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. 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 3-space (or N-space, for that matter), 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.

If we have a set M points in an N-dimensional space:

$\mathbf{X}_i=[x_{i1},x_{i2},\ldots,x_{iN}]$

where $i=1,2,\ldots M$ then the correlation integral $C (r)$ is calculated by:

$C(r)=\frac{g}{M^2}$

where g is the total number of pairs of points which have a distance between them that is less than or equal to distance r. As the number of points tends to infinity, and the distance between them tends to zero, the correlation integral, for small values of r, will take the form:

$C(r) \sim r^\nu$

If the number of points is sufficiently large, and evenly distributed, a plot of the correlation integral versus r 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, et. al. (1983) is the main reference for this technique, and gives the results of such estimates for a 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 chaotic and truly random behavior. For example, in the "Sun in Time" article, 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.

References

P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica, 9D, 189-208. (LINK)

Sonett, C., Giampapa, M., and Matthews, M. (Eds.) (1992). The Sun in Time. University of Arizona Press. ISBN 0816512973.

Correlation dimension

See also

References

See also: Correlation dimension, Box-counting dimension, Chaos theory, Hausdorff dimension, Sun, Sunspot, Takens' theorem, Information dimension