Recurrence plots, cross recurrence plots

Norbert

Recurrence Plots: Usually, real data are added with noise and, thus, invariant properties of phase space embedding can not well determined. furthermore, a higher dimensional phase space reconstruction is not easy to imagine and can not be visualized. a recurrence plot visualizes the recurrence of the phase space trajectory into a two-dimensional matrix and allows an assertion of the phase space embedding.
Cross recurrence plot: A cross recurrence plot compares the phase space reconstructions of two different systems. it allows an easy evaluation of epochs of a similar progress of the both phase space trajectories. it is an extension of the simple recurrence plot.

Evaluation of recurrence plots

The following complexity measures can be estimated from RPs and from CRPs. for the latter, it is easy to generalize the interpretations of these measures.

RR: the recurrence rate of a RP is the probability of the recurrence of a random selected state. it is equal to the correlation integral, of a time series.
The measures DET, L and ENT are based on diagonal line elements in a RP. a diagonal line element occurs when the both phase space trajectories run parallel to them self. stochastic behavior causes none or very short diagonals, whereas deterministic behavior causes longer diagonals.
DET: the determinism of a RP is the quota of the diagonal line elements on the total amount of recurrence points and, thus, is a measure for determinism (or predictability) in the system.
L: the mean length of the diagonal line elements in a RP is the averaged time two pieces of the phase space trajectory are close to each other (i.e. the mean of the length of all diagonal lines). this can be interpreted as the mean prediction time and its inverse is correlated with the positive largest Lyapunov exponent.
ENT: the entropy means the Shannon entropy of the histogram of diagonal line lengths and reflects the complexity of the deterministic structure in the system.
TREND: a drift in a dynamical system causes a fall off in recurrences away from the main diagonal. the value trend measures this effect by diagonal wise computation of diagonal recurrence density and its linear relation to the time distance of these diagonals to the main diagonal.
TT: a vertical and horizontal line element, resp., in the RP occur, when the phase space trajectory lingers(?) at the same state for longer time (i.e. the trajectory is 'trapped', laminar state). analogous the mean length of the diagonal line elements, the trapping time is the mean of all vertical lines in the RP.