RECURRENCE PLOTS AND CROSS RECURRENCE PLOTS

Introduction To Cross and Joint Recurrence Plots

Googlefight CRP vs. JRP

Note: The following text is from 2000 and still has to be revised.

Following Takens' embedding theorem (1981) we reconstruct the phase space from a time series u_k by using an embedding dimension m and a time delay τ

x(t) = x_i = ( u_i, u_i+τ, …, u_i+(m-1)τ ),     t = iΔt,

whereas x(t) is the vector of reconstructed states in the phase-space at the time t. The choice of m and τ should base on usually methods for detecting these parameters like method of false nearest neighbours (for m) and mutual information (for τ), which ensures the entire covering of all free parameters and avoiding autocorrelated effects (Kantz and Schreiber, 1997). Eckmann et al. (1987) introduced the recurrence plot R_i,j (Fig. 1). This is an square array, that visualizes when a state in the phase space will nearly recur. Every point of the phase space trajectory x_i ( i = 1 … N; N = n − (m − 1) τ  ) is tested whether it is close to another point of the trajectory x_j, i.e. the distance between these two points is less than a specified threshold ε. In this case the value one (a black dot, recurrence point) in a N × N-array

R_{i, j} = Θ ( ε_i − || x_i − x_j||)

(Θ(⋅) is the Heaviside-function) is placed otherwise the value zero (a white dot) is placed. Such recurrence plot represents the behaviour of a single trajectory in the phase space with typical long-scale and small-scale structures. A recurrence point contains no information about the special state.

Figure 1: Recurrence plot of the Southern Oscillation Index (SOI). The diagonal structures represent the deterministic dynamics of the El Ni no/ Southern Oscillation (ENSO). From the distance between diagonal structures one can reveal cyclicities of recurrence of a especially state of ENSO.

As an extension of these recurrence plots, the recurrence quantification analysis (RQA), was introduced by Zbilut and Webber (1992). It defines measures for diagonal segments in a recurrence plot, recurrence rate, determinism, averaged length of diagonal structures, entropy and trend. The recurrence rate is the ratio of all recurrent states (recurrence points) to all possible states and is the probability of recurrence of a special state. Stochastic behaviour causes very short diagonals, whereas deterministic behaviour causes longer diagonals. Therefore, the ratio of recurrence points forming diagonal structures to all recurrence points is a measure for the determinism in the system. Diagonal structures show the range in which a piece of the trajectory is rather close to another piece of the trajectory at different time. The diagonal length is the time how long they will be close to each other and can be interpreted as the mean prediction time. The inverse of this measure is correlated with the Lyapunov-exponent (Eckmann, 1987). The entropy means the Shannon entropy of the histogram of line segment lengths and reflects the complexity of the deterministic structure in the system. Stationary systems will deliver homogeneous recurrence plots, instationary systems causes changes in the distribution of recurrence points in the plot which is visible by brightened areas. For example, a simple drift causes a brightening of the recurrence plot away from the main diagonal to edges. The value trend measures this effect by diagonalwise computation of diagonal recurrence density and its linear relation to the time distance of these diagonals to the main diagonal.

Analogous to Webber and Zbilut (1998) we expanded the method of recurrence plot to the method of cross recurrence plot, which compares the dynamical behaviour of two time series, which are embedded in phase space (Marwan, 1999). Here two time series are simultaneously embedded in the phase space. The test for closeness of each point of first trajecory x_i (i = 1 … N_x) with each point of second trajectory y_j (j = 1 … N_y) results in a N_x × N_y-array

CR_{i, j} = Θ ( ε_i − || x_i − y_j||),    x_i, y_i, ∈ ℜ^m,

which is called cross recurrence plot. Long diagonal structures show similar phase space behaviour of both time series. We have modified some RQA measures for quantifying the phase space similarity. The recurrence rate RR, determinism DET and the averaged diagonal line length L were determined as a function of the distance from the main diagonal, e.g. the diagonal-wise recurrence rate would be

It is therefore possible to assess the similarity in dynamics depending on a certain delay. The second time series was embedded with positive and negative sign, what gives us the type of a possible correlation.

An example illustrates the application of our method. We use a linear correlated noise process (auto-regressive process) and couple it non-linear with the x-component of a Lorenz-System x_n (the quadratic exponent in the coupling term) to a second order autoregressive process (Fig. 2)

y_n = 1.095 y_{n − 1} − 0.400 y_{n − 2} + 0.700 ξ_n + 0.300 x_n²,

(where ξ is white noise and x is normalized to standard deviation σ). The coupling is realized without any lag.

