Pointwise correlation dimension (PD2)
Skinner's PD2 algorithm for the pointwise correlation dimension
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
- 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.
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.
y = PD2(x,mmin,mmax,tau,skip);
Correlation dimension, Correlation integral,
Largest Lyapunov exponent (LLE), Momentary largest Lyapunov exponent, Lyapunov spectrum,
Pointwise correlation integral.
Skinner et al.