TCSE16

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2 Time Series t { x(t } N1 t =t " x(t 0 { 0,x(t 0 + t,...x(t 0 + (N 1t } t { } N1 t =t 0 x(t x(t x(t Random Variable Stochastic Process t x(t N1 x(t { } t =t 0 t { } N1 t =t 0 Statistical Moment t { } N1 t =t 0 Statistical Model x n = f (x n1,x n2,... + " n { x n } " n { } State-Space Model { x n } { X } X n = F ( X n1, X n 2,... x n x n = h( X n

3 Real World Theoretical World Real World Theoretical World

4 Real World Theoretical World x(t x(t x (t = x(t,x(t 1,...x(t D +1 { } { } x(t + = F( x (t F(a x + b y = af( x + bf( y time series prediction x ˆ (t + = f ( x (t x(t + Wold z(t x(t { } { y(t } { } { z(t } = { y(t } + { x(t }

5 x(t = (t + $ a i (t " i, $ a i < " i =1 (t i =1 x(t = c i x(t i + $(t, c i < " i =1 ( x(t = (t + a i (t " i $ i =1 = (t + a 1 (t "1 + a 2 (t " 2 + %%% x(t "1 = (t "1 + a 1 (t " 2 + a 2 (t " 3 + % %% x(t " 2 = (t " 2 + a 1 (t " 3 + a 2 (t " 4 + % %% " " i =1 x(t + (a 1 x(t 1 + (a 2 (a 1 a 1 x(t 2 +" "" Auto-Regressive " x(t = c i x(t i + $(t, c i < " % i =1 p p x(t = c i x(t i + $(t i =1 p x ˆ (t = c i x(t i i =1 " i =1 (t (t 2 " Linear Predictor c 1,c 2,...c p, 2 { x(p, x(p +1,...x(1, x(0, x(1, x(2,... } N 1 lim $ c N " N (t'2 1,c 2,...c p t'=1 2

6 x n = a 1 x n1 + a 2 x n a M x nm + w n w n x n1, x n1,...x nm w 2 n = { x n a 1 x n1 a 2 x n2... a M x nm } 2 " x n+1 a i x n i { a m ;m =1,2,...M} { x 1m ;m =1,2,...M } M x s = " a m x sm + s m =1 { } a m ;m =1,2,...M 1 a 1 z a 2 z 2 "" " a M z M = 0 lim s x s = 0 x s = " a m x sm + s a M 0 m =1 s" M

7 (y(t,y' (t, y' ' (t,... Packard N H, Crutchfield J P, Farmer J D, Shaw R S, Phys. Rev. Letters 45 (

8 v t = (y t, y t",...y t(m1" (x(t, x(t + " (x(t,dx/dt(t Z t h x t = h(z t (t = 1,2,...N x t X i h : R d R X i = (x i, x i+,...x i+( m"1 (i = 1,2,...N " (m "1 Embedding immersion

9 A h R d R F Rk R m for m > 2d F(z t = h(z t,h(z t +,...h(z t +( m"1 R m ( 2d +1 R k Takens F, Detecting strange attractors in turbulence, Lecture Note in Mathematics 893 (Springer, Berlin, 1981 pp H.G.E. Hetchel, I Procaccia, Fractal nature of turbulence as manifested in turbulent diffusion, Phys. Rev. A27 ( dx = "x + "y dt dy = xz + rx y dt dz = xy bz dt = 3 b =1 r = 26.5 =10 b = 8 3 " r = 28

10 Lorenz-z Lorenz-y Lorenz-x % " ' $ & % % $ " z x y $ $ " % $ " " & $ $& " & % t %& $ & " t ' $ & % % $ Lorenz-zxs Lorenz-yzs Lorenz-xy " "& t % " x z y $ % $ " " $ $ % % $ " " ($ ( (" x = 10, r = = "t Delayed2 " $ $ $ $ = 2"t x(t $ Delayed10 % & Delayed5 % ' ( " $ $ " % = 5"t " " " " $ $ x % x(t+2d x(t+d t = 0.01 " " $ x(t+5d Delayed1 8 " , b = 28 3 時間遅れ座標系での表示 % " y $ = 10"t x(t $ Delayed20 % % " " " " % " $ = 20"t x(t $ Delayed50 " % = 50"t " $% $ x(t+50d x(t+20d x(t+10d $ $ % % $ $ $ $% " " $ $ x(t " % " " " $ = 10, r = x(t $ " % 8 " , b = 28 3 " $ $ x(t " &

