Partition-Based Nonstationary Covariance Estimation Using the Stochastic Score Approximation

Abstract

We introduce computational methods that allow for effective estimation of a flexible nonstationary spatial model when the field size is too large to compute the multivariate normal likelihood directly. In this method, the field is defined as a weighted spatially varying linear combination of a globally stationary process and locally stationary processes. Often in such a model, the difficulty in its practical use is in the definition of the boundaries for the local processes, and therefore, we describe one such selection procedure that generally captures complex nonstationary relationships. We generalize the use of a stochastic approximation to the score equations in this nonstationary case and provide tools for evaluating the approximate score in $O(n\hspace{0.17em}\text{log}\hspace{0.17em}n)$ operations and O(n) storage for data on a subset of a grid. We perform various simulations to explore the effectiveness and speed of the proposed methods and conclude by predicting average daily temperature. Supplementary materials for this article are available online.

Keyword:

• Circulant embedding
• Computational efficiency
• Gridded data
• Prediction
• Spectral density
• Temperature

Funding

This material is based upon work supported by NSF Research Network on Statistics in the Atmosphere and Ocean Sciences (STATMOS) through grants DMS-1106862 and DMS-1107046 as well as NSF-DMS Grant Numbers 1406016, 1613219, and 1723158. It was also funded partially through NSF grant 570235. Research reported in this publication was supported by the National Institutes of Health under award number R01ES027892. This material was based upon work partially supported by the National Science Foundation under Grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Supplementary Materials

Appendices A-F provide further details on auxiliary details of the estimation method presented in this manuscript. These include: Appendix A compares dependent and independent sampling schemes in score approximation. Appendix B details the derivatives of the quasi-Matén spectral density. Appendix C describes the non-linear solver used for parameter estimation. Appendix D demonstrates the flexibility of partitions created via our method compared to other methods. Appendix E explores use of BIC as a proxy variable for the unknowable Rand Index by studying their correlation. Appendix F demonstrates the BIC of selected partitions plotted by the p-value cutoff used in its generation.

Additional information

Funding

This material is based upon work supported by NSF Research Network on Statistics in the Atmosphere and Ocean Sciences (STATMOS) through grants DMS-1106862 and DMS-1107046 as well as NSF-DMS Grant Numbers 1406016, 1613219, and 1723158. It was also funded partially through NSF grant 570235. Research reported in this publication was supported by the National Institutes of Health under award number R01ES027892. This material was based upon work partially supported by the National Science Foundation under Grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.