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

Abstract

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, 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. Fig. 5. Log-log regression for the standard version of the box-count estimator and the datasets in the lower row of Figure 1. Only the points marked with filled circles are used when fitting the regression line.

Deterministic Chaos, El Niño Southern Oscillation, and Seasonal Influenza Epidemics

“…Differentiable graphs have topological dimension d, while non-differentiable graphs have fractal dimension, which have values between the topological dimension d and d + 1 (Gneiting et al, 2012). Fractal time series, which are not differentiable, can be characterized using the Hurst exponent (H), which measures long range dependence or memory.…”

Validation of CMIP5 multimodel ensembles through the smoothness of climate variables

“…(1) where u0(C, H), for the scale parameter, C!0, and the smoothness parameter, H # (0,1) (Gneiting et al, 2012 and Taqqu, 1994;Genton et al, 2007). The index H determines the smoothness of the self-similar process, and it has been widely used as a measure of surface roughness for various natural phenomena, such as the surface of soil, surface height measurements of computer chips, etc.…”

“…[see Mandelbrot and Wallis (1969) and Adler (1981) for some application examples]. Note that the quantity, 2H, is essentially the fractal index for the mean zero isotropic Gaussian process, Z (Gneiting and Schlather, 2004;Gneiting et al, 2012). A locally self-similar process is more general than a selfsimilar process, in the sense that it does not fully specify the variogram but only in a region around the origin.…”

Market Efficiency, Roughness and Long Memory in PSI20 Index Returns: Wavelet and Entropy Analysis

“…(2) The Hall-Wood estimator is a version of the box-counting estimator, this being obtained from (10) (see Gneiting et al [37]). In fact, considering the boxes of size (scale) ε that intersect with the linearly interpolated data graph {(t, X t ) : t = i n , i = 0, 1, .…”

“…We consider different approaches, such as Hall-Wood, Genton and box-counting estimators (see [23][24][25]37]), applied to the PSI20 log-prices and to the estimated series of integrated variance. The reference value for the fractal dimension is 1.5, which stands for the absence of either local persistence or local anti-persistence.…”