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 et al. (1987); Marwan et al. (2007)) 2 $2$ $M$ $d$ $M$ $i$ $x(i)\in M$ $R(i,j)=\{\begin{array}{l}1, d(x(i),x(j))<r(i,j)0, otherwise.\end{array}$ $R(i,j)=1$ $(i,j)$ $R(i,j)=0$ $(i,j)$ $r(i,j)$ Zbi lut and $b^{l}ebber$ (1992) $r(i,j)=r$ Eckmann et al, (1987) $r(i, j)$ $i$ $k$
$0$ 50 $#_{l}^{y_{r^{1}}}i$ $\grave$ 151 $0$ $0$ 100 50 100 50 $0$ 60 100 $0$ 50 100 $0$ 50 100 $\not\in \#_{\iota^{ }}$ $\Re,\hslash_{\langle}J$ $*t_{\wedge} $ 1: ( ) ( ) $M$ $d$ 1 2 $d(x(i),x(j))= x(i)-x(j) $ $($ $1)_{\text{ }}$ 2 $M$ $n\iota$ $x_{k}(i)$ $x(i)$ $k$ $d$
152 $d(x(\dagger),x(j))=j\leqq\sqrt{\sum_{k\overline{-}1}^{t1l}(x_{k}(i)-x_{k}(j))-}$ $\circ$ 2 (Hirata et al. (2008) ;1 Thiel et al. $(2004a))$ $\circ$ (Faure and (1998) :Thiel et al. $(2004b)$ ) 2 3 4 5 6 7 2. 1992 (Webber and Zbilut, 1994; Marwan et al., 2002; Marwan et al., 2009) 3 1 (Webber and Zbilut, $1992)$ $2$ $($Marwan et al., $2002)$ $3$ (Marwan et al., 2009) $l$ $D(l)=\{(i, j),$ $f=1,2,$ $\ldots,$ $n-1,$ $j=i+1,l+2,\ldots,$ $n (1-R(i-1,j-1))(1-R(l+l, j+t)) \prod_{k\underline{-}0}^{l-1}r(i+k, j+k)=1\}$ 4 1 (DET) $\sum l D(l) $ $DET= \frac{l\geq 2}{\sum_{l\geq 1}l D(l) }$
$ $ A 153 $ $ A DET $L$ 2 $\sum l D(l) $ $L= \frac{/\geq 1}{\sum_{\prime\geq 1} D(l) }$ $L$ 3 $L_{mas}= \max\{l D(l)\neq\emptyset\}$ $L_{1,\iota_{t}\tau x}$ $\emptyset$. 4 $p(l)= D(l) / \sum_{/\geq 1} D(l) $ $ENTR=- \sum_{l\geq 1}p(l)\log p(l)$ 3. 1 3.1 (Hirata and Aihara, $\circ$ 2011) $p$ $p^{2}$ 2
154 $\mathfrak{l}2_{(l}$ 2 2 lyt$d= \frac{1}{2}(n-1)(n-2)$ $p^{2\text{ }}$ $n_{d}$ $m_{d}$ 2 2 $m_{d}$ $m_{d}p^{2\text{ }}$ $m_{c},p^{2}(1-p^{2})$ $z_{d}= \frac{n_{d}-m_{d}p^{2}}{\sqrt{m_{d}p^{2}(1-p^{2})}}$ $2_{/1}$ 1 $0$ $Z_{d}$ 1 $P$ $0.58$ $p$ $0.001$ 3.2 Devaney (Devaney, l989) (Hirata and Aihara, $2010a$ ) Devaney Hirata and Aihara (2010b) 1
155 4. (Casdagli, 1997; Stark, 1999; Hegger et al., 2000) $i$ $1997)_{\text{ }}$ $($Casdagl, Hirata et al. (2008) Tanio et al. (2009) Lorenz 63 Henon 2 Lorenz 63 Henon 10 3 Lorenz 63 Hirata et al. (2008) 2 Lorenz 63
$0$ 5 156 10 15 20 2: Lorenz 63 ( ) ( ) 25 20 15 10 5 $0$ $0$ 5 10 15 20 25 3 :Lorenz 63 Henon
157 5. 2 1 $($Zbilut et al., 1998; $2002)_{\text{ }}$ Marwan and Kurths, $M$ $i$ 2 $x(i),y(i)\in M$ $C(i,j)=\{\begin{array}{l}1, d(x(i),y(j))<r(i,j)0, otherwise.\end{array}$ 2 $\circ$ 1 (Romano et al., 2004) 2 2 2 $x_{1}(i)\in M_{1}$, $x_{2}(i)\in M_{2}$ $R_{1}(j,j),$ $R_{2}(i,j)$ $J(i,j)=R_{1}(i,j)R_{2}(l, j)$ 2 Hirata and Aihara (2010b) 3 2 3 6. (Victor and Purpura, 1997; Hirata and Aihara, 2009; Suzuki et al., 2010) (Suzuki et al., 2010)
158 2 ( 4 ) 3 1 ( ) Victor and Purpura(1997) (Suzuki et al., 2010) time 4: 2
$\cross$ 159 $k$ 5 :Rossler ( ) ( ) 5 Rossler 6 2 3. 2 Devaney consistent
