Correlation integral

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 $\vec{x}(i)$, ε 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 τ the time delay.

The correlation integral is used to estimate the correlation dimension.

An estimator of the correlation integral is the correlation sum:

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

See also

• Recurrence quantification analysis

References

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