Correlation integral

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

$C(\varepsilon )=\lim _{N\rightarrow \infty }{\frac {1}{N^{2}}}\sum _{\stackrel {i,j=1}{i\neq j}}^{N}\Theta (\varepsilon -||{\vec {x}}(i)-{\vec {x}}(j)||),\quad {\vec {x}}(i)\in {\mathbb {R}}^{m},$

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

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

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

The correlation integral is used to estimate the correlation dimension.

An estimator of the correlation integral is the correlation sum:

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

See also

• Recurrence quantification analysis

References

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