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1 総説 Convergent cross mapping 1, * 2, ** An introduction to convergent cross mapping: a novel method for causality detection in ecological time series Shin-Ichiro Nakayama 1, *, Masato S. Abe 2, ** and Hiroshi Okamura 1 1 National Research Institute of Fisheries Science, Fisheries Research Agency, 2 Department of General Systems Studies, Graduate School of Arts and Sciences, University of Tokyo Convergent cross mapping Granger Abstract: Ecologists frequently need to detect causalities among events from time series data. It can be difficult to detect causality from time series data created from complex, deterministic, nonlinear systems, which are universal. Convergent cross mapping is a novel method for detecting causality in such situations. This review explains the mechanism, how to use the method, and future issues with this method. Keywords: Causality, deterministic dynamical systems, Granger causality, non-linearity, time series analyses * shinichironak@affrc.go.jp ** Present address: JST, ERATO, A B B A A B A B 4 A B B A B A B A 241

2 2 X, Y X t = f(x t-1 ), Y t = g(x t-1, Y t-1 ) X Y Granger Granger 1969 Granger Granger transfer entropy, Schreiber 2000 Granger Pereda et al Convergent cross mapping CCM Sugihara et al Granger Science 2 CCM 2.1. CCM CCM CCM 5 Granger CCM 6 CCM CCM 2. Convergent cross mapping CCM 2 2 CCM model-free CCM 2.1. 理論的背景 CCM t X t X t = f(x t-1 ) 242

3 1 CCM a Y T b X T t Y t Y t = g(x t-1,y t-1 ) T Y T 1a 1. T T 1 2 P P 1 2. P Y P Y T Y t = g(x t-1,y t-1 ) T 1 P 1 T P T X T 1b 1. T T 1 2 Q Q 1 2. P X Q X T Y t = g(x t-1,y t-1 )T Q T 1 Q 1 T 1 Q 1 T 1 Q 1 T Q X t = f(x t-1 ) CCM 2 A B

4 1. A B B A A B 2. B A A B A B AB CCM model-free cross-mapping 埋め込み (embedding) X, Y 2 t Lorenz Lorenz a CCM Takens Takens 1981 Sauer et al b Sugihara CCM 2 X = {X 1, X 2,..., X T } Y = {Y 1, Y 2,..., Y T } 2 M X M Y embedding dimension, E X M X τ 1 E M X t 1 (E 1)τ t 1 X t 2 (E 1)τ t 2 X x(t 1 ) x(t 2 ) (E 1)τ 2 E = 2 τ = Cross mapping X Y 2 X Y X Y Y X Y 2 m (E 1)τ m n (E 1)τ n X Y M Y 2 y(m) 244

5 2 a dx/dt = -10X+10Y, dy/dt = -XZ+28X-Y, dz/dt = XY-8/3Z X Y Z b X τ = 0.1, E = 3 τ = 1 y(n) x(m) x(n) M X 2 M Y 2 X Y X Y Y X Y CCM M X Y M Y X cross mapping M Y X M X Y X Y X Y CCM X Y Sugihara et al CCM mapping M X Y 1. M X Y t x(t) E+1 x(t 1 ), x(t 2 ),, x(t E+1 ) x(t) E x(t 1 ), x(t 2 ),..., x(t E+1 ) M Y y(t 1 ), y(t 2 ),, y(t E+1 ) Y t Y t y(t) E+1 M X Y t Ŷ t 2 245

6 w i,j 3 u t,i = exp{-d[x(t), x(t i )]/d[x(t), x(t 1 )]} d[x(a), x(b)] 2 x(a), x(b) 3. Ŷ t Ŷ Y ρ mapping 2.4. 埋め込み次元 Eと時間遅れτの決め方 CCM E τ CCM E CCM simplex projection, Sugihara and May 1990 Clark et al A = {X 1, X 2,..., X t 1 } B = {X 2, X 3,..., X t } B A 1 2. E τ M A B 3. B CCM E τ τ τ = X Y cross mapping cross mapping cross mapping CCM 3.1. 結果の解釈 CCM L L = 10 M Y X {y(t), y(t+1),..., y(t+9)} M Y X t t X ˆX X ρ - ρ 3 L = 5, 10, 15,..., 200 M Y X - ρ X Y L M Y X 3a convergence CCM convergence X Y Z M Y X M X Y M X M Y L 3b L Z X M Z M X M Y X convergence

