Direct sample estimates of multidimensional quadratic statistical functions: application to the anisotropic KPZ equation

Abstract

We suggest a class of direct sample estimates for the two-point quadratic statistical functions of multidimensional data, which includes: estimates of the sample autocovariance function (AcF), sample mean square increment (also, structure) function, and the estimate of the power spectrum. The central estimate for the class is the sample AcF, which is constructed as to represent the finite Fourier transform of the periodogram estimate of the spectrum and is positive semidefinite. The estimate explicitly account for the anisotropy of the fields in all spatial directions and is illustrated on two examples: the morphology of the Grab nebula and the surface roughness generated as a solution of the anisotropic Kardar-Parisi-Zhang equation. We also provide an expression of the covariance of the sample AcF in the case of data assumed to be drawn from a two-dimensional Gaussian field with a known mean.