Correlation sum

In chaos theory, the '''correlation sum''' is the estimator of the correlation integral, which reflects the mean probability that the states at two different times are close:

:<math>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>

where <math>N</math> is the number of considered states <math>\vec{x}(i)</math>, <math>\varepsilon</math> is a threshold distance, <math>\| \cdot \|</math> a norm (e.g. Euclidean norm) and <math>\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>\vec{x}(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1)),</math>

where <math>u(i)</math> is the time series, <math>m</math> the embedding dimension and <math>\tau</math> the time delay.

The correlation sum is used to estimate the correlation dimension.

==References== * {{cite journal | author=P. Grassberger and I. Procaccia | title=Measuring the strangeness of strange attractors | journal=Physica | year=1983 | volume=9D| issue=1–2 | pages=189–208 | doi=10.1016/0167-2789(83)90298-1|bibcode = 1983PhyD....9..189G }}

Category:Chaos theory Category:Dynamical systems Category:Dimension theory