## Correlation dimension

Purpose

Description

Macro Synopsis

Modules

Related Functions

References

### Purpose

Correlation dimension

### Description

Being one of the characteristic invariants of nonlinear system
dynamics, the correlation dimension gives a measure of complexity for the
underlying attractor of the system. To detect the saturation value of the
correlation dimension, the function plots the computed correlation
dimension as a function of the embedding dimension.

Mathematically, the correlation dimension is a special case of the generalized dimension,
and it is given by

with being the probability to find a point of the attractor
within the i-th subcube of phase space.
(Phase space is subdivided into disjunctive cubes of side length `r`.)
The number `M(r)` of cubes that contain attractor points, is related to
the dimension `D` of the attractor :

The correlation dimension is closely related to the correlation integral by
Grassberger and Procaccia, which allows an accurate and efficient numerical
computation of the attractor's dimension. (See Correlation integral.)

The parameters are:
- Delay
`tau` for phase space reconstruction (see Introductory chapter to nonlinear dynamics) as number of samples,
- Minimum embedding dimension
`mmin` and
- Maximum embedding dimension
`mmax`, as a range for the calculation
of the correlation dimension,
- Lower relative radius
`r0` and
- Upper relative radius
`r1`, between which the correlation dimension is calculated as the derivate of the log C(r)/log(r) plot. Thus `r0` and `r1` should both lie within the so-called "scaling region'' of the attractor.

The result shows a plot of the estimated correlation dimension versus the embedding dimension. Ideally, the graph should converge against the actual correlation dimension.

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 = CorrDimension(x,tau,mmin,mmax,r0,r1);`

signal x,y;

int tau,mmin,mmax;

float r0,r1;

### Modules

Nonlinear

### Related Functions

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

### References

Grassberger/Procaccia [23], Vandenhouten [21]