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338 857 255 Tel : 48 858 3577, Fax : 48 858 3716 Email : tohru@ics.saitama-u.ac.jp URL : http://www.nls.ics.saitama-u.ac.jp/ tohru / / p.1/66

(time series) ( 225 ) / / p.2/66

/ / p.3/66

?? / / p.3/66

1.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 1 t.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 t / / p.4/66

1.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 1 t.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 1. t / / p.4/66

1.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 1 t.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 1. 2. t / / p.4/66

1 2 1 2 1 1 1 1 Power 1 Power 1 1 1 1 1 1 2.5.1.15.2.25.3.35.4.45.5 Frequency 1 2.5.1.15.2.25.3.35.4.45.5 Frequency, / / p.5/66

x(n) x(3) x(2) x(1) n / / p.6/66

x(n) x(3) x(2) x(1) n x(n) x(n + 1) / / p.6/66

x(n) x(3) x(2) x(1) n / / p.7/66

x(n) x(3) x(n + 1) x(2) x(1) (x(1), x(2)) n x(n) / / p.7/66

x(n) x(3) x(n + 1) (x(2), x(3)) x(2) x(1) (x(1), x(2)) n x(n) / / p.7/66

x(n) x(3) x(n + 1) (x(2), x(3)) x(2) x(1) (x(1), x(2)) n x(n) / / p.7/66

x(n) x(3) x(n + 1) (x(2), x(3)) x(2) x(1) (x(1), x(2)) n x(n) / / p.7/66

x(n) x(3) x(n + 1) (x(2), x(3)) x(2) x(1) (x(1), x(2)) n x(n) x(n) x(n + 1) / / p.7/66

1 1.9.9.8.8.7.7.6.6 y(t+1).5 y(t+1).5.4.4.3.3.2.2.1.1.1.2.3.4.5.6.7.8.9 1.1.2.3.4.5.6.7.8.9 1 / / p.8/66

1.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 1 t.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 t / / p.9/66

1.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 1 t.9.8.7.6???.5.4.3.2.1 5 1 15 2 25 3 35 4 45 5 t x(n + 1) = 4x(n)(1 x(n)) / / p.9/66

1 1.9.9.8.8.7.7???.6.5.4 y(t+1).6.5.4.3.3.2.2.1.1 5 1 15 2 25 3 35 4 45 5 1 t.1.2.3.4.5.6.7.8.9 1 1.9.9.8.8.7.7.6.6???.5.4 y(t+1).5.4.3.3.2.2.1.1 5 1 15 2 25 3 35 4 45 5 t.1.2.3.4.5.6.7.8.9 1 x(n + 1) = 4x(n)(1 x(n)) / / p.9/66

Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke / / p.1/66

Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke / / p.1/66

Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke / / p.1/66

Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke? / / p.1/66

x(n + 1) = 4x(n)(1 x(n)) 1. 1 2. 3. 2 / / p.11/66

1. 2. x(n + 1) = fµ(x(n)), x(n) R d, n Z (difference equation) (ordinary differential equation) (delay differential equation) (partial differential equation) x(n + 1) = f(x(n)) ẋ(t) = f(x(t)) ẋ(t) = f(x(t), x(t τ)) (autonomous system) ẋ(t) = f(x(t)) (non-autonomous system) ẋ(t) = f(x(t), t) (Input Output System) ẋ(t) = f(x(t), u(t)) / / p.12/66

1. x(n + 1) = fµ(x(n)), x(n) R d, n Z x() d x(n) 2. (a) (fixed point) (b) (limit cycle) (c) (torus) (d) (strange attractor) / / p.13/66

k R k /Z k R/Z (k 2) 1 k d λ i < (i=1,..., d) λ 1 = λ i < (i=2,...,d) λ i = (i=1,..., k) λ i < (i=k+1,.., d) λ i > (i=1,.., m 1) λ m = λ i < (i=m+1,.., d) / / p.14/66

1. (Orbital Instability) 2. (Long-term unpredictability and short-term predictability) 3. (Self-similarity) 4. (Non-periodicity) 5. (Boundedness) / / p.15/66

