Correlation integral

In chaos theory, the correlation integral is the mean probability that the states at two different times are close: In chaos theory, the correlation integral is the mean probability that the states at two different times are close: 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): where u ( i ) {displaystyle u(i)} is the time series, m {displaystyle m} the embedding dimension and τ {displaystyle au } the time delay. The correlation integral is used to estimate the correlation dimension. An estimator of the correlation integral is the correlation sum: