## Pointwise correlation integral

Purpose

Description

Macro Synopsis

Modules

Related Functions

References

### Purpose

Pointwise correlation integral

### Description

The "Pointwise correlation integral" is a time-resolving variant of the correlation integral.
Unlike the latter, which gives a sort of time-average over the whole signal,
the pointwise correlation integral yields a value for each sample (point in time) of the signal.

The pointwise correlation integral is calculated as

Here r denotes the radius of a phase space neighbourhood.
and are the phase space vectors
of delay coordinates, N is the length of the signal.
is the Heavyside-function, given by

"Pointwise correlation integral" is closely related to Pointwise correlation dimension (PD2).
(In the same way as Correlation integral is related to Correlation dimension.)

Parameters of "Pointwise correlation integral" are

- delay
`tau` of coordinates in samples,
- embedding dimension m for phase space reconstruction and
- relative radius r.

Note that since this function operates on a 'pure' time series, the scale and the shift of the given signal do not affect the result.

### Macro Synopsis

`y = PointCorrIntegral(x,tau,m,r);`

signal x,y;

int tau,m;

float r;

### Modules

Nonlinear

### Related Functions

Correlation dimension, Correlation integral,
Largest Lyapunov exponent (LLE), Momentary largest Lyapunov exponent, Lyapunov spectrum,
Pointwise correlation dimension (PD2).

### References

Pawelzik [49]