#### 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.