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| In [[chaos theory]], the '''correlation integral''' is the mean probability that the states at two different times are close:
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| :<math>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 \Bbb{R}^m,</math>
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| 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]]):
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| :<math>\vec{x}(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1)),</math>
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| where <math>u(i)</math> is the time series, <math>m</math> the embedding dimension and <math>\tau</math> the time delay.
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| The correlation integral is used to estimate the [[correlation dimension]].
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| An estimator of the correlation integral is the [[correlation sum]]:
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| :<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 \Bbb{R}^m.</math>
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| ==See also==
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| *[[Recurrence quantification analysis]]
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| ==References==
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| * {{cite journal | author=P. Grassberger and I. Procaccia | title=Measuring the strangeness of strange attractors | journal=Physica | year=1983 | volume=9D| pages=189–208 | doi=10.1016/0167-2789(83)90298-1}} [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983PhyD....9..189G&db_key=PHY (LINK)]
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| [[Category:Chaos theory]]
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Latest revision as of 17:50, 7 April 2014
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