title: Hierarchical factor models for large spatially misaligned data: a low-rank predictive process approach.
abstract: This article deals with jointly modeling a large number of geographically referenced outcomes observed over a very large number of locations. We seek to capture associations among the variables as well as the strength of spatial association for each variable. In addition, we reckon with the common setting where not all the variables have been observed over all locations, which leads to spatial misalignment. Dimension reduction is needed in two aspects: (i) the length of the vector of outcomes, and (ii) the very large number of spatial locations. Latent variable (factor) models are usually used to address the former, although low-rank spatial processes offer a rich and flexible modeling option for dealing with a large number of locations. We merge these two ideas to propose a class of hierarchical low-rank spatial factor models. Our framework pursues stochastic selection of the latent factors without resorting to complex computational strategies (such as reversible jump algorithms) by utilizing certain identifiability characterizations for the spatial factor model. A Markov chain Monte Carlo algorithm is developed for estimation that also deals with the spatial misalignment problem. We recover the full posterior distribution of the missing values (along with model parameters) in a Bayesian predictive framework. Various additional modeling and implementation issues are discussed as well. We illustrate our methodology with simulation experiments and an environmental data set involving air pollutants in California.
authors: Division of Biostatistics, University of Minnesota, Minnesota, USA. renxx014@umn.edu
authors: Qian
authors: Q
authors: Ren
authors: Sudipto
authors: S
authors: Banerjee
mesh_terms: D000393:Air Pollutants; D002140:California; D003198:Computer Simulation; D004784:Environmental Monitoring; D006801:Humans; D015233:Models, Statistical; D062211:Spatio-Temporal Analysis; D013269:Stochastic Processes
chemical_list: D000393:Air Pollutants
references: Environmetrics. 2011 Dec;22(8):997-1007
references: 22298952
references: Biostatistics. 2005 Oct;6(4):539-57
references: 15843593
references: Biostatistics. 2003 Oct;4(4):569-82
references: 14557112
references: Biometrics. 2003 Dec;59(4):762-9
references: 14969453
references: Biometrics. 2006 Jun;62(2):446-57
references: 16918908
references: Comput Stat Data Anal. 2007 May 1;51(8):3631-3653
references: 18443645
references: J R Stat Soc Series B Stat Methodol. 2007 Nov 1;69(5):817-838
references: 20981244
references: JAMA. 2006 Mar 8;295(10):1127-34
references: 16522832
references: J Am Stat Assoc. 2010 Jun 1;105(490):506-521
references: 20676229
references: J R Stat Soc Series B Stat Methodol. 2008 Sep 1;70(4):825-848
references: 19750209
references: Comput Stat Data Anal. 2009 Jun 15;53(8):2873-2884
references: 20016667