Recurrence plot
is the state of the system (or its phase space trajectory).
Natural processes can have a distinct recurrent behaviour, e.g. periodicities (as seasonal or Milankovich cycles), but also irregular cyclicities (as El Niño Southern Oscillation, heart beat intervals).
Moreover, the recurrence of states, in the meaning that states are again arbitrarily close after some time of divergence, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems (cf.
The recurrence of states in nature has been known for a long time and has also been discussed in early work (e.g. Henri Poincaré 1890).
One way to visualize the recurring nature of states by their trajectory through a phase space is the recurrence plot, introduced by Eckmann et al.
[1] Often, the phase space does not have a low enough dimension (two or three) to be pictured, since higher-dimensional phase spaces can only be visualized by projection into the two or three-dimensional sub-spaces.
One frequently used tool to study the behaviour of such phase space trajectories is then the Poincaré map.
Another tool, is the recurrence plot, which enables us to investigate many aspects of the m-dimensional phase space trajectory through a two-dimensional representation.
At a recurrence the trajectory returns to a location (state) in phase space it has visited before up to a small error
Mathematically, this is expressed by the binary recurrence matrix where
An alternative, more formal expression is using the Heaviside step function
Alternative recurrence definitions consider different distances
is available, the phase space can be reconstructed, e.g., by using a time delay embedding (see Takens' theorem): where
However, phase space reconstruction is not essential part of the recurrence plot (although often stated in literature), because it is based on phase space trajectories which could be derived from the system's variables directly (e.g., from the three variables of the Lorenz system) or from multivariate data.
The visual appearance of a recurrence plot gives hints about the dynamics of the system.
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 small-scale structures in recurrence plots contain information about certain characteristics of the dynamics of the underlying system.
For example, the length of the diagonal lines visible in the recurrence plot are related to the divergence of phase space trajectories, thus, can represent information about the chaoticity.
[3] Therefore, the recurrence quantification analysis quantifies the distribution of these small-scale structures.
[4][5][6] This quantification can be used to describe the recurrence plots in a quantitative way.
Applications are classification, predictions, nonlinear parameter estimation, and transition analysis.
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.
[3] More recent applications use recurrence plots as a tool for time series imaging in machine learning approaches and studying spatio-temporal recurrences.
[6] The main advantage of recurrence plots is that they provide useful information even for short and non-stationary data, where other methods fail.
Cross recurrence plots consider the phase space trajectories of two different systems in the same phase space:[7] The dimension of both systems must be the same, but the number of considered states (i.e. data length) can be different.
Cross recurrence plots compare the occurrences of similar states of two systems.
They can be used in order to analyse the similarity of the dynamical evolution between two different systems, to look for similar matching patterns in two systems, or to study the time-relationship of two similar systems, whose time-scale differ.
Joint recurrence plots can be used in order to detect phase synchronisation.