/ / p.16/66

= / / p.16/66

= / / p.16/66

= / / p.16/66

= ( ) / / p.16/66

1. 2. ( ) 3. 4. (a) (b) (c) (d) / / p.17/66

y(t + 1) y(t + 2) KS / / p.18/66

x(n + 1) = fµ(x(n)) + η(n) y(n) = g(x(n)) + ξ(n) x(n) : n f : k µ : η(n) : ξ(n) : g : y(n) : n y(n) f? / / p.19/66

x(n + 1) = fµ(x(n)) + η(n) y(n) = g(x(n)) + ξ(n) x(n) : n f : k µ : η(n) : ξ(n) : g : y(n) : n YES! y(n) f? / / p.19/66

v(t) = {, d dt, dy2 (t),..., dm dt 2 dt m (Packard et al.,198) } / / p.2/66

v(t) = {, d dt, dy2 (t),..., dm dt 2 dt m (Packard et al.,198) } : (O. Rössler, 1976) dx = y z dt dy = x + ay dt dz = bx cz + xz dt (y, dy/dt) / / p.2/66

4 35 3 25 z(t) 2 15 1 5 1 5 5 1 15 1 5 5 x(t) 1 15 / / p.21/66

8 z(t) 4 35 3 25 2 15 1 6 4 2 2 4 5 6 1 5 5 1 15 1 5 5 x(t) 1 15 8 1 12 5 1 15 2 25 3 35 4 t / / p.21/66

8 z(t) 4 35 3 25 2 15 1 6 4 2 2 4 5 6 1 5 5 1 15 1 5 5 x(t) 1 15 8 1 12 5 1 15 2 25 3 35 4 t 8 6 4 2 2 4 6 8 1 12 8 6 4 2 2 4 6 8 1 12 x(t) / / p.21/66

8 z(t) 4 35 3 25 2 15 1 6 4 2 2 4 5 6 1 5 5 1 15 1 5 5 x(t) 1 15 8 1 12 5 1 15 2 25 3 35 4 t 8 1 6 8 4 6 2 4 2 2 dy/dt 4 2 6 4 8 6 1 8 12 8 6 4 2 2 4 6 8 1 12 x(t) 1 12 1 8 6 4 2 2 4 6 8 / / p.21/66

8 z(t) 4 35 3 25 2 15 1 6 4 2 2 4 5 6 1 5 5 1 15 1 5 5 x(t) 1 15 8 1 12 5 1 15 2 25 3 35 4 t 8 1 6 8 4 6 2 4 2 2 dy/dt 4 2 6 4 8 6 1 8 12 8 6 4 2 2 4 6 8 1 12 x(t) 1 12 1 8 6 4 2 2 4 6 8 : = / / p.21/66

(Takens, 1981; Packard et al.,198) v(t) = {, y(t + τ), y(t + 2τ),...,y(t + (m 1)τ)} y(t + 2τ) y(3) y(2) y(1) t 1. v(1) v(3) v(2) y(t + τ) 2. τ / / p.22/66

v(t) = {, y(t + τ), y(t + 2τ),...,y(t + (m 1)τ)} : (O. Rössler, 1976) 8 6 4 2 2 4 6 8 1 12 5 1 15 2 25 3 35 4 t 4 35 3 12 25 1 8 z(t) 2 6 15 1 5 1 5 5 1 15 1 5 5 x(t) 1 15 x(t+2τ) 4 2 2 4 6 8 8 6 4 2 2 4 6 8 1 12 x(τ) 1 1 2 x(t+τ) / / p.23/66

(Takens, 1981) d M C 1 f : M M C 1 g : M R 1 m = 2d + 1 V : M R m V(x) = (g(x), g( f(x)), g( f 2 (x)),...,g( f m 1 (x))) =1 1& m = k + 1 ( ) M A m > 2d m > 2D Embedology (Sauer et al.,1991) m > D 2 (Ding et al, 1994) / / p.24/66

(Whitney, 1936) A d A R 2d+1 ( embedding ) C 1 C 1 F 1 1(one-to-one)+ (immersion)) = A F(A) / / p.25/66

