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:

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

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 $\Theta (\cdot )$ 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 sum is used to estimate the correlation dimension.

See also

• Recurrence quantification analysis