Applied Mathematics and Computation 264 (2015) 116–131 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc How to calculate the Hausdorff dimension using fractal structures M. Fernández-Martínez a,∗ , M.A. Sánchez-Granero b a Centro Universitario de la Defensa de San Javier (University Centre of Defence at the Spanish Air Force Academy), MDE-UPCT, C/Coronel López Peña, s/n, 30720, Santiago de la Ribera, Murcia, Spain b Department of Mathematics, Universidad de Almería, 04120 Almería, Spain article info MSC: 28A78 28A80 11K55 37F35 54E99 Keywords: Fractal Fractal structure Fractal dimension Hausdorff dimension Self-similar set Open set condition abstract In this paper, we provide the first known overall algorithm to calculate the Hausdorff dimen- sion of any compact Euclidean subset. This novel approach is based on both a new discrete model of fractal dimension for a fractal structure which considers finite coverings and a the- oretical result that the authors contributed previously in [14]. This new procedure combines fractal techniques with tools from Machine Learning Theory. In particular, we use a support vector machine to decide the value of the Hausdorff dimension. In addition to that, we artifi- cially generate a wide collection of examples that allows us to train our algorithm and to test its performance by external proof. Some analyses about the accuracy of this approach are also provided. © 2015 Elsevier Inc. All rights reserved. 1. Introduction The word fractal, which derives from the Latin term frangere (that means “to break”), provided a novel concept in mathematics since Benoît Mandelbrot first introduced it in the early eighties [23]. Since then, both the study and the identification of fractal patterns have become more and more important due to the large number of applications to diverse scientific fields where fractals have been found, including computation, physics, economics and statistics among them (see [10,11,15,32]). The tool that has mainly been applied in these areas to study fractals is the fractal dimension. Indeed, since some remarkable works such as [5] and [24] appeared in scientific literature, the fractal dimension has become the key invariant providing useful information about the level of irregularity that a certain system presents when being examined with enough detail. Most of the empirical applications of fractal dimension have been carried out on Euclidean spaces by means of the so-called box-counting dimension since it may be easily estimated. However, the Hausdorff dimension does provide the most accurate results since its technical definition is quite general. It is based on a measure which is the actual reason for which it presents the best theoretical properties. But it can be hard or even impossible to calculate in practical applications. This justifies why only a few attempts to calculate it have been made in the last years. Indeed, there exist only some partial results which allow us to explicitly calculate the Hausdorff dimension of some kind of sets and under certain conditions. They include Julia sets (see [17,21]) as well as strict self-similar sets. The case of self-similar sets becomes especially interesting since a few algorithms have been provided to calculate their Hausdorff dimension. Hence, if the corresponding iterated function system satisfies the ∗ Corresponding author. Tel.: +34 968189913. E-mail addresses: manuel.fernandez-martinez@cud.upct.es, fmm124@gmail.com (M. Fernández-Martínez), misanche@ual.es (M.A. Sánchez-Granero). http://dx.doi.org/10.1016/j.amc.2015.04.059 0096-3003/© 2015 Elsevier Inc. All rights reserved.