correlation sum

{{One source|date=November 2007}}

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:

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

where $N$ is the number of considered states $\backslash vec\{x\}(i)$, $\backslash varepsilon$ is a threshold distance, $\backslash |\; \backslash cdot\; \backslash |$ a norm (e.g. Euclidean norm) and $\backslash Theta(\; \backslash cdot\; )$ 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):

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

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

The correlation sum is used to estimate the correlation dimension.