|Spatial Processes with Stochastic Heteroscadesticity
Wenying Huang, Ph.D Candidate, Department of Statistics, Colorado State University
Wednesday, June 4, 2008
11 a.m.; 006 Statistics
Stationary Gaussian processes are widely used in spatial data modeling and analysis. Stationarity is a relatively restrictive assumption regarding spatial association. By introducing stochastic volatility into a Gaussian process, we propose a stochastic heteroscedastic process (SHP) with conditional nonstationarity. That is, conditional on a latent Gaussian process, the SHP is a Gaussian process with non-stationary covariance structure. Unconditionally, the SHP is a stationary non-Gaussian process. The realizations from SHP are versatile and can represent spatial inhomogeneities. The unconditional correlation of SHP offers a rich class of correlation functions which can also allow for a smoothed nugget effect.
For maximum likelihood estimation, we propose to apply importance sampling in the likelihood calculation and latent process estimation. The importance density we constructed is of the same dimensionality as the observations. When the sample size is large, the importance sampling scheme becomes infeasible and/or inaccurate. A low-dimensional approximation model is developed to solve the numerical difficulties. We develop two spatial prediction methods: PBP (plug-in best predictor) and PBLUP (plug-in best linear unbiased predictor). Empirical results with simulated and real data show improved out-of-sample prediction performance of SHP modeling over stationary Gaussian process modeling.
We extend the single-realization model to SHP model with replicates. The spatial replications are modeled as independent realizations from a SHP model conditional on a common latent process. A simulation study shows substantial improvements in parameter estimation and process prediction when replicates are available. In a example with real atmospheric deposition data, the SHP model with replicates outperforms the Gaussian process model in prediction by capturing the spatial volatilities.