Correlation integral

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

$C(\epsilon )=\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N^2}\sum _{\stackrel{i,j=1}{i\ne j}}^N\mathrm{\Theta }(\epsilon -\Vert \overrightarrow{x}(i)-\overrightarrow{x}(j)\Vert ),\overrightarrow{x}(i)\in {\mathbb{ℝ}}^m,$

where $N$ is the number of considered states $\overrightarrow{x}(i)$, $\epsilon$ is a threshold distance, $\Vert \cdot \Vert$ a norm (e.g. Euclidean norm) and $\mathrm{\Theta }(\cdot )$ 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):

$\overrightarrow{x}(i)=(u(i),u(i+\tau ),\dots ,u(i+\tau (m-1))),$

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:

$C(\epsilon )=\frac{1}{N^2}\sum _{\stackrel{i,j=1}{i\ne j}}^N\mathrm{\Theta }(\epsilon -\Vert \overrightarrow{x}(i)-\overrightarrow{x}(j)\Vert ),\overrightarrow{x}(i)\in {\mathbb{ℝ}}^m.$

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. Bibcode: 1983PhyD....9..189G.  (LINK)