7 3 CCM L ρ - ρ CCM 2 CCM CCM 95 a 2 b 0, 1.5 η(t) r x (t) = 3.1x(t 1)(1 x(t 1))e 0.3η (t) r y (t) = 2.9y(t 1)(1 y(t 1))e 0.36η (t) x(t) = 0.4x(t 1)+max(r x (t),0), y(t) = 0.35y(t 1)+max(r y (t),0) 2 CCM η(t) r x r y L x y cross mapping L cross mapping 3.2. サロゲートデータを用いた類推の正確性の検定 - ρ 0 - ρ - ρ 2 CCM x(t) x s (t) Sugihara et al Fourier Transform Iterative Amplitude Adjusted Fourier Transform Thiel et al twin surrogate twin surrogate N i, j = 1,, N recurrence matrix 4 Θ( ) < 0 Θ =

8 Θ = 1 x(i) x(j) Thiel recurrence matrix 2 x(i) x(j) δ 0 1 recurrence matrix Thiel et al., k = 1, 2,..., N k R i,k = R j,k x(i) x(j) twin twin 0 1 recurrence matrix twin 3. i x s (i) 1 x(i) x s (1) = x(i) 4. x s (j) = x(m) x(m) twin x s (j+1) = x(m+1) x(m) twin x(n) x s (j+1) = x(m+1) x s (j+1) = x(n+1) twin 3 twin recurrence matrix Step4 twin δ δ Thiel et al δ = δ = twin surrogate Thiel et al Verdes 2005 Lizier et al X t Y t x(t) y(t) CCM ρ -S ρ P ρ 3.3. convergenceの検定 convergence CCM ρ - L 3.1. convergence - ρ CCM Sugihara et al convergence convergence convergence Clark et al CCM multispatial CCM 6.3. Clark et al CCM Sugihara et al L 3.1. bootstrapping L convergence bootstrapping L CCM L - ρ Clark et al L - ρ L - ρ convergent 4. 3 CCM CCM CCM CCM 248

9 Ye et al Simplex projection S-map CCM R C++ redm 時系列の準備 CCM Sugihara et al Sugihara et al CCM 埋め込み次元時間遅れの決定と非線形性のチェック 2.4. E τ CCM E τ 0 S-map Tsonis et al. 2015library library X T library AR Sugihara 1994 Hsieh et al CCMによる解析結果の解釈検定 cross-mapping L ρ - L - ρ twin surrogate ρ convergence Granger Granger CCM Granger Granger CCM 5.1. Granger 因果性テスト 2 X Y U U Y X U Y X Granger Granger CCM 2 5 N(μ, σ 2 ) μ σ 2 X Y b c X Y 249

10 6 X X Y Y X Y σ 2 X σ 2 Y X Y (1+d 2 )σ 2 X +b 2 σ 2 Y c 2 σ 2 X +(1+a 2 )σ 2 Y X Y X X X Y X Granger 5.2. CCM CCM Sugihara et al X Y M Y X Granger X t X t 1 X t 2 X X Y Y Granger Y X CCM Hirata and Aihara 2010 recurrence plot CCM CCM Y X Y X b 08 b X t X t-1 Y t M X Y M Y M X Y 6.1. 適用例 Sugihara et al CCM 1 Didinium nasutum Paramecium aurelia Lotka-Volterra CCM Sardinops 250

11 sagax Engraulis mordax 2 Matsuda et al Chavez et al CCM Sugihara et al Web of Science CCM Wang et al van Nes et al Fan et al Web of Science Heskamp et al Huffaker and Fearne 今後の課題 : ノイズに対する頑健性について CCM CCM CCM CCM multispatial CCM Clark et al AR a t AR(1) 9 r K t r, K 10 r = 0.5, K = 1000, e r = e K = 0 e r = e K = 2 Ricker 11 o t 12 σ p σ o 100 a t o t CCM 101 E = 2 τ = 1 L 100 L = 100 twin surrogate S ρ 5 e r = e K = 2 a t x t x t a t 4 a t x t true positive x t a t true negative 95 e r = e K = 0 a t x t x t a t true negative 95 CCM e r = e K = 2 σ p = σ o = 0 x t a t 2 Sugihara et al M X M Y 251

12 4 σ p σ o σ o = 0 σ o = 0.1 σ o = 0.2 σ o = 0.4 a a t x t b x t a t CCM CCM 6.3. CCMを用いた発展的手法 :multispatial CCM CCM Sugihara et al Clark et al CCM multispatial CCM n E n-e+1 multispatial CCM m m(n-e+1) CCM multispatial CCM CCM CREST Chih-hao Hsieh CCM Granger CCM twin surrogate 252

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リカレンスプロット : 時系列の視覚化を越えて (マクロ経済動学の非線形数理)

リカレンスプロット : 時系列の視覚化を越えて (マクロ経済動学の非線形数理) 1768 2011 150-162 150 : Recurrence plots: Beyond visualization of time series Yoshito Hirata Institute of Industrial Science, The University of Tokyo voshito@sat. t.u\cdot tokvo.ac.ip 1 1. 1987 (Eckmann