The simplest measure is the recurrence rate, which is the density of recurrence points in a recurrence plot:
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 recurrence plot of minimal length :
where is the frequency distribution of the lengths of the diagonal lines. 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 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 is 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 system and the trapping time, measuring the average length of 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 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. 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.
The probability that a diagonal line has exactly length can be estimated from the frequency distribution with . The Shannon entropy of 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 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. More exactly, let us consider the recurrence rate in a diagonal line parallel to LOI of distance k (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 recurrence plot. 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 synchronisation between different systems (using recurrence plots or cross recurrence plots).
Read more about this topic: Recurrence Quantification Analysis
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