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:
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},}
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 sum is used to estimate the correlation dimension.
See also Recurrence quantification analysis
References P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica. 9D (1–2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.