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 , where N is the number of considered states x → ( i ) , ε is a threshold distance, | | ⋅ | | a norm (e.g. Euclidean norm) and Θ ( ⋅ ) 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 ) ) , where u ( i ) is the time series, m the embedding dimension and τ the time delay.
The correlation sum is used to estimate the correlation dimension.
