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 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, of being less noisy when only a small number of points is available, and is often in agreement with other calculations of dimension.