2011/01/07 by Tilmann Gneiting, Hana Ševčíková, Donald B. Percival · 1 voice · 4 citations
Mathematics · #stat.ME
paper · pdf · doi:10.1214/11-sts370
published as Statistical Science 2012, Vol. 27, No. 2, 247-277 · Published in at http://dx.doi.org/10.1214/11-STS370 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv published 2011/01/07 · arxiv created 2012/07/25 · arxiv updated 2012/07/25
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 ℝd 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.