In
chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
, the correlation sum is the estimator of the
correlation integral In chaos theory, the correlation integral is the mean probability that the states at two different times are close:
:C(\varepsilon) = \lim_ \frac \sum_^N \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m,
where N is the n ...
, which reflects 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
) and
the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
. If only a
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
is available, the phase space can be reconstructed by using a time delay embedding (see
Takens' theorem
In the study of dynamical systems, 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. The reconstruction preserves the prope ...
):
:
where
is the time series,
the embedding dimension and
the time delay.
The correlation sum is used to estimate the
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.
For example, if we have a set of random points on ...
.
See also
*
Recurrence quantification analysis Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space tr ...
References
*
Chaos theory
Dynamical systems
Dimension theory
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