Correlation integral

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

C (ε) = lim_{N \to \infty} \frac{1}{N^{2}} \sum_{\overset{i, j = 1}{i \neq j}}^{N} Θ (ε - ‖ \vec{x} (i) - \vec{x} (j) ‖), \vec{x} (i) \in ℝ^{m},

where $N$ is the number of considered states $\vec{x} (i)$ , $ε$ is a threshold distance, $‖ \cdot ‖$ a norm (e.g. Euclidean norm) and $Θ (\cdot)$ 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 + τ), \dots, u (i + τ (m - 1))),

where $u (i)$ is the time series, $m$ the embedding dimension and $τ$ the time delay.

The correlation integral is used to estimate the correlation dimension.

An estimator of the correlation integral is the correlation sum:

C (ε) = \frac{1}{N^{2}} \sum_{\overset{i, j = 1}{i \neq j}}^{N} Θ (ε - ‖ \vec{x} (i) - \vec{x} (j) ‖), \vec{x} (i) \in ℝ^{m} .

References

P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica. 9D (1–2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1. (LINK)

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References