Correlation integral
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In chaos theory, the correlation integral is the mean probability that the states at two different times are close:
C ( ε ) = lim N → ∞ 1 N 2 ∑ i ≠ j i , j = 1 N Θ ( ε − ‖ x → ( i ) − x → ( j ) ‖ ) , x → ( i ) ∈ R m , {\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 N {\displaystyle N} is the number of considered states x → ( i ) {\displaystyle {\vec {x}}(i)}, ε {\displaystyle \varepsilon } is a threshold distance, ‖ ⋅ ‖ {\displaystyle \|\cdot \|} a norm (e.g. Euclidean norm) and Θ ( ⋅ ) {\displaystyle \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):
x → ( i ) = ( u ( i ) , u ( i + τ ) , … , u ( i + τ ( m − 1 ) ) ) , {\displaystyle {\vec {x}}(i)=(u(i),u(i+\tau ),\ldots ,u(i+\tau (m-1))),}
where u ( i ) {\displaystyle u(i)} is the time series, m {\displaystyle m} the embedding dimension and τ {\displaystyle \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 ( ε ) = 1 N 2 ∑ i ≠ j i , j = 1 N Θ ( ε − ‖ x → ( i ) − x → ( j ) ‖ ) , x → ( i ) ∈ R m . {\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}.}