# 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 ... The real utility of the correlation dimension is in determining the (possibly fractional) dimensions of fractal objects ... Hausdorff dimension, the box-counting dimension, and the information dimension) but the correlation dimension has the advantage of being ...

### Famous quotes containing the word integral:

Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.
Michel de Montaigne (1533–1592)