RECURRENCE PLOTS AND CROSS RECURRENCE PLOTS

Dynamical Invariants Derived from Recurrence Plots

Correlation entropy and correlation dimension

The lengths of diagonal lines in an RP are directly related to the ratio of determinism or predictability inherent to the system. Suppose that the states at times i and jt are neighbouring, i.e. R_{i, j} = 1. If the system behaves predictably, similar situations will lead to a similar future, i.e. the probability for R_{i+1, j+1} = 1. is high. For perfectly predictable systems, this leads to infinitely long diagonal lines (like in the RP of the sine function). In contrast, if the system is stochastic, the probability for R_{i+1, j+1} = 1. will be small and we only find single points or short lines. If the system is chaotic, initially neighbouring states will diverge exponentially. The faster the divergence, i.e. the higher the Lyapunov exponent, the shorter the diagonals.

Prototypical examples of recurrence plots of a (A) stochastic system (white noise), (B) fully deterministic, oscillating system (sine function) and (C) chaotic system (Rössler oscillator in chaotic regime).

At first, we recall the definition of the second order Rényi entropy (correlation entropy). Let us consider a trajectory x(t) in a bounded d-dimensional phase space; the state of the system is measured at time intervals τ. Let {1,2,...,M(ε)} be a partition of the attractor in boxes of size ε. Then p(i₁,...,i_l) denotes the joint probability that x(t =1τ) is in the box i₁, x(t=2τ) is in the box i₂, ..., and x(t = lτ) is in the box i_l. The 2^nd order Rényi entropy is then defined as (Renyi, 1970; Grassberger & Procaccia, 1983)

$K_2 = - \lim_{\tau \rightarrow 0} \ \lim_{\varepsilon \rightarrow 0} \ \lim_{\l \rightarrow \infty} \frac{1}{l \tau} \sum_{i_1,\ldots,i_l} p^2(i_1,\ldots,i_l).$

Roughly speaking, this measure is directly related to the number of possible trajectories that the system can take for l time steps in the future. If the system is perfectly deterministic in the classical sense, there will be only one possibility for the trajectory to evolve and hence K₂=0. In contrast, for purely stochastic systems the number of possible future trajectories increases to infinity so fast, that K₂→∞. Chaotic systems are characterised by a finite value of K₂, as they belong to a category between pure deterministic and pure stochastic systems. Also in this case the number of possible trajectories diverges but not as fast as in the stochastic case. The inverse of K₂ has units of time and can be interpreted as the mean prediction time of the system.

The sum of the probabilities p(i₁,...,i_l) can be approximated by the probability p_t(l) of finding a sequence of l points in boxes of size ε centred at the points x(t), …, x(t+(l-1)) :

$\frac{1}{N}\sum_{t=1}^N p_{i_1(t),\ldots,i_l(t+ (l-1) \Delta t)} \approx \frac{1}{N}\sum_{t=1}^N p_t(l).$

Moreover, p_t(l) can be expressed by means of the recurrence matrix

$p_t(l) = \lim_{N \to \infty}\frac{1}{N}\sum_{s=1}^N \prod_{k=0}^{l-1}\mathbf{R}_{t+k,s+k}.$

Using these relation, we find an estimator for the second order Réenyi entropy by means of the RP (Thiel et al., 2003)

$K_2(l)= -\frac{1}{l\,\Delta t} \ln \left( p_c(l) \right)=-\frac{1}{l\,\Delta t} \ln \left(\frac{1}{N^2}\sum_{t,s=1}^N \prod_{k=0}^{l-1} \mathbf{R}_{t+k,s+k}\right),$

where p^c(l) is the probability to find a diagonal of at least length l in the RP.

On the other hand, the l-dimensional correlation sum can be used to define K₂ (Grassberger & Procaccia, 1983a). This definition of K₂ can also be expressed by means of RPs and yields the following fundamental relationship (Thiel et al., 2003):

$\ln p^c(l) \sim \varepsilon^{D_2} e^{-K_2(\varepsilon)\tau},$

where p^c(l) is the cumulative probability distribution of diagonal lines in the RP, i.e. the probability of finding a diagonal in the RP of at least length l. D₂ is the correlation dimension of the system under consideration (Grassberger & Procaccia, 1983). Therefore, in a logarithmic presentation of p^c(l) over l the slope of the lines corresponds to –K₂τ for large l, which is independent of ε for a rather large range in ε.

