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.
y = CorrDimension(x,tau,mmin,mmax,r0,r1);
Correlation integral, Largest Lyapunov exponent (LLE),
Momentary largest Lyapunov exponent, Lyapunov spectrum,
Pointwise correlation dimension (PD2), Pointwise correlation integral.
Grassberger/Procaccia , Vandenhouten