Recurrence plot
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In descriptive statistics, a recurrence plot (RP) is a graph of
- <math>x(i)\approx x(j),\,<math>
showing i on a horizontal axis and j on a vertical axis, where x is a time series.
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 arbitrarily close after some time, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems. The recurrence of states in nature has been known for a long time and has also been discussed in early publications (e.g. recurrence phenomena in cosmic-ray intensity, Monk 1939).
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 <math>i<math> at a different time <math>j<math> 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 recurrence plot. Such an RP can be mathematically expressed as
- <math>R(i,j) = \Theta(\varepsilon - || \vec{x}(i) - \vec{x}(j)||), \quad \vec{x}(i) \in \mathbf{R}^m, \quad i, j=1, \dots, N,<math>
where <math>N<math> is the number of considered states <math>\vec{x}(i)<math>, <math>\varepsilon<math> is a threshold distance, <math>|| \cdot ||<math> a norm (e.g. Euclidean norm) and <math>\Theta( \cdot )<math> the Heaviside step function.
Close returns plots are similar to recurrence plots. The difference is that the relative time between recurrences is used for the y-axis (instead of absolute time).
Recent developments allow the quantification of recurrence plots (Zbilut et al., 1998) and, thus, the study of transitions or interesting nonlinear parameters in the data.
RP-daily_sunshine_duration_Den_Helder.png
References
- Eckmann J.P., Kamphorst S.O., Ruelle D. (1998) "Recurrence Plots of Dynamical Systems", Europhysics Letters, 5, 973-977 (1987).
- Zbilut JP, Giuliani A, Webber CL Jr (1998) "Detecting deterministic signals in exceptionally noisy environments using cross-recurrence quantification", Physics Letters A 246: 122–128.