Everything about Correlation Dimension totally explained
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've a set of random points on the
real number line between 0 and 1, the correlation dimension will be ν=1, while if they're distributed on say, a triangle embedded in 3-space (or m-space), the correlation dimension will be ν=2. This is what we'd 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 (for example 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
»
where
is the total number of pairs of points which have a distance between them that's less than distance
(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
, will take the form:
»
If the number of points is sufficiently large, and evenly distributed, a
Log-log graph of the correlation integral versus
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 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.
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