Английская Википедия:Correlation integral
In chaos theory, the correlation integral is the mean probability that the states at two different times are close:
- <math>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 \mathbb{R}^m,</math>
where <math>N</math> is the number of considered states <math>\vec{x}(i)</math>, <math>\varepsilon</math> is a threshold distance, <math>\| \cdot \|</math> a norm (e.g. Euclidean norm) and <math>\Theta( \cdot )</math> 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):
- <math>\vec{x}(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1))),</math>
where <math>u(i)</math> is the time series, <math>m</math> the embedding dimension and <math>\tau</math> the time delay.
The correlation integral is used to estimate the correlation dimension.
An estimator of the correlation integral is the correlation sum:
- <math>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 \mathbb{R}^m.</math>
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