## Pointwise correlation dimension (PD2)

Purpose

Description

Tips

Macro Synopsis

Modules

Related Functions

References

### Purpose

Skinner's PD2 algorithm for the pointwise correlation dimension

### Description

Also referred to as PD2, the pointwise correlation dimension
is given by the formula:

Here r denotes the radius of a phase space neighborhood around x.
and are the phase space
coordinates, delayed by . N is the length of the signal.
is the Heavyside-function, given by

PD2 is a pointwise (i.e. time-resolving) measure, displayed as a function
of time. It is well suited to examine instationary signals.

The parameters are

- the minimum embedding dimension
`mmin`,
- the maximum embedding dimension
`mmax`,
- the coordinate delay
`tau` (no. of samples) for phase space
reconstruction and
- the
`skip` represents the scanning resolution used by the
algorithm. (The PD2 formula is evaluated every skip-th sample only.)

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.

### Tips

You can use "Pointwise correlation dimension (PD2)" to detect phase transitions in instationary time
series of dynamic systems. Sudden leaps of the average PD2 indicate a change
of the system's dynamic complexity.

For the dependencies between correlation integral and dimension, refer to
Correlation dimension and Correlation integral.

### Macro Synopsis

`y = PD2(x,mmin,mmax,tau,skip);`

signal x,y;

int mmin,mmax,tau,skip;

### Modules

Nonlinear

### Related Functions

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

### References

Skinner et al. [48]