On the Estimation of Pointwise Dimension

TL;DR

This paper introduces a new estimator for fractal dimension called pointwise dimension, addressing limitations of existing methods by capturing local dimensional variations and avoiding reliance on limiting behavior from finite data.

Contribution

It proposes a novel pointwise dimension estimator that overcomes correlation dimension's insensitivity to local dimensional differences and finite data limitations.

Findings

The estimator effectively captures local dimensional variations.

It addresses the 'dimension blindness' problem of correlation dimension.

Potential applications include analyzing complex fractal structures.

Abstract

Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation dimension. There are two major difficulties in estimation based on this method. The first is the insensitivity of correlation dimension itself to differences in dimensionality over data, which we term "dimension blindness". The second comes from the reliance of the method on the inference of limiting behavior from finite data. We propose pointwise dimension as an object for estimation in response to the dimension blindness of correlation dimension. Pointwise dimension is a local quantity, and the distribution of pointwise dimensions over the data contains the information to which correlation dimension is blind. We use a "limit-free" description of pointwise dimension to develop a new estimator. We conclude by discussing potential applications of our estimator as well as some challenges it raises.