Recurrence quantification analysis
It quantifies the number and duration of recurrences of a dynamical system presented by its phase space trajectory.
[1] The recurrence quantification analysis (RQA) was developed in order to quantify differently appearing recurrence plots (RPs), based on the small-scale structures therein.
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 space region for some time.
is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem): where
However, pPhase space reconstruction is not essential part of the RQA (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 main advantage of the RQA is that it can provide useful information even for short and non-stationary data, where other methods fail.
It is widely used in physiology, but was also successfully applied on problems from engineering, chemistry, Earth sciences etc.
[2] Further extensions and variations of measures for quantifying recurrence properties have been proposed to address specific research questions.
RQA measures are also combined with machine learning approaches for classification tasks.
It is almost equal with the definition of the correlation sum, where the LOI is excluded from the computation.
This measure is called determinism and is related with the predictability of the dynamical system, because white noise has a recurrence plot with almost only single dots and very few diagonal lines, whereas a deterministic process has a recurrence plot with very few single dots but many long diagonal lines.
The number of recurrence points which form vertical lines can be quantified in the same way:[6] where
The averaged diagonal line length[5] is related with the predictability time of the dynamical system and the trapping time, measuring the average length of the vertical lines,[6] is related with the laminarity time of the dynamical system, i.e. how long the system remains in a specific state.
[6] Because the length of the diagonal lines is related on the time how long segments of the phase space trajectory run parallel, i.e. on the divergence behaviour of the trajectories, it was sometimes stated that the reciprocal of the maximal length of the diagonal lines (without LOI) would be an estimator for the positive maximal Lyapunov exponent of the dynamical system.
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.
The Shannon entropy of this probability,[5] reflects the complexity of the deterministic structure in the system.
The last measure of the RQA quantifies the thinning-out of the recurrence plot.
The trend is the regression coefficient of a linear relationship between the density of recurrence points in a line parallel to the LOI and its distance to the LOI.
This latter relation should ensure to avoid the edge effects of too low recurrence point densities in the edges of the recurrence plot.
The measure trend provides information about the stationarity of the system.
-recurrence rate, the other measures based on the diagonal lines (DET, L, ENTR) can be defined diagonal-wise.
[8] 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 allow detecting, e.g., chaos-chaos transitions.
[9][1] Note: the choice of the size of the window can strongly influence the measure trend.
