On the computation of the correlation integral for fractal dimension estimation

Dimension reduction is a powerful technique which transforms data from a high-dimensional to a low-dimensional space. Usually, it requires fixing the intrinsic dimension (ID) of the low-dimensional subspace in advance. Fractal dimension is a global ID estimation method, which studies the geometry of the data set. The correlation dimension is a common method to find the fractal dimension, but its practical implementation is far from straightforward, since the correlation integral needs to be estimated for a ball of radius tending to 0. The aim of this paper is to develop approaches to approximate the correlation integral in this limit. Experimental results on real world and simulated data are used to demonstrate the algorithms and compare to other methodology. A simulation study which verifies the effectiveness of the proposed methods is also provided.