On the Estimation of Pointwise Dimension
This work addresses a fundamental challenge in fractal analysis for researchers in nonlinear dynamics and data science, though it appears incremental as it builds on existing methods.
The paper tackles the problem of estimating fractal dimension by addressing the 'dimension blindness' of the popular Grassberger-Procaccia method, proposing pointwise dimension as a local alternative and developing a new estimator based on a limit-free description.
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.