Correlation integral

In chaos theory, the correlation integral is the mean probability that the states at two different times are close:

where ${\displaystyle N}$ is the number of considered states ${\displaystyle {\vec {x}}(i)}$, ${\displaystyle \varepsilon }$ is a threshold distance, ${\displaystyle ||\cdot ||}$ a norm (e.g. Euclidean norm) and ${\displaystyle \Theta (\cdot )}$ 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):

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