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Correlación integral
En la teoría del caos, la integral de correlación es la probabilidad media de que los estados en dos tiempos diferentes estén cercanos:
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Correlation integral
From Wikipedia, the free encyclopedia
In chaos theory , the correlation integral is the mean probability that the states at two different times are close:
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{\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},}
where
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{\displaystyle N}
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{\displaystyle {\vec {x}}(i)}
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{\displaystyle \varepsilon }
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{\displaystyle \|\cdot \|}
Euclidean norm ) and
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{\displaystyle \Theta (\cdot )}
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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{\displaystyle {\vec {x}}(i)=(u(i),u(i+\tau ),\ldots ,u(i+\tau (m-1))),}
where
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{\displaystyle u(i)}
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{\displaystyle m}
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{\displaystyle \tau }
The correlation integral is used to estimate the correlation dimension .
An estimator of the correlation integral is the correlation sum :
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{\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}.}
