STOCHASTIC APPROXIMATION OF SCORE FUNCTIONS FOR GAUSSIAN PROCESSES

• 成果类型: Article
• 署名作者: Stein, Michael L.; Chen, Jie; Anitescu, Mihai
• 署名单位: University of Chicago; United States Department of Energy (DOE); Argonne National Laboratory
• 刊物名称: ANNALS OF APPLIED STATISTICS
• ISSN/ISSBN: 1932-6157
• DOI: 10.1214/13-AOAS627
• 发表日期: 2013
• 页码: 1162-1191
• 关键词: fixed-domain asymptotics; parameter-estimation; likelihood; estimator; algorithm; matrix; trace

摘要

We discuss the statistical properties of a recently introduced unbiased stochastic approximation to the score equations for maximum likelihood calculation for Gaussian processes. Under certain conditions, including bounded condition number of the covariance matrix, the approach achieves O(n) storage and nearly O(n) computational effort per optimization step, where n is the number of data sites. Here, we prove that if the condition number of the covariance matrix is bounded, then the approximate score equations are nearly optimal in a well-defined sense. Therefore, not only is the approximation efficient to compute, but it also has comparable statistical properties to the exact maximum likelihood estimates. We discuss a modification of the stochastic approximation in which design elements of the stochastic terms mimic patterns from a 2(n) factorial design. We prove these designs are always at least as good as the unstructured design, and we demonstrate through simulation that they can produce a substantial improvement over random designs. Our findings are validated by numerical experiments on simulated data sets of up to 1 million observations. We apply the approach to fit a space-time model to over 80,000 observations of total column ozone contained in the latitude band 40 degrees-50 degrees N during April 2012.