1 1 A g 1 1 1 1 / / p.26/66

A R 1 y 1 (t) / / p.27/66

A R 2 y 2 y 2 S y 1 y 1 / / p.28/66

A R 3 y 3 y 2 y 3 y 2 S y 1 y 1 / / p.29/66

R 3 S S / / p.3/66

R m d 1 d 2 D I = d 1 + d 2 m D I D I < D I < m > d 1 + d 2 d 1 = d 2 = d m > 2d / / p.31/66

(Sauer et al.,1991) A R k d m > 2d R k R m C 1 g = (g 1, g 2,...,g m ) R m / / p.32/66

(Sauer et al.,1991) A R k d m > 2d R k R m C 1 g = (g 1, g 2,...,g m ) R m / / p.32/66

(Sauer et al.,1991) A R k D m m > 2D R k R m C 1 1. A 2. A / / p.33/66

(Takens, 1981) d M C 1 f : M M C 1 g : M R 1 m = 2d + 1 V : M R m V(x) = (g(x), g( f(x)), g( f 2 (x)),...,g( f m 1 (x))) / / p.34/66

(Sauer et al.,1991) Φ R k U A U A D m > 2D τ> A pτ(3 p m) τ 2τ U C 1 V(x) = (g(x), g(φ τ (x)), g(φ 2τ (x)),...,g(φ (m 1)τ (x))) 1. A 1 1 2. A / / p.35/66

(Sauer et al.,1991) f R k U A U A D m > 2D p m p p A p p/2 Df p U C 1 g 1. A V(x) = (g(x), g( f(x)), g( f 2 (x)),...,g( f m 1 (x))) 2. A / / p.36/66

R n S ( ) D D = lim ɛ log N(ɛ) log 1 ɛ N(ɛ) S ɛ n [, 1] ( ) n 1 ɛ = N(ɛ) = 3 n D = 1 3 ( ) n 1 ɛ = N(ɛ) = 2 n D = log 2 3 log 3 =.6392... / / p.37/66

x(n + 1) = fµ(x(n)) + η(n) y(n) = g(x(n)) + ξ(n) x f x V V F = V f V 1 v F v f F / / p.38/66

: dx dt dy dt dz dt = σx + σy = xz + rx y = xy bz 1. 2. 3. σ = 1, r = 28, b = 8/3 / / p.39/66

1 5 4 z(t) 3 2 2 15 1 1 5 3 2 1 1 2 3 2 15 1 5 x(t) 5 1 15 2 5 1 15 2 1 2 3 4 5 6 7 8 9 1 1 x(t) / / p.4/66

2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+4) y(t+8) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+2) y(t+4) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+4) y(t+8) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+2) y(t+4) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+4) y(t+8) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+2) y(t+4) / / p.41/66

2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+4) y(t+8) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+2) y(t+4) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+4) y(t+8) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+2) y(t+4) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+4) y(t+8) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+1) y(t+2) 2 1 1 2 2 1 1 2 2 15 1 5 5 1 15 2 y(t+2) y(t+4) / / p.42/66

1.5 x 14 1.5.5 1 1.5 2 2.5 5 1 15 2 25 3 t 48[kHz] 12[bits] 1 / / p.43/66

x 1 4 x 1 4 x 1 4 2 2 1.5 2 1.5 1.5 1 1 1.5.5.5 y(t+2).5 y(t+1).5 y(t+22).5 1 1.5 1 1 2 1.5 1.5 2.5 3 2 2 x 1 4 2 1 y(t+1) 1 2 3 2 1 1 x 1 4 2 2.5 4 2 x 1 4 y(t+5) 2 3 2 1 1 2 x 1 4 2.5 4 2 x 1 4 y(t+11) 2 3 2 1 1 2 x 1 4 y(t+3) x 1 4 2 1.5 1.5.5 1 1.5 y(t+4) x 1 4 2 1.5 1.5.5 1 1.5 x 3 (t) x 1 4 1.5 1.5.5 1 1.5 2 1.5 1.5 2 2 x 1 4 2.5 4 2 x 1 4 y(t+15) 2 3 2 1 1 2 x 1 4 2.5 4 2 x 1 4 y(t+2) 2 3 2 1 1 2 x 1 4 x 2 (t).5 1 1.5 2 2 1 x 1 (t) 1 2 x 1 4 / / p.44/66

