RECURRENCE PLOTS::Recurrence Quantification Analysis, Definition

Quantification of Recurrence Plots (Recurrence Quantification Analysis)

Definition

Recurrence Quantification Analysis – The recurrence quantification analysis (RQA) is a method of nonlinear data analysis which quantifies the number and duration of recurrences of a dynamical system presented by its state space trajectory.

A quantification of recurrence plots was developed by Zbilut and Webber Jr. (Zbilut and Webber Jr., 1992; Webber Jr. and Zbilut, 1994) and extended with new measures of complexity by Marwan et al. (2002). Measures which base on diagonal structures are able to find chaos-order transitions (Trulla et al., 1996), measures based on vertical (horizontal) structures are able to find chaos-chaos transitions (laminar phases, Marwan et al., 2002).

These measures can be computed in windows along the main diagonal. This allows us to study their time dependence and can be used for the detection of transitions (Trulla et al., 1996). Another possibility is to define these measures for each diagonal parallel to the main diagonal separately (Marwan and Kurths, 2002). This approach enables the study of time delays, unstable periodic orbits (UPOs; Lathrop and Kostelich, 1989; Gilmore, 1998), and by applying to cross recurrence plots, the assessment of similarities between processes (Marwan and Kurths, 2002).

Measure	Definition
Recurrence rate $RR$	The percentage of recurrence points in an RP: $$ RR = \frac{1}{N^2}sum_{i,j=1}^N R_{i,j} $$ Corresponds to the correlation sum.
Determinism $DET$	The percentage of recurrence points which form diagonal lines: $$ DET = \frac{\sum_{l=l_{\min}}^N l P(l)}{\sum_{l=1}^N l P(l)} $$ $P(l)$ is the histogram of the lengths $l$ of the diagonal lines.
Laminarity $LAM$	The percentage of recurrence points which form vertical lines: $$ LAM = \frac{\sum_{v=v_{\min}}^N v P(v)}{\sum_{v=1}^N v P(v)} $$ $P(v)$ is the histogram of the lengths $v$ of the vertical lines.
Ratio $RATIO$	The ratio between $DET$ and $RR$: $$ RATIO = N^2 \frac{\sum_{l=l_{\min}}^N l P(l)}{\left(\sum_{l=1}^N l P(l)\right)^2} $$
Averaged diagonal line length $L$	The average length of the diagonal lines: $$ L = \frac{\sum_{l=l_{\min}}^N l P(l)}{\sum_{l=l_{\min}}^N P(l)} $$
Trapping time $TT$	The average length of the vertical lines: $$ TT = \frac{\sum_{v=v_{\min}}^N v P(v)}{\sum_{v=v_{\min}}^N P(v)} $$
Longest diagonal line $L_\max$	The length of the longest diagonal line $$ L_\max = \max\left( \{l_i; \ i=1,\ldots, N_l\}\right) $$
Longest vertical line $V_\max$	The length of the longest vertical line $$ V_\max = \max\left( \{v_i; \ i=1,\ldots, N_v\}\right) $$
Divergence $DIV$	The inverse of $L_\max$ $$ DIV = \frac{1}{L_\max} $$ Related with the KS entropy of the system, i.e. with the sum of the positive Lyapunov exponents.
Entropy $ENTR$	The Shannon entropy of the probability distribution of the diagonal line lengths $p(l)$: $$ ENTR = -\sum_{l=l_\min}^N p(l) \ln p(l) $$
Trend $TREND$	The paling of the RP towards its edges: $$ TREND = \frac{\sum_{i=1}^\tilde{N}(i-\tilde{N}/2)(RR_i - \langle RR_i\rangle) } { \sum_{i=1}^\tilde{N} (i-\tilde{N}/2)^2} $$

$N$ – number of points on the phase space trajectory
$N_l$ – number of diagonal lines in the recurrence plot
$N_v$ – number of vertical lines in the recurrence plot
$P(l), P(v)$ – histogram of the line lengths of diagonal/ vertical lines
$\tilde{N}$ – maximal number of diagonals parallel to the LOI which will be considered for the calculation of $TREND$

» Recurrence quantification analysis in Wikipedia

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Measure	Definition
Recurrence rate \(RR\)	The percentage of recurrence points in an RP: $$ RR = \frac{1}{N^2}sum_{i,j=1}^N R_{i,j} $$ Corresponds to the correlation sum.
Determinism \(DET\)	The percentage of recurrence points which form diagonal lines: $$ DET = \frac{\sum_{l=l_{\min}}^N l P(l)}{\sum_{l=1}^N l P(l)} $$ \(P(l)\) is the histogram of the lengths \(l\) of the diagonal lines.
Laminarity \(LAM\)	The percentage of recurrence points which form vertical lines: $$ LAM = \frac{\sum_{v=v_{\min}}^N v P(v)}{\sum_{v=1}^N v P(v)} $$ \(P(v)\) is the histogram of the lengths \(v\) of the vertical lines.
Ratio \(RATIO\)	The ratio between \(DET\) and \(RR\): $$ RATIO = N^2 \frac{\sum_{l=l_{\min}}^N l P(l)}{\left(\sum_{l=1}^N l P(l)\right)^2} $$
Averaged diagonal line length \(L\)	The average length of the diagonal lines: $$ L = \frac{\sum_{l=l_{\min}}^N l P(l)}{\sum_{l=l_{\min}}^N P(l)} $$
Trapping time \(TT\)	The average length of the vertical lines: $$ TT = \frac{\sum_{v=v_{\min}}^N v P(v)}{\sum_{v=v_{\min}}^N P(v)} $$
Longest diagonal line \(L_\max\)	The length of the longest diagonal line $$ L_\max = \max\left( \{l_i; \ i=1,\ldots, N_l\}\right) $$
Longest vertical line \(V_\max\)	The length of the longest vertical line $$ V_\max = \max\left( \{v_i; \ i=1,\ldots, N_v\}\right) $$
Divergence \(DIV\)	The inverse of \(L_\max\) $$ DIV = \frac{1}{L_\max} $$ Related with the KS entropy of the system, i.e. with the sum of the positive Lyapunov exponents.
Entropy \(ENTR\)	The Shannon entropy of the probability distribution of the diagonal line lengths \(p(l)\): $$ ENTR = -\sum_{l=l_\min}^N p(l) \ln p(l) $$
Trend \(TREND\)	The paling of the RP towards its edges: $$ TREND = \frac{\sum_{i=1}^\tilde{N}(i-\tilde{N}/2)(RR_i - \langle RR_i\rangle) } { \sum_{i=1}^\tilde{N} (i-\tilde{N}/2)^2} $$

Recurrence Plots and Cross Recurrence Plots

Quantification of Recurrence Plots (Recurrence Quantification Analysis)

Definition