Correlation integral

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:


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