Fractal dimension
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In fractal geometry, the fractal dimension is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales.
One way to define the dimension is to ask, how many balls of radius ε it would take to cover the fractal; and how does this number scale as ε is made smaller? This gives a quantity
- <math>D = \lim_{\epsilon \rightarrow 0} \frac{\log N(\epsilon)}{\log \epsilon}</math>
which is more or less the Hausdorff dimension.
Closely related to this is the box-counting dimension, which considers, if the space were divided up into a grid of boxes of size ε, how does the number of boxes scale that would contain part of the attractor? Again,
- <math>D_0 = \lim_{\epsilon \rightarrow 0} \frac{\log N(\epsilon)}{\log \epsilon}</math>
Other dimension quantities include the information dimension, which considers how the average information needed to identify an occupied box scales, as the scale of boxes gets smaller:
- <math>D_1 = \lim_{\epsilon \rightarrow 0} \frac{-\langle \log p_\epsilon \rangle}{\log \epsilon}</math>
and the correlation dimension, which is perhaps easiest to calculate,
- <math>D_2 = \lim_{\epsilon \rightarrow 0, M \rightarrow \infty} \frac{g_\epsilon / M^2}{\log \epsilon}</math>
where M is the number of points used to generate a representation of the fractal or attractor, and gε is the number of pairs of points closer than ε to each other.
Rényi dimensions
The last three can all be seen as special cases of a continuous spectrum of generalised or Rényi dimensions of order α, defined by
- <math>D_\alpha = \lim_{\epsilon \rightarrow 0} \frac{\frac{1}{1-\alpha}\log(\sum_{i} p_i^\alpha)}{\log \epsilon}</math>
where the numerator in the limit is the Rényi entropy of order α. The Rényi dimension with α=0 treats all parts of the support of the attractor equally; but for larger values of α increasing weight in the calculation is given to the parts of the attractor which are visited most frequently.
An attractor for which the Rényi dimensions are not all equal is said to be a multifractal, or to exhibit multifractal structure. This is a signature that different scaling behaviour is occurring in different parts of the attractor.
It should be noted that practical dimension estimates are very sensitive to numerical or experimental noise, and particularly sensitive to limitations on the amount of data. Claims based on fractal dimension estimates, particularly claims of low-dimensional dynamical behaviour, should always be taken with a handful of salt — there is an inevitable ceiling, unless very large numbers of data points are presented.