/ / p.45/66

/ / p.46/66

Orbital instability ɛ(t) = ɛ()e λt ɛ() ɛ() : λ : / / p.47/66

x(n + 1) = 4x(n)(1 x(n)) x() =.1.1 + 1 8 1.9.8.7.6 x(t).5.4.3.2.1 5 1 15 2 25 3 35 4 t / / p.48/66

ɛ() ɛ(t) = ɛ()e λt / / p.49/66

Lorenz (1969) Tong & Lim (198) Priestly (1985) Sano & Sawada (1985) Eckmann et al (1985,86) Farmer & Sidorowich (1987) Broomhead & Lowe (1987) Casdagli (1989) Sugihara & May (199) Mees (1991) Mees et al.(1993) Judd & Mees (1995) Lapedes & Farber (1987) Weigend et al (199) Wolpert & Miall (199) Sauer (1993) Cao et al. (1995)... Sugihara & May (199) Tsonis & Elsner (1992) Ikeguchi & Aihara (1993,97), MDL log log semi log / / p.5/66

= / / p.51/66

/ / p.52/66

/ / p.52/66

/ / p.52/66

/ / p.52/66

/ / p.52/66

/ / p.52/66

/ / p.52/66

/ / p.53/66

ẑ(t) z(t) P (z(t) z)(ẑ(t) ẑ) R 1 = t=1 P P (z(t) z) 2 (ẑ(t) ẑ) 2 t=1 t=1 ẑ(t) z(t) E 1 = P (z(t) ẑ(t)) 2 t=1 P (z(t) z) 2 t=1 / / p.54/66

z(t) = z(t + 1) z(t), ẑ(t) = ẑ(t + 1) z(t) z(t) ẑ(t) ( ) R 2 = P ( z(t) z)( ẑ(t) ẑ) t=1 P P ( z(t) z) 2 ( ẑ(t) ẑ) 2 t=1 t=1 z(t) ẑ(t) S e = 1 P t=1 I( z(t) ẑ(t)) P / / p.55/66

(Sugihara & May, 1991) (Wales, 1991) / / p.56/66

1 1 8 9 8 Correlation (%) 6 4 2 Henon Ikeda Measles Chickenpox Cobalt Correlation (%) 7 6 5 Henon Ikeda Measles 4 3 2 1 2 3 4 5 6 7 8 9 1 p 2 2 3 4 5 6 7 8 m / / p.57/66

1/ f α (Tsonis & Elsner, 1992) ɛ(t) ɛ()e λt log ɛ(t) λt ɛ(t) t H log ɛ(t) H log t = log log, semi log / / p.58/66

(TI & Aihara, 1994,97) 1. 2. z(t) : ẑ(t) : z(t) = z(t + 1) z(t) : ẑ(t) = ẑ(t + 1) z(t) : {z(t), ẑ(t)} R 1 { z(t), ẑ(t)} R 2 R 1, R 2 R 1 1, R 2 / / p.59/66

1 (a) 1 (b) predicted.5.5 1 predicted 1 2 actual 2 actual (c).2 (d) predicted.1.2 predicted.2.3.2 actual actual / / p.6/66

Correlation (%) 1 95 9 85 8 75 7 65 6 55 Correlation (%) 1 95 9 85 8 Correlation (%) 1 9 8 7 6 5 4 3 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 Embedding dimension Embedding dimension Embedding dimension (a) (b) A.dat (c) 1 1 8 8 8 Correlation (%) 6 4 2 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 Embedding dimension Embedding dimension Correlation (%) 6 4 2 Correlation (%) 6 4 2 2 4 2 3 4 5 6 7 8 9 Embedding dimension (d) (e) NYEX (f) / / p.61/66

1. 2. 3. (a) (b) 4. / / p.62/66

/ / p.63/66

23 ( ) / / p.64/66

1. ( ) 2. ( ) / / p.65/66

15 8 8 ( ) 17: 723 A4 ( [ ]) / / p.66/66

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