11 Takens F, Dynamical systems and turbulence, Warwick, 1980, edited by D. Rand and L.S. Young, Lecture Notes in Mathematics No.898 (Springer, Berlin, 1981 p.366. Mane R, ibid, p.230. d > 2d A ( s ˆ (n,ˆ s (n + jt,ˆ s (n + (j +1T,...ˆ s (n + j(d 1 s ˆ (n + jt s ˆ (n + (j +1T C L ( = s = 1 N 1 N s ˆ (n + jt s ˆ (n + (j +1T N m=1 N [ ˆ s (m + " s ] ˆ s (m " s m=1 ˆ s (m 1 N N [ s ˆ (m " s ] 2 m=1 [ ]

12 a. b. average mutual information Fraser A M, Swinney H L, "Independent coordinates for strange attractors from mutual information," Phys. Rev. A33 ( I AB (a i,b k = log 2 P AB (a i, b k P A (a i P B (b k I ˆ AB (T = P AB (a i,b k I AB (a i,b k a i,b k A = { s ˆ (n}, B = s ˆ (n + T ˆ I (T = { } N ' n=1 P(ˆ s (n, s ˆ (n + T log 2 " P(ˆ s (n, s ˆ (n + T P(ˆ s (np(ˆ s (n +T $ %& P AB (a,b = P A (ap B (b I AB (a,b = 0 I ˆ (T T = T m T = T m P(ˆ s (n, ˆ s (n + T P( ˆ s (np(ˆ s (n + T P(x n, P(x n" P(x n,x n"

13 =10, b = 8 " , r = 28 3 t = 0.01 " 50 ( $t " 0.5 "18 ( $t " 0.18 ~ 1 ~ 20 ~ 50 x n ~ x n" ~ x n2" Grassberger- Procaccia P. Grassberger, I. Procaccia, Characterization of strange attractors, Physical Review Letters 50, (1982. P. Grassberger, I. Procaccia, Measuring the strangeness of strange attractors, Physica D 9, (1983. M " 2 Y 2 ( = P i i=1 d d Y 2 (

14 C( X i, X j H(x " H(x = 1 x 0 $ 0 x < 0 X i P i C i (" N C( = 1 $ H( " X N 2 i " X j i,j=1 i j N C i ( " 1 $ N H( X X i j j=1 M 2 Y 2 ( " P i = P i C i ( = C( i=1 M i=1 P = (p 1,p 3,p 3,...p M Q = (q 1, q 2,q 3,...q M D 2 = lim logy 2 (" logc(" = lim " 0 log" "0 log" $ P Q n " %& M m=1 n' p m q m ( P Q sup " max 1mM p m q m 1 n a z = a z

15 X n (x n, x n+1,...x n+d"1 x 1, x 2,...x m,... C d ( " 1 N H( X N % 2 i X j i,j=1 i$ j 2 = N(N 1 C d ( " D 2 (d N N %% i=1 j= i+1 D 2 (d = lim "0 logc d ( log H( X i X j " 0 1 < < 2 logc D 2 (d = lim d ( "0 log ] C d ( = C 1 ( d = d D 2 (d = d ] D 2 (d < d D 2 lim D 2 (d d" for d 0 < d

16 LeBaron LeBaron LeBaron B, Forecast improvements using a volatility index, J. Appl. Econometrics 7 ( LeBaron B, Nonlinear forecasts for the S&P stock index, Santa Fe Institute Studies in the Science of Complexity (M. Casdagli and S. Eubank eds. Addison-Wesley x n = f (x n1 x n = f (x n1 x ˆ n = f (x n1 (n' < n x n' 1 " x ˆ n1 x n x n' x n = f (x n1,x n 2 (n' < n x n' 1 " x n1 x n' 2 " x ˆ n 2 x n x n'

17 p>>1 x n = f (x n1,x n 2,...x n p = (x n"1,x n" 2,...x n" p LeBaron n = 1 p 2 x p n" j j =1 LeBaron Farmer Farmer J D, Sidorawich J J, Predicting chaotic time series, Phys. Rev. Letters 59 (

18 Yule Yule G U, Philos. Trans. Roy. Soc. London A 226 ( Pandit S M, Yu S -M, Time Series and System Analysis with Applications (Wiley, New York, x(t i x (t : x 1 (t = x(t,x 2 (t = x(t ",...x d = x(t (d 1" x (t + T = f T ( x (t f T t+2t, t+3t,... F T x (t + nt F T (x(t + (n "1T F n T (x(t T'=nT F T

19 Farmer x(t + T x (t A. x (t x (t' x (t x (t' < " B. x (t' x(t' +T x (t' (domain (range x(t' +T x pred (t,t = x(t' +T x (t',x(t' +T 1 " =< [x pred (t,t x(t + T] 2 2 > x pred (t,t =< x > N D d h S T T max E = (T 1 " x =< (x" < x > 2 2 > x t char

20 x 1 (t 1,...x d (t 1 x 1 (t 1 + T x 1 (t k,...x d (t k x 1 (t k + T a 0 + a 1 x a d x d = x 1 (t +T " " =< [x pred (t,t x(t + T] 2 > E = " (T x x =< (x" < x > 2 > E = E = x pred (t,t =< x > " N D d h S T T max t char

21 The first regime < I(T >= ln S ht The second regime: > S "1 d>>d E<<1 " N 1 D < S 1 E Ce (m+1kt N " (m+1 / D E(T max = 1 T max = ln N kd x n+1 = F T,1 ( x n = a 0 + a 1 x 1 + a 2 x a d x d d (1,j,k (1, j ( x n+ = A n (dx n"( j"1 x n"(k "1 + B n (dx n"( j"1 + C n 1(d j,k=1 d j=1

22 x 0 = 1 x 1 x 2 x 3 x 4 x 5 w 1 w 2 w 3 w 4 w 5 w 0 y N & y = %" w n x n ( $ ' n = 0 (X 1 % (X = $ 1 + exp("x &% X 1 "$ m-1 m M " X i (m w ij j " Y Im-1 Im "$ %&$ "$ Y = F NN ( X

23 Output( y x1 x2 """ """ (2 y = w (1 " j (" w ij x i $y (2 = w (1 " j ' w kj j i $x k j y = a i f i (x i xi

24 X Y = f ( X F( X, Y = 0 F( X j, 2 ( Y j j j X j x 1,x 2,...x n (x 1,...x m x m +1 (x 2,...x m +1 x m +2 (x n"m"1,...x n"1 x n (x n"m,...x n x n +1 x im1 x i x i +1

25 x 1,...x n x 3,...x n x 2,...x n1 x 1,...x n2 t 1 " y 1 (x 1,x 2,x 3 " t n2 " y n2 (x n2,x n1,x n dy (i dt = f i (y 1, y 2, y 3 (i = 1,2,3 F = n2 * j =1 $ & % 3 * i =1 $ & % dy (i dt '' f i (y (1, y (2, y (3 (( NN y (1 y (1 (t j w dy (1 dt t j F, F w t j NN y (2 NN y (3 y (2 (t j y (3 (t j NN f1 f 1 dy (2 dt dy (3 dt t j t j dy (i dt = f i (y 1, y 2, y 3 (i = 1,2,3 NN f 2 NN f 3 f 2 f 3

26 "$%&'(* y = f ( x g( x N " a n n ( x $ n = ,-./0%& n ( x a n ,1234 5"6$%&'(* y = f ( x g( x " f NN ( x I % $ w i & ( v i ' x + w ,78/0%& & ( v i ' x ,1234 w i v i i=1 v i = (v i 0, v i i 1,...v M x = (1,x 1,...x M "$ Observation data Data consisting of annual sun-spot activity From year 1700 to year 1988:

27 Neural network system for reconstruction We then reconstruct 3D dynamical system as described earlier with L = 3 t = 1 year " = 1 Training length = 100 years t obs = 1, Total patterns = 100 Neural network structure = 3 layer (1-9-1 & (3-9-1 Obs. data used for training = 100 patterns (100 years Testing period length = 30 years Training period Sun-spot activity

28 Prediction skill Sun-spot activity Mean RMSE for 15 cases

29 Sun-spot activity Prediction period Sun-spot activity RMSE over prediction period

30 "$%&'( *"&'( +,-./0* "2. 7:;<.= "$%&'(./0* *"&'(./0* g?o$:;hoijk "$%&'($*+,-./01( /$89:; QR+ST UVWX >LMN Y $[\]^_ `>LGHabc

31 "$%&'(*%+,-. /0123"$ slope GBHIF -.2.0J2.1 14(0"$ slope GBHIF -.1.8~ :;</=' "$%&'(*+ 3$4&56*78&1./01 2-,----- Z.S[\]%&'(*+ QCCCC&3$4&56*78&1 QCCCC&3$4&56*78&1./01 @STU*+&VWXY1,:,: : -9-M: R &qx ^_`abc1cd-9--,e-9--:&mfg hic1&jkl=cccd&m1nop

32 "$%&'(* +,"-./ &7788&9: PQRSTUVWX7778&YZ[ \J]<^_`a& ]AbJ_`a& "$%&'(*+,-./01, :; %&'(*+,<=>???> BC( DEFG (HA= IJKLG

33 "$%&'(*+,-./01, %&'(*+,:;<===< BCDE (F?; GHIJE "$%&'(*+',-./ $/345 slope 88?>;D= 88?:;D= 889>;D= 889:;D= 88?>;<= 88?:;<= 889>;<= 889:;<= D 2.07 for " = 10,b = 8/3,r = 30 D 2.03 for a = 0.2,b = 0.2,c = 5.7

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