Recurrence quantification analysis
Recurrence quantification analysis (RQA) is a method of
nonlinear data analysis(cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase spacetrajectory.
The recurrence quantification analysis was developed in order to quantify differently appearing
recurrence plots (RPs) based on the small-scale structures therein. Recurrence plots are tools which visualise the recurrence behaviour of the phase spacetrajectory of dynamical systems. They mostly contain single dots and lines which are parallel to the mean diagonal ("line of identity", LOI) or which are vertical/horizontal. Lines parallel to the LOI are referred to as "diagonal lines" and the vertical structures as "vertical lines". Because an RP is usually symmetric, horizontal and vertical lines correspond to each other, and, hence, only vertical lines are considered. The lines correspond to a typical behaviour of the phase space trajectory: whereas the diagonal lines represent such segments of the phase space trajectory which run parallel for some time, the vertical lines represent segments which remain in the same phase spaceregion for some time.
If only a
time seriesis available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem):
where is the time series, the embedding dimension and the time delay.
The RQA quantifies the small-scale structures of recurrence plots, which present the number and duration of the recurrences of a dynamical system. The measures introduced for the RQA were developed heuristically between 1992 and 2002 (Zbilut & Webber 1992; Webber & Zbilut 1994; Marwan et al. 2002). They are actually
measures of complexity. The main advantage of the recurrence quantification analysis is that it can provide useful information even for short and non-stationary data, where other methods fail.
RQA can be applied to almost every kind of data. It is widely in
physiology, but was also successfully applied on problems from engineering, chemistry, Earth sciencesetc.
The simplest measure is the recurrence rate, which is the density of recurrence points in an RP:
The recurrence rate corresponds with the probability that a specific state will recur. It is almost equal with the definition of the
correlation sum, where the LOI is excluded from the computation.
The next measure is the percentage of recurrence points which diagonal lines in the RP of minimal length :
where is the
frequency distributionof the lengths of the diagonal lines. This measures is called determinism and is related with the predictabilityof the dynamical system, because white noisehas an RP with almost only single dots and very few diagonal lines, whereas a deterministic processhas an RP with very few single dots but many long diagonal lines.
The amount of recurrence points which form vertical lines can be quantified in the same way:
where is the frequency distribution of the lengths of the vertical lines, which have at least a length of . This measure is called laminarity and related with the amount of
laminar phases in the system ( intermittency).
The lengths of the diagonal and vertical lines can be measured as well. The averaged diagonal line length
is related with the "predictability time" of the dynamical systemand the trapping time, measuring the average lengthof the vertical lines,
is related with the "laminarity time" of the dynamical system, i.e. how long the system remains in a specific state.
Because the length of the diagonal lines is related on the time how long segments of the
phase spacetrajectory run parallel, i.e. on the divergencebehaviour of the trajectories, it was sometimes stated that the reciprocalof the maximal length of the diagonal lines (without LOI) would be an estimator for the positive maximal Lyapunov exponentof the dynamical system. Therefore, the maximal diagonal line length or the divergence
are also measures of the RQA. However, the relationship between these measures with the positive maximal Lyapunov exponent is not as easy as stated, but even more complex (to calculate the Lyapunov exponent from an RP, the whole frequency distribution of the diagonal lines has to be considered). The divergence can have the trend of the positive maximal Lyapunov exponent, but not more. Moreover, also RPs of white noise processes can have a really long diagonal line, although very seldom, just by a finite probability. It is obvious that therefore the divergence cannot reflect the maximal Lyapunov exponent.
probabilitythat a diagonal line has exactly length can be estimated from the frequency distribution with . The Shannon entropyof this probability,
reflects the complexity of the deterministic structure in the system. However, this entropy depends sensitively on the bin number and, thus, may differ for different realisations of the same process, as well as for different data preparations.
The last measure of the RQA quantifies the paling of the RP. The trend is the number of the linear relationship between the density of recurrence points in a line parallel to the LOI and its distance to the LOI. More exact, let us consider the recurrence rate in a diagonal line parallel to LOI of distance ("diagonal-wise recurrence rate"):
then the trend is defined by
with as the average value and . This latter relation should ensure to avoid the edge effects of too low recurrence point densities in the edges of the RP. The measure "trend" provides information about the stationarity of the system.
Similar to the diagonal-wise defined recurrence rate, the other measures based on the diagonal lines (DET, L, ENTR) can be defined diagonal-wise. These definitions are useful to study interrelations or
synchronisationbetween different systems (using recurrence plots or cross recurrence plots).
Instead of computing the RQA measures of the entire recurrence plot, they can be computed in small windows moving over the recurrence plot along the LOI. This provides time-dependent RQA measures which are able to detect, e.g., chaos-chaos transitions (Marwan et al. 2002). Note: the choice of the size of the window can strongly influence the measure "trend".
Recurrence plot, a powerful visualisation tool of recurrences in dynamical (and other) systems.
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