If we represent the slope of the curves for large l in dependence on ε a plateau can be found for chaotic systems. The value of this plateau determines K₂. If the system is not chaotic, we have to consider the value of the slope for a sufficiently small value of ε.

The relationship between K₂ and RPs also allows to estimate D₂ from p^c(l). Considering the relationship between K₂ and RPs for two different thresholds ε and ε + Δε and dividing both of them, we get

$D_2(\varepsilon) = \frac{\ln\left(\frac{p_c(\varepsilon,l)}{p_c(\varepsilon+\Delta\varepsilon,l)}\right)}{ \ln\left(\frac{\varepsilon}{\varepsilon+\Delta\varepsilon}\right)},$

which is an estimator of the correlation dimension D₂ (Grassberger, 1983).

K2 and D2 calculated by means of RPs for the Bernoulli map.

K₂ and D₂ examplary calculated for the chaotic Bernoulli map, x_n+1 = 2x_nmod(1). (A) Total number of diagonal lines of at least length l in the RP of the Bernoulli map. Each histogram is computed for a different threshold ε, from 0.000436 (bottom) to 0.0247 (top). 10,000 data points have been used for the computation. (B) Estimate of K₂ in dependence on ε. We find K₂ = 0.6733, which is in good accordance with the values found by others (e.g. Ott, 1993). (C) Estimate of D₂ in dependence on ε. We obtain a value of 0.9930, which is also close to the theoretical value of D₂=1.

Generalised mutual information (generalised redundancies)

The mutual information quantifies the amount of information that we obtain from the measurement of one variable on another. It has become a widely applied measure to quantify dependencies within or between time series (auto and cross mutual information). The time delayed generalised mutual information (redundancy) I_q(τ) of a system x_i is defined by (Rényi, 1970)

$I_q^{\vec x}(\tau) = 2 H_q - H_q(\tau).$

H_q is the q^th-order Rényi entropy of x_i and H_q(τ) is the q^th-order joint Rényi entropy of x_i and x_i+τ

$H_q=-\ln\sum\limits_{k}p_k^q,\qquad H_q(\tau)=-\ln\sum\limits_{k,l}p_{k,l}^q(\tau),$

where p_k is the probability that the trajectory visits the k^th box and p_k,l(τ) is the joint probability that x_i is in box k and x_i+τ is in box l. Hence, for the case q=2 we can use the recurrence matrix to estimate H₂

$H_2=-\ln \left(\frac{1}{N^2}\sum_{i,j=1}^N\mathbf R_{i,j}\right)$

and H_q(τ)

$H_2(\tau) = -\ln \left(\frac{1}{N^2}\sum_{i,j=1}^N\mathbf R_{i,j}\mathbf R_{i+\tau,j+\tau}\right) = -\ln \left(\frac{1}{N^2}\sum_{i,j=1}^N\mathbf{JR}^{\vec x, \vec x}_{i,j}(\tau)\right),$

where JR_{i, j}(τ) denotes the delayed joint recurrence matrix. Then, the second order generalised mutual information can be estimated by means of RPs (Thiel et al., 2003)

$I_2^{\vec x}(\tau)= \ln \left(\frac{1}{N^2}\sum\limits_{i,j=1}^N \mathbf{JR}_{i,j}^{\vec x, \vec x}(\tau)\right) - 2 \ln \left(\frac{1}{N^2} \sum\limits_{i,j=1}^N \mathbf{R}_{i,j}\right).$

References

P. Grassberger & I. Procaccia, Measuring the strangeness of strange attractors, Physica D, 9(1-2), 189-208, 1983.
P. Grassberger & I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal, Physical Review A, 9(1-2), 2591-2593, 1983a.
Grassberger, P., Generalized Dimensions of Strange Attractors, Physics Letters A, 97(6), 227-230, 1983.
E. Ott, E. 1993, Chaos in Dynamical Systems, Cambridge University Press
A. Rényi, Probability theory, North-Holland (appendix), 1970
M. Thiel, M. C. Romano & J. Kurths, Analytical Description of Recurrence Plots of white noise and chaotic processes, Izvestija vyssich ucebnych zavedenij/ Prikladnaja nelinejnaja dinamika - Applied Nonlinear Dynamics, 11(3), 20-30, 2003.
N. Marwan, M. C. Romano, M. Thiel & J. Kurths, Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329, 2007.

References

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