 Recurrence plot

In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for a given moment in time, the times at which a phase space trajectory visits roughly the same area in the phase space. In other words, it is a graph of
showing i on a horizontal axis and j on a vertical axis, where is a phase space trajectory.
Contents
Background
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). 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. Poincaré recurrence theorem). 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).
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. Higherdimensional phase spaces can only be visualized by projection into the two or threedimensional subspaces. However, Eckmann's tool enables us to investigate the mdimensional phase space trajectory through a twodimensional representation of its recurrences. Such recurrence of a state at time i and a different time j is pictured within a twodimensional 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 τ the time delay.
Caused by characteristic behaviour of the phase space trajectory, a recurrence plot contains typical smallscale structures, as single dots, diagonal lines and vertical/horizontal lines (or a mixture of the latter, which combines to extended clusters). The largescale 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 smallscale 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 yaxis (instead of absolute time).
The main advantage of recurrence plots is that they provide useful information even for short and nonstationary data, where other methods fail.
Extensions
Multivariate extensions of recurrence plots were developed as cross recurrence plots and joint recurrence plots.
Cross recurrence plots consider the phase space trajectories of two different systems in the same phase space (Marwan & Kurths 2002):
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 simultaneous 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 timerelationship of two similar systems, whose timescale differ (Marwan & Kurths 2005).
Joint recurrence plots are the Hadamard product of the recurrence plots of the considered subsystems (Romano et al. 2004), e.g. for two systems and the joint recurrence plot is
In contrast to cross recurrence plots, joint recurrence plots compare the simultaneous occurrence of recurrences in two (or more) systems. Moreover, the dimension of the considered phase spaces can be different, but the number of the considered states has to be the same for all the subsystems. Joint recurrence plots can be used in order to detect phase synchronisation.
Example
See also
 Recurrence quantification analysis, a heuristic approach to quantify recurrence plots.
 Recurrence period density entropy, an informationtheoretic method for summarising the recurrence properties of both deterministic and stochastic dynamical systems.
 Selfsimilarity matrix
 Poincaré plot
References
 J. P. Eckmann, S. O. Kamphorst, D. Ruelle (1987). "Recurrence Plots of Dynamical Systems". Europhysics Letters 5 (9): 973–977. Bibcode 1987EL......4..973E. doi:10.1209/02955075/4/9/004. http://iopscience.iop.org/02955075/4/9/004/.
 N. Marwan, M. C. Romano, M. Thiel, J. Kurths (2007). "Recurrence Plots for the Analysis of Complex Systems". Physics Reports 438 (56): 237. Bibcode 2007PhR...438..237M. doi:10.1016/j.physrep.2006.11.001.
 N. Marwan (2008). "A historical review of recurrence plots". The European Physical Journal  Special Topics 164 (1): 3–12. Bibcode 2008EPJST.164....3M. doi:10.1140/epjst/e2008008291. http://www.springerlink.com/content/5412256211633127/.
External links
Categories: Plots (graphics)
 Signal processing
 Dynamical systems
 Visualization (graphic)
 Chaos theory
 Scaling symmetries
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