ESTIMATION OF FRACTAL INDEX AND FRACTAL DIMENSION OF A GAUSSIAN PROCESS BY COUNTING THE NUMBER OF LEVEL CROSSINGS

Abstract

Abstract. The fractal index α and fractal dimension D of a Gaussian process are characteristics that describe the smoothness of the process. In principle, smoother processes have fewer crossings of a given level, and so level crossings might be employed to estimate α or D. However, the number of crossings of a level by a non‐differentiable Gaussian process is either zero or infinity, with probability one, so that level crossings are not directly usable. Crossing counts may be rendered finite by smoothing the process. Therefore, we consider estimators that are based on comparing the sizes of the average numbers of crossings for a small, bounded number of different values of the smoothing bandwidth. The averaging here is over values of the level. Strikingly, we show that such estimators are consistent, as the size of the smoothing bandwidths shrinks to zero, if and only if the weight function in the definition of ‘average’ is constant. In this important case we derive the asymptotic bias and variance of the estimators, assuming only a non‐parametric description of covariance, and describe the estimators' numerical properties. We also introduce a novel approach to generating Gaussian process data on a very fine grid.