Correlation sum

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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:

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

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):