Correlation integral explained

In chaos theory, the correlation integral is the mean probability that the states at two different times are close:

C(\varepsilon) = \lim _{N} (1)/(N ^{2}) \sum _{\stackrel{i,j=1} {i ≠ j}} ^{N} \Theta(\varepsilon - \| \vec{x}(i) - \vec{x}(j)\|), \vec{x}(i) \in R ^{m,}

where

N is the number of considered states \vec{x}(i) , \varepsilon is a threshold distance, \| ⋅ \| a norm (e.g. Euclidean norm) and \Theta( ⋅ ) the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem):\vec{x}(i) = (u(i), u(i + \tau), \ldots, u(i + \tau(m -1 ))),

where

u(i) is the time series, m the embedding dimension and \tau the time delay.

The correlation integral is used to estimate the correlation dimension.

An estimator of the correlation integral is the correlation sum:

C(\varepsilon) = (1)/(N ^{2}) \sum _{\stackrel{i,j=1} {i ≠ j}} ^{N} \Theta(\varepsilon - \| \vec{x}(i) - \vec{x}(j)\|), \vec{x}(i) \in R ^{m.}

See also

• Recurrence quantification analysis

References

• P. Grassberger and I. Procaccia . Measuring the strangeness of strange attractors . Physica . 1983 . 9D. 189–208 . 10.1016/0167-2789(83)90298-1. 1983PhyD....9..189G. (LINK)