to study how to explore the characterstc of ruoff uder mult-scalg so that the more accurate model ca be establshed. The wavelet trasform s a kd of sgal tme-frequecy aalyss method. It has the characterstc of mult-resoluto aalyss. It ca mafest the local characterstc of tme doma ad frequecy doma. It s also a sort of localzed tme-frequecy aalyss method that both of tme frame ad frequecy frame ca be chaged whle the sze of frame s fxed. By meas of the wavelet trasform, t s very effectve to acheve the adustg regulato of complcated tme ad to resolve evolvemet characterstc of tme at dfferet tme scalg. The local doma forecastg method of chaos s a very useful method, whch s well developed recet years [5][6]. Ufortuately, t s ot so vald to predct attractors wth hgh-embedded dmesos by usg local doma forecastg method. Ths paper develops a ew method for dsposg ruoff tme wth hgh-embedded dmeso based o wavelet trasform theory. Frstly, o-statoary ruoff tme s decomposed to statoary tme ad stochastc tme. Secodly, dfferet models are employed to forecast them respectvely. Especally, by studyg chaos characterstc of stochastc tme, a olear chaos dyamcs-forecastg model s put forward to dspose ruoff tme wth hgh-embedded dmesos. Fally, the forecastg result s recostructed based o wavelet theory, ad the the forecastg result of orgal ruoff tme s acheved. The method s hgh precso ad feasble through example test. Theory of wavelet decomposto ad recostructo Assumg L ( R), ts Fourer trasform s ˆ ( ϖ ). (t) s defed as a basc wavelet or

2 Proceedgs of the 5th WSEAS/IASME It. Cof. o SYSTEMS THEORY ad SCIENTIFIC COMPUTATION, Malta, September 5-7, 5 (pp75-79) mother wavelet f the followg full recostructo codto s met: ˆ ( ω) C = dω < () R ω The followg equato () s obtaed after the basc wavelet (t) s stretched ad traslated. a, b( t) s amed a wavelet. t b a, b = ( ) () a a a, b R; a Where a ad b are dexes for stretch ad traslato respectvely. If let a = a, b = ka b, the the correspodg dscrete wavelet s:, k = a ( a t kb ) (3) It s ecessary to chage the parameters a ad b so that wavelet trasform possesses chageable tme-resoluto ad frequecyresoluto, whch accommodate the ostatoary sgal. Let a =, b = k, the the bary wavelet s obtaed:, k = ( t k ) (4) The bary wavelet trasform s: W f ( k) =< f, ( k) > (5) * = f ( t k) dt R The correspodg verse trasform s: f = W f ( x) ( t k) Z dt (6) The bary wavelet does ot destroy the sgal traslato varable tme doma because t oly has dscrete parameter a but cotuous parameter b tme doma. The sgal ca be decomposed to hgh frequecy ad low frequecy because of the dfferet scalg. I ths paper, ruoff tme s decomposed ad recostructed by usg bary wavelet based o the above theory. Mallat algorthm [7] s adopted to decompose ad recostruct sgal. The sgal s decomposed step by step. The low frequecy sgal havg bee acqured at prevous step s decomposed to hgh frequecy ad low frequecy every step. After the Nth step has bee fulflled, the orgal sgal s decomposed to: X = + + L+ N + A N (7) Where,, L, N are the hgh frequecy sgals that are acqured at the frst step ad at the secod step utl the Nth step respectvely. A N s the low frequecy sgal that s acqured at the Nth step. If,, L, N ad A N ca be forecasted, the orgal sgal wll be able to be forecasted by meas of the recostructo algorthm. 3 The wavelet decomposto ad characterstc aalyss of ruoff tme The sgal ca be decomposed to statoary low frequecy ad olear hgh frequecy by usg wavelet decomposto algorthm. For the ruoff of draage area Eos state, the orgal s decomposed through fve steps. The results are showed fgure (a-f). Fgure (a) shows the curve of the low frequecy, whle fgure (b-f) show the curves of hgh frequecy. 