On the Performance of Box-Counting Estimators of Fractal Dimension

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Box-counting estimators are popular for estimating fractal dimension. However, very little is known of their stochastic properties, despite increasing statistical interest in their application. We show that, if the irregular curve to which the estimators are applied is modelled by a Gaussian process, concise formulae may be developed for asymptotic bias and variance of box-counting estimators. These formulae point to critical differences between a naive form of the box-counting estimator, based directly on the capacity definition of fractal dimension, and a regression-inspired version of that estimator.

Exploration of Fractal Characteristics in Crash Data

“…Furthermore, the larger the fractal dimension estimate, the more complicated or more fractal the system is. In this study, two methods are used to estimate the fractal dimension of the trend, the Hall-Wood and variogram Hall and Wood (1993) introduced this fractal dimension estimator. Let u denote the range of the data and consider a box of size (width) ε k = 2 k -K , where k = 0, 1, .…”

The Dagum and auxiliary covariance families: Towards reconciling two-parameter models that separate fractal dimension and the Hurst effect

“…(4) refers to the issue of scaling laws, which describe the way in which rather elementary measurements vary with the size of the measurement unit, and we refer to Hall and Wood [34] for a detailed analysis of the relation between the fractal index α and the fractal dimension d, as well as to the previous work in [33] on Gaussian index-β random fields, with β = α/2 in this case.…”

A note on decoupling of local and global behaviours for the Dagum Random Field

“…Thus, estimation of α is linked with that of the fractal dimension D. Eq. (5) refers to the issue of scaling laws, which describe the way in which rather elementary measurements vary with the size of the measurement unit, and we refer to Hall and Wood [16] for a detailed analysis of the relation between the fractal index α and the fractal dimension D.…”

“…It is certainly true that the assumption of self-similarity is reasonable for many natural phenomena; at the same time, owing to an evident lack of methodological alternatives, only very few scientists have constructively criticized a somehow mechanical use of this assumption [16,20]. More recently, Gneiting and Schlather [14] have presented and analysed the natural counterbalance of self-affinity, the so-called decoupling.…”