// Workers AI · dad joke modeDoes correlation integral have friends? It integrates well with others.

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

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

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

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

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

The correlation integral is used to estimate the correlation dimension.

An estimator of the correlation integral is the correlation sum:

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

See also

• Recurrence quantification analysis

References

• P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica. 9D (1–2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1. (LINK)