In chaos theory, the correlation integral is the mean probability that the states at two different times are close:
formula_1
where formula_2 is the number of considered states formula_3, formula_4 is a threshold distance, formula_5 a norm (e.g. Euclidean norm) and formula_6 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):
formula_7
where formula_8 is the time series, formula_9 the embedding dimension and formula_10 the time delay.
The correlation integral is used to estimate the correlation dimension.
An estimator of the correlation integral is the correlation sum:
formula_11