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

• Published in: Statistical science 2012-05, Vol.27 (2), p.247-277
• Main Authors: Gneiting, Tilmann, Ševčíková, Hana, Percival, Donald B.
• Format: Article
• Language: English
• Subjects: Box-count; Covariance; Data lines; Estimation methods; Estimators; Fractal dimensions; Fractals; Gaussian process; Hausdorff dimension; madogram; Outliers; power variation; robustness; sea-ice thickness; Simulation; smoothness; Spatial data; Statistics; Stochastic processes; Time series; variation estimator; variogram
• ISSN: 0883-4237; 2168-8745
• DOI: 10.1214/11-sts370

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 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, boxcount, 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.