﻿

# 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.[1][2][3]

For example, 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 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, the box-counting dimension, and the information dimension) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, and is often in agreement with other calculations of dimension.

For any set of N points in an m-dimensional space

$\vec x(i)=[x_1(i),x_2(i),\ldots,x_m(i)], \qquad i=1,2,\ldots N$

then the correlation integral C(ε) is calculated by:

$C(\varepsilon)=\lim_{N \rightarrow \infty} \frac{g}{N^2}$

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). 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:

$C(\varepsilon) \sim \varepsilon^\nu \,$

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;[1] 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.[4]

As an example, in the "Sun in Time" article,[5] the method was used to show that the number of sunspots on the sun, 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.

## Notes

1. ^ a b Peter Grassberger and Itamar Procaccia (1983). "Measuring the Strangeness of Strange Attractors". Physica D: Nonlinear Phenomena 9 (1‒2): 189‒208. Bibcode 1983PhyD....9..189G. doi:10.1016/0167-2789(83)90298-1.
2. ^ Peter Grassberger and Itamar Procaccia (1983). "Characterization of Strange Attractors". Physical Review Letters 50 (5): 346‒349. Bibcode 1983PhRvL..50..346G. doi:10.1103/PhysRevLett.50.346.
3. ^ Peter Grassberger (1983). "Generalized Dimensions of Strange Attractors". Physics Letters A 97 (6): 227‒230. Bibcode 1983PhLA...97..227G. doi:10.1016/0375-9601(83)90753-3.
4. ^ DeCoster, Gregory P.; Mitchell, Douglas W. (1991). "The efficacy of the correlation dimension technique in detecting determinism in small samples". Journal of Statistical Computation and Simulation 39: 221–229.
5. ^ Sonett, C., Giampapa, M., and Matthews, M. (Eds.) (1992). The Sun in Time. University of Arizona Press. ISBN 0-8165-1297-3.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• correlation dimension — An estimate of the fractal dimension which measures the probability that two points chosen at random will be within a certain distance of each other, and examines how this probability changes as the distance is increased. white noise will fill… …   Financial and business terms

• 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: where N is the number of considered states , ε is a threshold distance, a… …   Wikipedia

• Correlation integral — In chaos theory, the correlation integral is the mean probability that the states at two different times are close: where N is the number of considered states , ε is a threshold distance, a norm (e.g. Euclidean norm) and …   Wikipedia

• Dimension — 0d redirects here. For 0D, see 0d (disambiguation). For other uses, see Dimension (disambiguation). From left to right, the square, the cube, and the tesseract. The square is bounded by 1 dimensional lines, the cube by 2 dimensional areas, and… …   Wikipedia

• Dimension — Sur les autres projets Wikimedia : « Dimension », sur le Wiktionnaire (dictionnaire universel) Dans le sens commun, la notion de dimension renvoie à la taille ; les dimensions d une pièce sont sa longueur, sa largeur et sa… …   Wikipédia en Français

• Correlation Integral — The probability that two points are within a certain distance from one another. Used in the calculation of the correlation dimension. Bloomberg Financial Dictionary …   Financial and business terms

• Correlation spectroscopy — is one of several types of two dimensional nuclear magnetic resonance (NMR) spectroscopy. Other types of two dimensional NMR include J spectroscopy, exchange spectroscopy (EXSY), and Nuclear Overhauser effect spectroscopy (NOESY). Two dimensional …   Wikipedia

• Dimension fractale — Mesure de la dimension fractale de la côte de Grande Bretagne En géométrie fractale, la dimension fractale, D, est une grandeur qui a vocation à traduire la façon qu a un ensemble fractal de remplir l espace, à toutes les échelles. Dans le cas… …   Wikipédia en Français

• Dimension (data warehouse) — This article is about a dimension in a data warehouse. For other uses, see dimension (disambiguation). In a data warehouse, a dimension is a data element that categorizes each item in a data set into non overlapping regions. A data warehouse… …   Wikipedia

• Corrélation linéaire — Régression linéaire Pour les articles homonymes, voir Régression. Un exemple graphique En statistiques, étant donné un échantillon aléatoire …   Wikipédia en Français