## Correlation integral

Purpose

Description

Macro Synopsis

Modules

Related Functions

References

### Purpose

Correlation integral

### Description

The instantaneous state of a dynamical system is characterized
by a point in phase space. A sequence of such states subsequent in
time defines the phase space trajectory. If the system is governed by
deterministic laws, then after a while, it will arrive at a permanent
state regime. This fact is reflected by the convergence of ensembles of
phase space trajectories towards an invariant subset of phase space,
the attractor of the system.
The output of Correlation integral depends on the given radii of the phase space
neighbourhood and the embedding dimension and is plotted logarithmically
against log(r).
The Correlation integral given by Grassberger and Procaccia [24] reads as follows :

where r is the radius of the neighbourhood around the phase space point
,
N is the length of the signal, and denotes the Heaviside
function with

In a certain range of r (the so-called scaling region), C(r) behaves like

The dimensionality d of the attractor is therefore given by the slope of
log(C(r)) versus log(r).

Parameters:
- Delay parameter
`tau` for phase space reconstruction,
given in samples
- Embedding dimension m of phase space for reconstruction with
delay coordinates
- Minimum radius r0, relative to the span of the time series,
i.e. r must be between 0 and 1.
- Maximum relative radius r1
- No. n of radius steps (horizontal resolution)

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.

**Warning:** *Phase space calculations on long signals are computationally very expensive, and can lead to large response times of the program. Try to use small radii and low embedding dimensions if possible.*

### Macro Synopsis

`y = CorrIntegral(x,tau,m,r0,r1,n);`

signal x,y;

int tau,m;

float r0,r1;

int n;

### Modules

Nonlinear

### Related Functions

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

### References

Grassberger [24]