Estimating the fractal dimension of a point cloud - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:52:50Z http://mathoverflow.net/feeds/question/73931 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73931/estimating-the-fractal-dimension-of-a-point-cloud Estimating the fractal dimension of a point cloud DoubleJay 2011-08-29T00:36:19Z 2011-08-29T16:52:06Z <p>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?</p> <p>Thanks</p> http://mathoverflow.net/questions/73931/estimating-the-fractal-dimension-of-a-point-cloud/73935#73935 Answer by Matthew Kahle for Estimating the fractal dimension of a point cloud Matthew Kahle 2011-08-29T02:10:55Z 2011-08-29T02:10:55Z <p>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.</p> <p>Link: <a href="http://front.math.ucdavis.edu/1011.2258" rel="nofollow">http://front.math.ucdavis.edu/1011.2258</a></p> http://mathoverflow.net/questions/73931/estimating-the-fractal-dimension-of-a-point-cloud/73988#73988 Answer by Piotr Migdal for Estimating the fractal dimension of a point cloud Piotr Migdal 2011-08-29T16:52:06Z 2011-08-29T16:52:06Z <p>It depends what you want to measure. For real-life data box-counting dimension based on <a href="http://en.wikipedia.org/wiki/R%C3%A9nyi_entropy" rel="nofollow">Renyi entropy</a> (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).</p> <p>References, starting from the most accessible:</p> <ul> <li>Kinsner, W. 2005. <a href="http://dx.doi.org/10.1109/COGINF.2005.1532616" rel="nofollow">A unified approach to fractal dimensions</a></li> <li>Halsey, Thomas, Mogens Jensen, Leo Kadanoff, Itamar Procaccia, and Boris Shraiman. 1986. <a href="http://dx.doi.org/10.1103/PhysRevA.33.1141" rel="nofollow">Fractal measures and their singularities: The characterization of strange sets</a></li> <li>Chhabra, Ashvin, and Roderick Jensen. 1989. <a href="http://dx.doi.org/10.1103/PhysRevLett.62.1327" rel="nofollow">Direct determination of the $f(\alpha)$ singularity spectrum</a></li> </ul>