Correlation integral

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

C(\varepsilon) = \lim_{N \rightarrow \infty} \frac{1}{N^2} \sum_{\stackrel{i,j=1}{i \neq j}}^N \Theta(\varepsilon - || \vec{x}(i) - \vec{x}(j)||), \quad \vec{x}(i) \in \Bbb{R}^m,

where N is the number of considered states \vec{x}(i), \varepsilon is a threshold distance, || \cdot || a norm (e.g. Euclidean norm) and \Theta( \cdot ) 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) = \frac{1}{N^2} \sum_{\stackrel{i,j=1}{i \neq j}}^N \Theta(\varepsilon - || \vec{x}(i) - \vec{x}(j)||), \quad \vec{x}(i) \in \Bbb{R}^m.

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