Correlation integral
In chaos theory, the correlation integral is the mean probability that the states at two different times are close:
- [math]\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, }[/math]
where [math]\displaystyle{ N }[/math] is the number of considered states [math]\displaystyle{ \vec{x}(i) }[/math], [math]\displaystyle{ \varepsilon }[/math] is a threshold distance, [math]\displaystyle{ \| \cdot \| }[/math] a norm (e.g. Euclidean norm) and [math]\displaystyle{ \Theta( \cdot ) }[/math] 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):
- [math]\displaystyle{ \vec{x}(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1))), }[/math]
where [math]\displaystyle{ u(i) }[/math] is the time series, [math]\displaystyle{ m }[/math] the embedding dimension and [math]\displaystyle{ \tau }[/math] the time delay.
The correlation integral is used to estimate the correlation dimension.
An estimator of the correlation integral is the correlation sum:
- [math]\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. }[/math]
See also
References
- P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica 9D: 189–208. doi:10.1016/0167-2789(83)90298-1. Bibcode: 1983PhyD....9..189G. (LINK)
 |  |
