Open Access
May 2012 Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data
Tilmann Gneiting, Hana Ševčíková, Donald B. Percival
Statist. Sci. 27(2): 247-277 (May 2012). 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 $\mathbb{R}^{d}$ 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 $p=1$.


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Tilmann Gneiting. Hana Ševčíková. Donald B. Percival. "Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data." Statist. Sci. 27 (2) 247 - 277, May 2012.


Published: May 2012
First available in Project Euclid: 19 June 2012

zbMATH: 1330.62354
MathSciNet: MR2963995
Digital Object Identifier: 10.1214/11-STS370

Keywords: Box-count , Gaussian process , Hausdorff dimension , madogram , power variation , robustness , sea-ice thickness , smoothness , variation estimator , variogram

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.27 • No. 2 • May 2012
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