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.

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.

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Correlation sum

See also

References