Estimation and Prediction in Spatial Models With Block Composite Likelihoods

Abstract

This article develops a block composite likelihood for estimation and prediction in large spatial datasets. The composite likelihood (CL) is constructed from the joint densities of pairs of adjacent spatial blocks. This allows large datasets to be split into many smaller datasets, each of which can be evaluated separately, and combined through a simple summation. Estimates for unknown parameters are obtained by maximizing the block CL function. In addition, a new method for optimal spatial prediction under the block CL is presented. Asymptotic variances for both parameter estimates and predictions are computed using Godambe sandwich matrices. The approach considerably improves computational efficiency, and the composite structure obviates the need to load entire datasets into memory at once, completely avoiding memory limitations imposed by massive datasets. Moreover, computing time can be reduced even further by distributing the operations using parallel computing. A simulation study shows that CL estimates and predictions, as well as their corresponding asymptotic confidence intervals, are competitive with those based on the full likelihood. The procedure is demonstrated on one dataset from the mining industry and one dataset of satellite retrievals. The real-data examples show that the block composite results tend to outperform two competitors; the predictive process model and fixed-rank kriging. Supplementary materials for this article is available online on the journal web site.

Key Words:

• Gaussian process
• GPU
• Large datasets
• Parallel computing
• Spatial statistics

SUPPLEMENTARY MATERIALS

Appendix: Score function and Hessian.

Datasets, CPU, and GPU examples of code.

ACKNOWLEDGMENTS

We thank the Statistical and Applied Mathematical Sciences Institute (SAMSI) for support during the program on space–time analysis (2009–2010). We also thank NVIDIA for supporting us with graphics cards. Rana Gruber provided the joints data, while Noel Cressie and Gardar Johannesson made the TCO data acquired by NASA available to us. Brian Reich was supported by National Science Foundation grant number 1107046.