Recurrence plot

A recurrence plot (from the Latin recurrere "recurring") is a modern method of nonlinear data analysis . The return property is typical of deterministic dynamic systems ( chaos , non-linear dynamics ) and is also reflected in many natural processes, such as B. El Niño , Milanković cycles or sunspot cycles . Recurrence does not mean that the original state occurs exactly again, but that it is just hit again as precisely as required. Already Poincaré had postulated the infinite recurrence of states.

The method of recurrence plots was introduced by Eckmann in 1987. It is used to represent higher-dimensional phase space trajectories .

description

The recurrence plot is a square matrix with two time axes. In this matrix those time pairs are represented by black dots, the states of which are almost the same, i.e. H. when the corresponding state has returned. The recurrence is usually determined from the distance between all pairings of the dates:

{\ displaystyle R (i, j) = \ Theta (\ varepsilon - || {\ vec {x}} (i) - {\ vec {x}} (j) ||).}

Here is the Heaviside function , the maximum distance and a norm , for example the Euclidean norm . ${\ displaystyle \ Theta}$ ${\ displaystyle \ varepsilon}$ ${\ displaystyle || \ cdot ||}$

The appearance of the recurrence plot is determined by the behavior of the phase space trajectory. A distinction is made between the small structures, such as individual points, diagonal lines or vertical lines, and the overall impression of the plot (texture).

Typical examples of recurrence plots (top row: time series; bottom row: associated recurrence plots). From left to right: uncorrelated stochastic noise (white noise), harmonic oscillation with two frequencies, chaos with a linear trend (logistic mapping) and auto-regressive process.

Recent developments allow a more extensive investigation of data through a quantitative evaluation of recurrence plots (Zbilut and Webber, 1992; Marwan et al., 2002).

A close returns plot is a recurrence plot with a slightly different way of plotting the return times. Here the y-axis does not correspond to the absolute time, but to the time difference (i.e. after which time the state returns).

Recurrence plot of the El Niño index.

literature

JP Eckmann, SO Kamphorst, D. Ruelle: Recurrence Plots of Dynamical Systems . In: Europhysics Letters . 5, 1987, pp. 973-977. doi : 10.1209 / 0295-5075 / 4/9/004 .
N. Marwan, MC Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems . In: Physics Reports . 438, No. 5-6, 2007. doi : 10.1016 / j.physrep.2006.11.001 .
N. Marwan: A historical review of recurrence plots . In: The European Physical Journal - Special Topics . 164, No. 1, 2008, pp. 3-12. doi : 10.1140 / epjst / e2008-00829-1 .

Web links

Recurrence Plot website