Correlation Integral

In chaos theory, the correlation integral is the mean probability that the states at two different times are close:

where is the number of considered states, is a threshold distance, a norm (e.g. Euclidean norm) and 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 is the time series, the embedding dimension and the time delay.

The correlation integral is used to estimate the correlation dimension.

An estimator of the correlation integral is the correlation sum:

Other articles related to "correlation":

Correlation Dimension
... In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension ... if we have a set of random points on the real number line between 0 and 1, the correlation dimension will be ν = 1, while if they are distributed on say, a ... The real utility of the correlation dimension is in determining the (possibly fractional) dimensions of fractal objects ...

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