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

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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 non-differentiable 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.

Keywords: Box-count; Fractal dimension; Gaussian process; Hausdorff dimension; Madogram; Power variation; Robustness; Sea-ice thickness; Smoothness; Variogram; Variation estimator


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