Chaotic Analysis on Precipitation Time Series of Sichuan Middle Part in Upper Region of Yangtze

Transcription

1 Natue and Science, (), 4, en, iang and Zhao, Chaotic Analysis on Pecipitation Tie Seies Chaotic Analysis on Pecipitation Tie Seies of Sichuan iddle Pat in Uppe Region of angtze Baohui en,, Chuan iang, Xiejing Zhao (.Agicultual Depatent, Key Open aboatoy of Agicultue Resouce and Envionent in angtze Rive Uppe Reaches, Chengdu, Sichuan 666, China, of Hydaulic Engineeing, Sichuan Univesity, Chengdu, Sichuan 665, China ) Abstact: Based on the intoduction of ain quantitative indees (coelation diension D and Kologoov entopy) in econstuction phase-space technique, accoding to pecipitation tie seies of Sichuan iddle pat in uppe egions of angtze, elationship between ebedding diension and coelation diension D is discussed and satuation coelation diension, iniu ebedding diension and Kologoov entopy ae calculated, that is, D = 5., = and k =.338. eanwhile, piay coponent analytic ethod (PCA) is applied to validate its chaotic chaacte and esult shows foecasting length fo this pecipitation tie seies should be less than.96 yeas. Thus, chaotic analysis on pecipitation tie seies povides a scientific gist fo pecipitation foecasting. [Natue and Science, 4,():74-78]. Key wods: coelation diension; Uppe egions of angtze; Piay coponent analysis (PCA); Kologoov entopy Intoduction ainly focusing on those anoalistic and nopeiodic acoscopical spatio-tepoal phenoenon, chaotic theoy is about eseach of special uddle behavio in dynaic syste evolving, which eveals the undelying egulations of those appeaing stochastic and out-of-ode behavio. Chaotic theoy, as a newly developent of nonbalancable Stat physics, coes up with new ethods of eseaching soe special and uddleheaded phenoenon. Aount of yealy pecipitation has diect effect on wate esouce eploitation and utility and is also closely elated to occuence and evolution of dought and flood. Theefoe, pecise foecast of annual pecipitation povides not only scientific gist fo wate esouce eploitation but also guidance fo calaity pecaution and salvage. Consideing pecipitation is uncetain o stochastic, we need fo scientific and easonable pedicting odels as well as pobe syste causing pecipitation as uncetain and stochastic and deteine effective phase-space, poviding foundation. fo aking dynaical foecasting odels. Factal theoy, aiing at eseaching factal chaacteistic of cople syste and deteining state space of tie seies, is an efficient tool fo eplaining coplicated dynaical syste and is widely used in field of physics, cheisty, biology, edicine, geology, weathe, hydology etc. In this thesis, accoding to pecipitation tie seies of Sichuan iddle pat in uppe egions of angtze, fo 953 to (Figue ), fo the view of coelation diension D and Kologoov entopy, egulation of pecipitation foation and evolution is discussed, and then PCA is applied to validate chaotic featue of pecipitation tie seies. Chaotic Systes Out-of-ode in chaotic theoy efes to those appeaing disode but not siply and noal chaos. Fo this eason, befoe thee is ipe identifying ethod in descibing featues of attacto, we can ecognize chaos fo the point of coelation diension D and Kologoov entopy. 74

2 Natue and Science, (), 4, en, iang and Zhao, Chaotic Analysis on Pecipitation Tie Seies pecipitation () tie(yea) Figue Pecipitation Seies in iddle Region Sichuan Povince of angtze Rive Up Reaches. Phase-space econstuct technique Ruelle poposes to change syste phase-space with a new n diensional phase-space (ebedding phase-space) esulting fo tie seies t and its continual oveent t+ τ, t+ τ,, t+ ( n )τ, and points out diension of econstuction phase-space should be oe than o equal to two ties plus of state space, that is d + () Take a scala tie seies,,, n in syste phase-space as an eaple. Supposing its diension d is, its diension of ebedding phase-space should be 3. If hee = 4,,, 3, 4 fos the fist vecto of a fou-diensional state space and then oving ight one step,, 3, 4, 5 fos the second vecto. Just do in the sae way,,, 3,, l fos the tie seies of econstuction phase-space.. Coelation diension D In tie seies of,, 3,, k, supposing is the absolute value of diffeence between two sectos, naely, = () i j Then which is in the ange of aiu and iniu is deteined. With the justification of, a goup of value of ln, lnc() and coelation diension can be calculated. The definition of coelation diension is as follows: () ( ) D = lilnc ln (3) l l l l () = H( ) = H( ) C i j l i j l i j (4) z > H () z = (5) z Whee: l is the total nube of state vectos, is sphee diaete with the cente of i o, i j is Euclid distance, H is the function of Heaviside step function. Nubes in Table coe fo (3),(4),(5), = { 3, 4, }, ebedding diension = { 4, 5,, 3}. Using nubes in Table, we can daw a plot of lnc( ) ~ ln. If line pat eits, we say this tie seies has the featue of factal, and slope of line is the coelation diension (coelation diension of the attacto). Figue shows the elation between ln C( ) and ln. Fo Figue, we see in chat of lnc( ) ~ ln, with diffeent ebedding diension, pat of linea coelation does eit, that is, pecipitation tie seies of Sichuan iddle pat in uppe egions of angtze has factal featue. Slope of line in each cuve is the coelation diension coesponding. Figue 3 shows the elation between coelation diension D and diffeent ebedding diension. In Figue 3, we see when =, coelation diension tends to be stable, naely satuate. So when D = 5., we say iniu ebedding diension = epesents effective feedo ate of dynaical syste. That is, when ebedding phase-space of pecipitation tie seies is, coelation diension of the attacto is 5.. Fo this aspect, we can daw a conclusion that pecipitation tie seies of iddle pat of Sichuan in uppe egions of angtze has chaotic featue. In ode to futhe oganize chaotic featue of this tie seies, PCA is intoduced. j 75

