Correlation sum

In chaos theory, the correlation sum is the estimator of the correlation integral, which reflects the mean probability that the states at two different times are close:

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

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 sum is used to estimate the correlation dimension.

See also

• Correlation integral
• Correlation dimension
• Recurrence quantification analysis