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:

where is the number of considered states , is a threshold distance, a norm (e.g. Euclidean norm) and 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):

where is the time series, the embedding dimension and the time delay.

The correlation sum is used to estimate the correlation dimension.

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