M. 160 $0$ $\}$ $t\mathfrak{d}$ 0.. $\infty$ $\Re$ 6:Rossler ( ) ( ) 7 3 1 4 5 6 (B) 21700249 C. Casdagli: Recurrence plots revisited, Physica $D,$ $108,12-44$ (1997). R. L. Devaney: An Introduction to Chaotic Dynamical Systems, $Addison\cdot Wesley$, Reading, Massachusetts, 1989. $J.\cdot P$. Eckmann, S. Oliffson Kamphorst, D. Ruelle: Recurrence plots of dynamical systems, Europhysics Letters, 5, 973-977 (1987). P. Faure, H. Korn: A new method to estimate the Kolmogorov entropy from recurrence plots: its application to neuronal signals, Physica $D,$ $122,265-279$ (1998). R. Hegger, H. Kantz, L. Matassini, T. Schreiber: Coping with nonstationarity by
Y. 161 overembedding, Physical Review Letters, 84, 4092-4095 (2000). Y. Hirata, S. Horai, K. Aihara: Reproduction of distance matrices from recurrence plots and its applications, European Physical Journal-Special Topics, 164, (2008). $13\cdot 22$ Y. Hirata, K. Aihara: Representing spike trains using constant sampling intervals, Journal of Neuroscience Methods, 183, $277\cdot 286$ (2009). Hirata, K. Aihara: Devaney s chaos on recurrence plots, Physical Review $E,$ $82$, 036209 (2010a). M. C. Romano, M. Thiel, J. Kurths, W. von Bloh: Multivariate recurrence plots, Physics Letters $A,$ $330,214-223$ (2004). J. Stark: Delay embeddings for forced systems. I. Deterministic forcing, Journal of Nonlinear Science, 9, $255\cdot 332$ (1999). S. Suzuki, Y. Hirata, K. Aihara: Definition of distance for marked point process data and its application to recurrence $plot\cdot based$ analysis of exchange tick data of foreign currencies, International Journal of Bifurcation and Chaos, 20, $3699\cdot 3708$ (2010). 1 M. Tanio, Y. Hirata, H. Suzuki: Reconstruction of driving forces through recurrence plots, Physics Letters $A,$ $373,2031-2040$ (2009). M. Thiel, M. C. Romano, J. Kurths: How much information is contained in a recurrence plot?, Physics Letters $A,$ $380,343\cdot 349(2004a)$. M. Thiel, M. C. Romano, P. L. Read, J. Kurths: Estimation of dynamical invariants without embedding by recurrence plots, Chaos, 14, 234-243 (2004b). J. Victor, K. $Pui\sim pura:metric\cdot space$ analysis of spike trains: theory, algorithms and
162 $\cdot$ application, Network 8, 127 164 (1997). C. L. Webber Jr., J. P. Zbilut: Dynamical assessment ofphysiological systems and states using recurrence plot strategies, Journal ofapplied Physiology, 76, (1994). $965\cdot 973$ J. P. Zbilut, C. L. Webber Jr.: Embeddings and delays as derived from quantification of recurrence plots, Physics Letters $A,$ $171,199-203$ (1992). J. P. Zbilut, A. Giuliani, C. L. Webber Jr.: Detecting deterministic signals in exceptionally noisy environments using $cross\cdot recurrence$ quantification, Physics Letters $A,$ $246,122-128$ (1998).