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

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 $p=1$.