3 Natue and Science, (), 4, en, iang and Zhao, Chaotic Analysis on Pecipitation Tie Seies Table Data Calculation Table of ~C()~ C() ln() ln(c()) =4 =5 =6 =7 =8 =9 = = = =3 Figue Relation between ln( ) and lnc().3 Piay coponent analysis (Pca) PCA is a newly poposed ethod in oganizing noise and chaos. The step of this ethod is as follows: Supposing a scala tie seies is,,, n, afte econstucting phase-space (ebedding diension is, and delay tie is τ ), ati ( l = n ( ) ) d l is foed: l = l l 3 l+ + n = l Calculate covaiance ati l T A d l l = l 76

4 Natue and Science, (), 4, en, iang and Zhao, Chaotic Analysis on Pecipitation Tie Seies and its eigenvalue i ( i =,,3, ) U i ( i,,3, ) λ and eigenvecto =, then ode the λ λ λ in descending sequence. Eigenvalue and eigenvecto is called piay secto. Su of all d eigenvalue γ is γ = λi. Chat of i and ( λ /γ ) i= ln i is called piay spectu. Piay spectu of noise, which is paallel to ais, is quite diffeent fo that of chaotic seial, which is a line acoss fied dot with negative slope. Figue 4 is the chat of piay spectu. Accoding to featue of spectu of chaotic tie seies, we can futhe deteine that pecipitation tie seies has chaotic featue. D Figue 3 Relation between and D Figue 4 Relation between i and PCA C ( ) K = ln (4).4 Kologoov entopy τ C () + Anothe ipotant inde of chaotic featue is Kologoov entopy, which povides uppe and lowe Whee: τ is delay tie, C () is the value of ange of aveage aount of infoation in unit tie. C ( ) when ebedding diension of phase-space is Geneally, fo a sequential syste, K = ; fo a, C + ( ) is the value of C () when ebedding stochastic syste, K =. When < K <, syste diension of phase-space is +. is a chaotic syste, and the bigge K is, the oe Choice of τ and is key to calculation of seious the degee of chaos is. Foula poposed by diension, inde and entopy. In application, we need to conside diension of ebedding phase-space as well Gassbege-Pocaccia algoith is: as τ which has bette siulating effect. 77

5 Natue and Science, (), 4, en, iang and Zhao, Chaotic Analysis on Pecipitation Tie Seies In theoy, when, K K. In fact, when is soewhat value, K tends to be stable and this stable value can be used as estiating value of K. K Figue 5 Relations between + and K Figue 5 shows the elation between k and ( +). Fo Figue 5, we know with the incease of, k tends to be stable and when ( + ) =, Kologoov entopy coes to satuation, that is, K =.338 ( > ). This data also indicates the chaotic featue of pecipitation tie seies of Sichuan iddle pat in uppe egions of angtze. / K shows pedictable length of this syste is.96 yeas. 3 Conclusion Coelation diension D = 5. and Kologoov entopy k =. 338 ae achieved by econstucting phase-space. Piay coponent analysis futhe validates the chaotic featue of pecipitation tie seies of Sichuan iddle pat in uppe egions of angtze and the ecipocal of Kologoov entopy tells us pedicting length of pecipitation tie seies should be to 3 yeas instead of long-te pediction, which povides scientific gist fo deteining length of pedicting peiod. Acknowledgeent This study is suppoted by HI-TECH Reseach and Developent Poga of China AA6Z363. Refeences [] Bellie S, et al. Is coelation diension a eliable indicato of low-diensional chaos in shot hydological tie seies? Wate esouces eseach, ;38(): [] Casdagli. Nonlinea pediction of chaotic tie seies. Phys D, 989,35: [3] Chen N, et al. Reseach on chaotic echanis of glacie outbust floods. Jounal of Natual Disastes, 999,8():48-5. [4] Gassbege P, et al. Estiation of the Koloogoov entopy fo a chaotic signal. Physic Rev A,983,8: [5] Isla S, et al. A possible eplanation fo low coelation diension estiates fo the atosphee, Jounal of Appl eteool, 993,3:3-8. [6] Jayawadena A. Analysis and pediction of chaos in ainfall and steaflow tie seies, Jounal of Hydol, 994;53:3-5. [7] v JH, et al. Chaotic tie seies analysis and its application, Wuhan Univesity pess, Wuhan, China,. [8] andelbot BB. The factal geoety of natue. San Fanclso. W H, Feean, 983:5. [9] Nicolis C. Is thee a cliate attacto? Natue 984;3: [] Pool R. Is thee soething stange about the weathe? Science, 989,43:4-44. [] Sheng ZHH, et al. Analysis quote about nonlinea dynaical syste, Science Pess, Being, China,. [] Wei, et al. The chaotic chaacteistics of annual pecipitation seies in Jiujiang, Jiangi Science, 998,6(3):4-45. [3] Xin HW. Factal theoy and its application, China Sci-Tech Univesity Pess, Hefei,