Approximate Likelihoods for Spatial Processes

• Activity Number: 308
• Type: Contributed
• Date/Time: Wednesday, August 14, 2002 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistics & the Environment*
• Author(s): Petrutza Caragea*+ and Richard Smith
• Affiliation(s): University of North Carolina, Chapel Hill and University of North Carolina
• Keywords: maximum likelihood ; approximation ; time series ; spatial statistics

Abstract

Many applications of spatial statistics to problems of environmental pollution involve evaluating a likelihood over samples of several hundred data locations. If the underlying field is Gaussian with some spatial covariance structure, this evaluation involves calculating the inverse and determinant of the covariance matrix. Although this is feasible for up to about 1000 observations, it is often troublesome for sample sizes larger than 100. Therefore, it is natural to consider using approximations to the likelihood function.

In this paper, we consider several approximate likelihoods based on grouping the observations into clusters and building an estimating function by accounting for variability both between and within clusters. This way the estimation problem becomes practical for virtually any size data sets. Theoretical results derived for an analogous time series problem allow us to compare three approximation schemes. Theoretical results derived for an analogous time series problem allow us to compare the three approximation schemes, and we illustrate the new method with simulations and with practical analysis of the U.S. fine particulate matter field.