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
The Correlation integral given by Grassberger and Procaccia  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
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).
- Delay parameter tau for phase space reconstruction,
given in samples
- Embedding dimension m of phase space for reconstruction with
- 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.
y = CorrIntegral(x,tau,m,r0,r1,n);
Correlation dimension, Largest Lyapunov exponent (LLE),
Momentary largest Lyapunov exponent, Lyapunov spectrum,
Pointwise correlation dimension (PD2), Pointwise correlation integral.