# Estimating the fractal dimension of a point cloud

I have finite set of geolocation point data, and I'd like to estimate the fractal dimension. I know there are several ways to do this, and some of them give different numbers. What is the most appropriate fractal dimension to look at and what method do you recommend I use to estimate it numerically?

Thanks

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Shouldn't the Hausdorff dimension (or box counting for that matter) of a finite set be zero? Is there some model you are using that these data points are giving you information about? – BSteinhurst Aug 29 '11 at 2:36
@BSteinhurst: For real-life data you never go $\epsilon\rightarrow0$, just to sizes which are 'small enough' comparing to the whole structure and 'large enough' comparing to the sampling. – Piotr Migdal Aug 29 '11 at 16:54
@Piotr, very true. However there is some model being applied that contains your assumption that you are seeing enough relevant length scales and that the numerical methods may depend on that model. – BSteinhurst Aug 29 '11 at 17:57

It depends what you want to measure. For real-life data box-counting dimension based on Renyi entropy (of order $q$) is a common choice. For some problems $q=1$ (Shannon entropy) or $q=2$ (collision entropy) may be privileged. You can plot fractal dimension for any $q$ (obtaining rather a function that a single number). Furthermore, you can make a Legendre transform obtaining so-called singularity spectrum (or calculate it directly, see links below).

References, starting from the most accessible:

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You might look at Robert MacPherson and Benjamin Schweinhart's recent preprint "Measuring Shape with Topology", where they use topological methods (i.e. persistent homology) to estimate fractal dimension for branched polymers, Brownian trees, and self-avoiding random walks.

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You also might want to check out the following paper which I think is really nice:

http://public.lanl.gov/jt/Papers/est-fractal-dim.pdf

I hope it helps

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