In
chaos theory, the correlation dimension (denoted by ''ν'') is a measure of the
dimensionality
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
of the space occupied by a set of random points, often referred to as a type of
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
.
For example, if we have a set of random points on the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
line between 0 and 1, the correlation dimension will be ''ν'' = 1, while if they are distributed on say, a triangle embedded in three-dimensional space (or ''m''-dimensional space), the correlation dimension will be ''ν'' = 2. This is what we would intuitively expect from a measure of dimension. The real utility of the correlation dimension is in determining the (possibly fractional) dimensions of fractal objects. There are other methods of measuring dimension (e.g. the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
, the
box-counting dimension, and the
information dimension
In information theory, information dimension is an information measure for random vectors in Euclidean space, based on the normalized entropy of finely quantized versions of the random vectors. This concept was first introduced by Alfréd Rény ...
) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, of being less noisy when only a small number of points is available, and is often in agreement with other calculations of dimension.
For any set of ''N'' points in an ''m''-dimensional space
:
then 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 ...
''C''(''ε'') is calculated by:
:
where ''g'' is the total number of pairs of points which have a distance between them that is less than distance ''ε'' (a graphical representation of such close pairs is the
recurrence plot In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i,
i.e., when the phase space trajectory visits roug ...
). As the number of points tends to infinity, and the distance between them tends to zero, the correlation integral, for small values of ''ε'', will take the form:
:
If the number of points is sufficiently large, and evenly distributed, a
log-log graph of the correlation integral versus ''ε'' will yield an estimate of ''ν''. This idea can be qualitatively understood by realizing that for higher-dimensional objects, there will be more ways for points to be close to each other, and so the number of pairs close to each other will rise more rapidly for higher dimensions.
Grassberger and
Procaccia introduced the technique in 1983;
the article gives the results of such estimates for a number of fractal objects, as well as comparing the values to other measures of fractal dimension. The technique can be used to distinguish between (deterministic) chaotic and truly random behavior, although it may not be good at detecting deterministic behavior if the deterministic generating mechanism is very complex.
As an example, in the "Sun in Time" article,
the method was used to show that the number of
sunspots on the
sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
, after accounting for the known cycles such as the daily and 11-year cycles, is very likely not random noise, but rather chaotic noise, with a low-dimensional fractal attractor.
See also
*
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 ...
*
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 ...
*
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 ...
*
Approximate entropy In statistics, an approximate entropy (ApEn) is a technique used to quantify the amount of regularity and the unpredictability of fluctuations over time-series data. For example, consider two series of data:
: Series A: (0, 1, 0, 1, 0, 1, 0, 1, 0, ...
Notes
{{Fractals
Chaos theory
Dynamical systems
Dimension theory
Fractals