3. The forecastg model of statoary tme Fgure (a) shows that the low frequecy has dstct statoary characterstc ad accordgly t ca be forecasted through AR () model. Assumg oe s x, x, L, xn, ad the the followg equato s obtaed: x = x + x + L+ x + ε t (8) Where the parameter set = [,, ] ca be estmated through least-squares algorthm accordg to the theory of tme. 3. The forecastg model of o-statoary tme Fgure (b-f) show that the hgh-frequecy s bore wth a very strog olear. It s ecessary to study specal forecastg models to deal wth the hgh frequecy. We employ the chaos theory to forecast them ths paper. Accordg to the chaos theory, we must decde whether the olear tme has chaos

3 Proceedgs of the 5th WSEAS/IASME It. Cof. o SYSTEMS THEORY ad SCIENTIFIC COMPUTATION, Malta, September 5-7, 5 (pp75-79) characterstc. Some methods, such as recostructo of phase space, Lyapuov expoetal spectrum ad coucto saturato expoetal, are commo use to decde chaos characterstc of tme. A a b c 5 dx = f( x, x,, xm),, m dt, L = L (9) Equato (9) s trasformed to a m-rak olear dfferetal equato va elmato method. ( m) () ( m ) x = f ( x, x, L, x ) () It correspods to tme evoluto: ( m ) t) = ( t), x, L, x ) () The above equato ca be dspersed to equato () wth substtutg dfferece equato for dfferetal. t) = ( t), t + τ), L, t + ( m ) τ)) () Equato () s amed recostructo phase space of tme delay. It retas the geometry structure, topology structure ad dyamcs characterstc wth orgal system a dfferetal homoeomorphsm sese. The theory s appled to recostructg the tme of fgure (f) two-dmeso ad threedmeso phase space. The recostructo dagrams are showed fgure ad fgure 3 respectvely d e Fg. Two-dmeso recostructo dagram of f Fg. Low frequecy ad hgh frequecy for fve steps The theory of recostructo of phase space s put forward by Takes 98 [8]. It s descrbed as followg: Assumg a m-dmeso self-cotrol dyamcs system ca be represet wth equato (9) Fg.3 Three-dmeso recostructo dagram of 5 Fgure ad Fgure 3 show that the tme possesses farly regular attractors structure. The maxmum Lyapuov expoetal of the tme

4 Proceedgs of the 5th WSEAS/IASME It. Cof. o SYSTEMS THEORY ad SCIENTIFIC COMPUTATION, Malta, September 5-7, 5 (pp75-79) s acqured to udge ts chaos characterstc by meas of the followg ways. It s well kow that postve Lyapuov expoetal meas chaos. That s sad that the mportat fucto of the expoetal les udgg chaos characterstc of the tme. The maxmum Lyapuov expoetal of the hgh frequecy sgal s acheved wth the same method adopted by Wolf [9]. The results are showed table. Table :The maxmum Lyapuov expoetal of hghfrequecy sgal Sgal λ Embedded max dmesos A chaos-forecastg model ca be establshed accordg to the chaos characterstc of the tme. The basc prcple of tradtoal local doma chaos forecastg method s as followg. Frstly, for oe tme { x, x, L, x N }, t ca be embedded to a -dmeso space by makg use of recostructo techology of phase space, ad the oe traectory of t - dmeso phase space ca be costructed wth equato (3). X = ( ) τ, ) τ, x ) X = ( ), ), x ) (3) M X = N ( ) ( ) N, ) N, xn ) Secodly, assumg x s a kow umber, t s ecessary to forecast x +τ (wthout loss of geeralty, let τ=). The latest traectory pot cotag formato of x s: X = ( x, x, L, x + ) (4) The ext traectory pot that eeds to be located -dmeso space s Xˆ ( xˆ, xˆ, L, xˆ ). Where the ew = xˆ + ˆ + formato X ca be regarded as oe forecastg result of x +. As a result, a mappg F wll be created as a predctor to acheve: X ˆ = F ( X ) +. The mappg F ca be acheved through the followg steps: () Accordg to equato (5), k of the earest pots amog N-+-(-)τ pots - dmeso away from X = ( x, x, L, x + ) ca be foud out. X X = [ ( x + x + ) ] (5) = Where equato (5) s the dstace equato -dmeso space. () Accordg to the teratve regularty of these k pots, oe polyomal P ca be employed to ft F. If the polyomal P has bee ascertaed, we ca employ equato (6) to acheve these coeffcets of P. Cosequetly, X + ca be obtaed: X ˆ + = F ( X ) m( F ( X ) P ( X )) k = m( X k + P ( X )) (6) The results are preferable by adoptg the algorthm metoed above whe we forecast the attractors wth low-embedded dmesos. However, t s ot satsfed whe we forecast the attractors wth hgh-embedded dmesos []-[3]. The ma reaso s that the hgher of embedded dmeso, the loger hstory of formato wll be eeded. I the dyamcs system wth postve Lyapuov expoetal spectrum, the loger the hstory s, the less effectve of hstory formato towards chaos tme s. The effect s expoetally decreased wth tme creasg. The large Lyapuov expoetal ewly added dmesos of recostructo set wll make the arthmetc sum of full Lyapuov expoetal spectrum crease, whle the dmesos of recostructo space crease. I order to reduce the arthmetc sum of full Lyapuov expoetal spectrum recostructo set, equato (5) ca be chaged to:

5 Proceedgs of the 5th WSEAS/IASME It. Cof. o SYSTEMS THEORY ad SCIENTIFIC COMPUTATION, Malta, September 5-7, 5 (pp75-79) + X [ ( ) ( ) ] X = x + x + = (7) 3.3 The test result Frstly, we forecast low frequecy ad hgh frequecy by meas of AR() model ad chaos model. Secodly, we costruct them to acheve the forecast result of the orgal tme based o the wavelet theory. Ruoff of Eos s forecasted ths paper ad the result s show fgure 4. Where the cotuous curve ad the dashed curve represet actual data ad forecastg data respectvely T 4 Cocluso Fg. 4 The result of forecastg It s dffcult to fd a effectve model to forecast strog olear ruoff. I ths paper, we decompose the tme to statoary tme ad stochastc tme through cosderg dfferet scalg based o wavelet trasform. Statoary tme ad stochastc tme are forecasted by meas of dfferet models respectvely. The wavelet trasform ad chaos aalyss are cosdered at the same tme the model of o-statoary. The result shows the method s powerful. Refereces: [] L Shuzhe, Lu Chaglg, You Baosha. The Chaos yamcs Research of aly Ruoff Tme Seres. Tawa Hydraulc. 44(), 996:3-3 [] I.Rodrguez-Iturbe & B.F de Power, Chaos Rafall. Water Resources Research, 5(7), 989: [3] Jag Chuawe, Qua Xazhag, Zhag Yogchua. A Research o the Chaos yamc Model of Ruoff. Joural of Hydrodyamcs. Ver B():,85-9 [4] Jag Chuawe, L Shua, u Soghua. Forecastg Ruoff Wth Hgher-Embedded mesos Usg MEP-Based Artfcal Neural Networks. Artfcal Itellgece Applcatos ad Iovatos ,5 [5] Mart Casdagl. Nolear Predcto of Chaotc Tme Seres. Physca,35,989 [6] Jag, Chuawe, L Tao. Forecastg Method Study o Chaotc Load Seres wth Hgh Embedded meso. Eergy Coverso ad Maagemet. 46(5),pp , 5 [7] Mallat S,Zhag S. Characterzato of Sgals from Multscale Edges. IEEE. Tras. o patter aalyss ad mache tellgece,99,4(7):7-73 [8] Takes.F. etectg Strage attractors turbulece, : yamcal Systems ad Turbulece,Warwck 98,Lecture Notes mathematcs, No.898,edted by.a.rad ad L.- S.Yog, Sprger-Verlay, Berl, 98: [9] Wolf A.etal. etermg Lyapuov Expoets From A Tme Seres. Physca.6,985(85-37) [] Ma Shexag, Lu Guzhog, Zeg Zhaohua. The Aalyss ad Forecastg of No-statoary Tme Seres Based o Wavelet Aalyss. Joural of Systems Egeerg.,(4)35-3 [] Jag Chuawe, Qua xazhag, Zhag Yogchua. Applcato of Chaos Theory Power System Load Forecastg. Joural of Wuha Trasportato Uversty. 3(6):999,68-6. [] Jag Chuawe, Hou Zha etal.a Improved Grey Procedure for Forecastg the Ruoff. Joural of Grey System,UK,3():,79-8. [3]Fracesco Ls, Vglo Vll. Chaotc Forecastg of scharge Tme Seres: A Case Study. JAWRA, 37():.