{"document_url": "https://en.wikipedia.org/wiki/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.