ON THE PERFORMANCE OF BOX-COUNTING ESTIMATORS OF FRACTAL DIMENSION

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.