Small Deviations of Gaussian Random Fields in $L_q$-Spaces

Small Deviations of Gaussian Random Fields in Lq–Spaces

We investigate small deviation properties of Gaussian random fields in the space Lq(R N , μ) where μ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby “thin” measures μ, i.e., those which are singular with respect to the N–dimensional Lebesgue measure; the so–called self–similar measures providing a class of typical examples. For a large class of random fi...

Skew-Gaussian Random Fields

Skewness is often present in a wide range of spatial prediction problems, and modeling it in the spatial context remains a challenging problem. In this study a skew-Gaussian random field is considered. The skew-Gaussian random field is constructed by using the multivariate closed skew-normal distribution, which is a generalization of the traditional normal distribution. We present an Metropolis...

Singularities in Gaussian Random Fields

Small Deviations of Smooth Stationary Gaussian Processes

We investigate the small deviation probabilities of a class of very smooth stationary Gaussian processes playing an important role in Bayesian statistical inference. Our calculations are based on the appropriate modification of the entropy method due to Kuelbs, Li, and Linde as well as on classical results about the entropy of classes of analytic functions. They also involve Tsirelson’s upper b...

Gaussian Process Random Fields

Gaussian processes have been successful in both supervised and unsupervised machine learning tasks, but their computational complexity has constrained practical applications. We introduce a new approximation for large-scale Gaussian processes, the Gaussian Process Random Field (GPRF), in which local GPs are coupled via pairwise potentials. The GPRF likelihood is a simple, tractable, and paralle...