Recurrence Plot - Detailed Description

Detailed Description

Eckmann et al. (1987) introduced recurrence plots, which can visualize the recurrence of states in a phase space. Usually, a phase space does not have a low enough dimension (two or three) to be pictured. Higher-dimensional phase spaces can only be visualized by projection into the two or three-dimensional sub-spaces. However, Eckmann's tool enables us to investigate the m-dimensional phase space trajectory through a two-dimensional representation of its recurrences. Such recurrence of a state at time i and a different time j is pictured within a two-dimensional squared matrix with black and white dots, where black dots mark a recurrence, and both axes are time axes. This representation is called recurrence plot. Such a recurrence plot can be mathematically expressed as

where N is the number of considered states, ε is a threshold distance, || · || a norm (e.g. Euclidean norm) and H the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem):

where u(i) is the time series, m the embedding dimension and the time delay.

Caused by characteristic behaviour of the phase space trajectory, a recurrence plot contains typical small-scale structures, as single dots, diagonal lines and vertical/horizontal lines (or a mixture of the latter, which combines to extended clusters). The large-scale structure, also called texture, can be visually characterised by homogenous, periodic, drift or disrupted. The visual appearance of an RP gives hints about the dynamics of the system.

The small-scale structures in RPs are used by the recurrence quantification analysis (Zbilut & Webber 1992; Marwan et al. 2002). This quantification allows to describe the RPs in a quantitative way, and to study transitions or nonlinear parameters of the system. In contrast to the heuristic approach of the recurrence quantification analysis, which depends on the choice of the embedding parameters, some dynamical invariants as correlation dimension, K2 entropy or mutual information, which are independent on the embedding, can also be derived from recurrence plots. The base for these dynamical invariants are the recurrence rate and the distribution of the lengths of the diagonal lines.

Close returns plots are similar to recurrence plots. The difference is that the relative time between recurrences is used for the -axis (instead of absolute time).

The main advantage of recurrence plots is that they provide useful information even for short and non-stationary data, where other methods fail.