About: 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: 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.