A class of stochastic volatility models for environmental applications

Abstract

Many environmental data sets have a continuous domain, in time and/or space, and complex features that may be poorly modelled with a stationary (in space and time) Gaussian process (GP). We adapt stochastic volatility modelling to this context, resulting in a stochastic heteroscedastic process (SHP), which is unconditionally stationary and non‐Gaussian. Conditional on a latent GP, the SHP is a heteroscedastic GP with non‐stationary (in space and time) covariance structure. The realizations from SHP are versatile and can represent spatial inhomogeneities. The unconditional correlation functions of SHP form a rich isotropic class that can allow for a smoothed nugget effect. We apply an importance sampling strategy to implement pseudo maximum likelihood parameter estimation for the SHP. To predict the process at unobserved locations, we develop a plug‐in best predictor. We extend the single‐realization SHP model to handle replicates across time of SHP realizations in space. Empirical results with simulated data show that SHP is nearly as efficient as a stationary GP in out‐of‐sample prediction when the true process is a stationary GP, and outperforms a stationary GP substantially when the true process is SHP. The SHP methodology is applied to enhanced vegetation index data and US NO3 deposition data for illustration.