Figure 2: Time series of a nonlinear related system consisting of a driven second-order autoregressive process, forced by the squared x-component of the Lorenz system.

The linear correlation analysis is not able to detect any significant coupling or correlation (Fig. 3b). In contrast, our new method finds a positive coupling with a lag of zero (Figs. 3c-e).

Figure 3a: Cross recurrence plot between the driven second-order autoregressive process and the forcing function (x-component of the Lorenz system).

Figure 3b-e: Crosscorrelation (b), RR (c), DET (d) and L (e) between the driven second-order autoregressive process and the forcing function (x-component of the Lorenz system). The solid lines show positive relation, dashed lines show negative relation. The point-dashed lines represent the significance levels. The crosscorrelation function find none significant correlation but the RR, DET and L functions show a positive relation with a lag of zero.

We prepare a test for significance with a set of N surrogate data. On the one hand these surrogates should reveal some features like our original data but on the other hand features which represent the randomness of possible correlation (stochastic processes). Linear correlated noise are such systems. We calculate a new surrogate time series for such system with the following iteration function (autoregressive process of 3rd order)

x_n = a₁ x_{n − 1} + a₂ x_{n − 2} + a₃ x_{n − 3} + b ξ_n ,

whereas ξ is white noise, n runs from 4 to the value of the desired length of the time series and a_k are coefficients which determine the behaviour of the system and allow us to adapt this stochastic system to our natural systems. After this calculation we apply our method to the natural data (e.g. SOI) and to the surrogate data and determine the introduced measures (RR, DET, L). We repeat this procedure (calculation of surrogate data and application of the method) for N-times. The result is a distribution of our measures for the unrelated systems. For each measure we obtain the (p-1)-quantiles for the error level of p (e.g. 10%), which is the one-side significance level for this error level. With this significance levels we assess the significance of the revealed measures of CRP and the interrelations of the natural systems.

An further interesting feature of cross recurrence plots is the line of synchronization (LOS). This line reveals the relationship of the both systems in the time domain.

Figure 4: Lines of synchronization in cross recurrence plots between harmonic functions, where the second function has a modification in the time domain (black: no modification, blue: small variation, red: stronger variation in the time domain of the second function). animated plot

A fitting of this LOS with a nonparametric function allows to re-synchronize both systems. However, both systems should have a similar dynamical evolution. Applications e. g. in adjustment of geophysical data which were gained from different boreholes or cores.

Another multivariate approach to recurrence plots was introduced by Romano et al. and is called joint recurrence plot (2004), and is simply the product of two (or more) recurrence plots of the data series

JR_{i, j} = Θ ( ε_x − || x_i − x_j||) ⋅ Θ ( ε_y − || y_i − y_j||),    x_i, ∈ ℜ^m,    y_i ∈ ℜⁿ,    i, j=1…N.

Such joint recurrence plots have the advantage, that the data can be different observables and can have different magnitudes. They can be used for the detection of phase synchronisation.

References

• CASDAGLI, M. C.: Recurrence plots revisited. In: Physica D 108 (1997), 12-44

• ECKMANN, J.-P., S. Oliffson Kamphorst, D. Ruelle: Recurrence Plots of Dynamical Systems. In: Europhysics Letters 4 (1987), 973-977

• MARWAN, N., M. Thiel, N. R. Nowaczyk: Cross Recurrence Plot Based Synchronization of Time Series. In: Nonlinear Processes in Geophysics 9 (2002), 325-331

• MARWAN, N., J. Kurths: Nonlinear analysis of bivariate data with cross recurrence plots. In: Physics Letters A 302 (2003), 299-307

• MCGUIRE, G., N. B.Azar, M.Shelhamer: Recurrence matrices and the preservation of dynamical properties. In: Physics Letters A 237 (1997), 43-47

• ROMANO, M., Thiel, M., Kurths, J., von Bloh, W.: Multivariate Recurrence Plots. In: Physics Letters A 330 (2004), 214-223

• SHOCKLEY, K., M. Butwill, J. P. Zbilut, C. L. Webber Jr.: Cross recurrence quantification of coupled oscillators. In: Physics Letters A 305 (2002), 59-69

• ZBILUT, J. P., C. L. Webber Jr.: Detecting deterministic signals in exceptionally noisy environments using cross-recurrence quantification. In: Physics Letters A 246 (1998), 122-128

Further references about RPs, RQA and their applications.

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