In chaos theory

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

, 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

Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

. If only a time series

Time series

In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

 is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem

Takens' theorem

In mathematics, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system...

):


where is the time series, the embedding dimension and the time delay.

The correlation integral is used to estimate the correlation dimension

Correlation dimension

In chaos theory, the correlation dimension 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....

.

An estimator of the correlation integral is the correlation sum

Correlation sum

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:...

: