Optimal low rank model for multivariate spatial data
Abstract
Massive spatial data have been observed in many disciplines such as environmental studies, earth science and ecology. Those massive spatial data present a computational challenge to statistical analysis because of the large covariance matrix involved. One of the most useful techniques for analyzing massive spatial data is to approximate the spatial process by a set of latent variables and latent functions since it simplifies the operation of the large covariance matrix. This leads to a low rank model. Various low rank models have been constructed for spatial data. However, the optimal low rank model is based on Karhunen-Loéve expansion since it minimizes the total mean square error. We proposed two algorithms to obtain Karhunen-Loéve expansion for spatial process and compared them with an existing one. Based on our efficient algorithm, we developed the Karhunen-Loéve Low Rank Model (KL Low Rank Model) for univariate spatial massive data and compared with predictive process Low Rank Model through simulations and real data analysis. The results show that KL Low Rank Model provides better predictive performance than predictive process Low Rank Models. KL Low Rank Model can be extended for multivariate spatial process. A major challenge in modelling multivariate spatial data is to specify appropriate covariance function which should be positive definite. There have been recent developments of the parametric covariance functions for multivariate spatial processes. However, usually a larger number of parameters need to be estimated and hence the complexity of the model would be increased significantly. In addition, because of the positive definite constraint, the unknown parameters have to satisfy certain constraints which further complicates the model. We model the marginal spatial process with KL Low Rank Model. Based upon the marginal KL Low Rank Model, a positive definite non-parametric cross covariance function can be specified (see Wang [2011]). There is no constraint on the marginal models to ensure a valid covariance function. The results from simulations and real data analysis show that this model outperforms.
Degree
Ph.D.
Advisors
Zhang, Purdue University.
Subject Area
Statistics
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