Correlation integral

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

${\displaystyle C(\varepsilon )=\lim _{N\rightarrow \infty }{\frac {1}{N^{2}}}\sum _{\stackrel {i,j=1}{i\neq j}}^{N}\Theta (\varepsilon -\|{\vec {x}}(i)-{\vec {x}}(j)\|),\quad {\vec {x}}(i)\in \mathbb {R} ^{m},}$

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):

${\displaystyle {\vec {x}}(i)=(u(i),u(i+\tau ),\ldots ,u(i+\tau (m-1))),}$

where ${\displaystyle u(i)}$ is the time series, ${\displaystyle m}$ the embedding dimension and ${\displaystyle \tau }$ the time delay.

The correlation integral is used to estimate the correlation dimension.

An estimator of the correlation integral is the correlation sum:

${\displaystyle C(\varepsilon )={\frac {1}{N^{2}}}\sum _{\stackrel {i,j=1}{i\neq j}}^{N}\Theta (\varepsilon -\|{\vec {x}}(i)-{\vec {x}}(j)\|),\quad {\vec {x}}(i)\in \mathbb {R} ^{m}.}$