Correlation Integral

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

:$C(\varepsilon) = \lim_ \frac \sum_^N \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m,$

where $N$ is the number of considered states $\vec(i)$, $\varepsilon$ 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):

:$\vec(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1))),$

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:

:$C(\varepsilon) = \frac \sum_^N \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m.$

See also

*Recurrence quantification analysis

References

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Chaos theory

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