## Recurrence plot

Purpose
Description
Tips
Example
Macro Synopsis
Modules
Related Functions
References

Recurrence plot.

### Description

Recurrence plots are a useful preprocessing tool since they provide a comprehensive view of the dynamic courses within a signal. A recurrence plot is a 2-dimensional N x N pattern of points where N is the number of embedding vectors obtained from the delay coordinates of the input signal.

A point i,j in this plot is set if

with being the set of M nearest neighbours of ; i.e. if is one of the nearest neighbours of and if it lies within a sphere of radius r around the current reference point.

Since the coordinates i and j represent points on a time axis, recurrence plots give information about the temporal correlation of phase space points.

From the occurrence of lines parallel to the diagonal in the recurrence plot it can be seen how fast neighboured trajectories diverge in phase space. Therefore, the average length of these lines is a measure of the reciprocal of the largest positive Lyapunov exponent.

Recurrence plots help revealing phase transitions and instationarities. Visible rectangular block structures with a higher density of points in the recurrence plot indicate phase transitions within the signal. If the texture of the pattern within such a block is homogeneous, stationarity can be assumed for the given signal within the corresponding period of time.
Parameters are:

• number M of neighbours
• step size s (i.e. size-1 samples are skipped between successive columns of the plot)
• embedding dimension m
• relative radius r

### Tips

Usually, the relative radius ranges from 0.05 to 0.2, and M is typically between 10 and 50.

### Example

Applying the function to an continuously instationary signal, you will get a plot with points accumulating near the diagonal. (e.g. Recurrence plot of the sample signal sinchirp.dpa)

### Macro Synopsis

y = RecurrPlot(x,M,s,m,r);
signal x,y;
int M,s,m;
float r;

Note that y is a 2D plot.

Nonlinear

### Related Functions

False nearest neighbors (FNN), Largest Lyapunov exponent (LLE), Mutual information.

### References

Eckmann et al. [54]