The estimation of the fractal dimension of a Gaussian field via Euler characteristic

Abstract

Fractal dimension is a useful characteristic for measuring the irregularity of random fields. In this work, we employ the Euler characteristic (EC) of the excursion sets of a field to estimate the fractal dimension of the random field using the roughness of the underlying realization. The EC of a non‐differentiable Gaussian field is either zero or infinity with probability one and hence we employ a weighted smooth version of the field. Under some weak assumptions, the estimator is consistent when the weight function is constant. The results are examined through a simulation study and applied on real data.