Spatial Field Reconstruction of Non-Gaussian Random Fields: The Tukey G-and-H Random Process

Abstract

A new class of models for non-Gaussian spatial random fields is developed for spatial field reconstruction in environmental and sensory network monitoring. The developed family of models utilises a class of transformation functions known as the Tukey g-and-h transformation to create a new class of warped spatial Gaussian process model which can support various desirable features such as flexible marginal distributions, which can be skewed and/or heavy-tailed. The resulting model is widely applicable for a wide range of spatial field reconstruction applications. To utilise the model for such applications in practice, we first need to derive the statistical properties of the new family of Tukey g-and-h random fields. We are then able to derive five different objectives to perform spatial field reconstruction. These include the Minimum Mean Squared Error (MMSE), Maximum A-Posteirori (MAP) and the Spatial-Best Linear Unbiased (S-BLUE) estimators as well as the Spatial Regional and Level Exceedance estimators. Extensive simulation results and real data examples show the benefits of using the Tukey g-and-h transformation as opposed to standard Gaussian spatial random fields as is classically utilised.