Correlation integral
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.