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In chaos theory, the correlation integral is the mean probability that the states at two different times are close:
where N {\displaystyle N} is the number of considered states x → {\displaystyle {\vec {x}}} , ε {\displaystyle \varepsilon } is a threshold distance, | | ⋅ | | {\displaystyle ||\cdot ||} a norm and Θ {\displaystyle \Theta } the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding :
where u {\displaystyle u} is the time series, m {\displaystyle m} the embedding dimension and τ {\displaystyle \tau } the time delay.
The correlation integral is used to estimate the correlation dimension.
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