Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data

TL;DR

This paper reviews and assesses various estimators of fractal dimension for time series and spatial data, recommending the madogram estimator for its efficiency and robustness based on extensive simulations and data analysis.

Contribution

It provides a comprehensive evaluation of fractal dimension estimators, introduces robust transect estimators for 2D data, and offers practical recommendations based on efficiency and robustness.

Findings

Madogram estimator is recommended for time series analysis.

Robust transect estimators are proposed for 2D lattice data.

Power variation with p=1 is particularly effective for estimating Hausdorff dimension.

Abstract

The fractal or Hausdorff dimension is a measure of roughness (or smoothness) for time series and spatial data. The graph of a smooth, differentiable surface indexed in $\mathbb{R}^d$ has topological and fractal dimension $d$. If the surface is nondifferentiable and rough, the fractal dimension takes values between the topological dimension, $d$, and $d+1$. We review and assess estimators of fractal dimension by their large sample behavior under infill asymptotics, in extensive finite sample simulation studies, and in a data example on arctic sea-ice profiles. For time series or line transect data, box-count, Hall--Wood, semi-periodogram, discrete cosine transform and wavelet estimators are studied along with variation estimators with power indices 2 (variogram) and 1 (madogram), all implemented in the R package fractaldim. Considering both efficiency and robustness, we recommend the use of the madogram estimator, which can be interpreted as a statistically more efficient version of the Hall--Wood estimator. For two-dimensional lattice data, we propose robust transect estimators that use the median of variation estimates along rows and columns. Generally, the link between power variations of index $p>0$ for stochastic processes, and the Hausdorff dimension of their sample paths, appears to be particularly robust and inclusive when $p=1$.