A New Construction of Covariance Functions for Gaussian Random Fields

We develop a new approach to creating covariance functions for Gaussian random fields via point processes on the complex plane. We present two approaches to construct valid covariance functions by exploiting Bochner’s theorem and then modeling the characteristic function of a covariance function. In particular, we use a complex point process (CPP) to model the Fourier coefficients and illustrate how to estimate the covariance function of a Gaussian random field model from data. We further illustrate our construction approaches and compare several algorithms via simulations. The methods are exemplified via applications to real-life research data in wheat yields and earthquake studies.