Generalized, quantile and constrained nonparametric regression for spatial data
Date
2021-08
Authors
Kim, Myungjin
Major Professor
Advisor
Wang, Lily
Dai, Xiongtao
Dutta, Somak
Kim, Jae-Kwang
Zhu, Zhengyuan
Committee Member
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Altmetrics
Abstract
The dissertation consists of three research projects to discuss some limitations in spatially varying coefficient models (SVCMs) for spatial data over complicated domains.
In the first project, we introduce generalized spatially varying coefficient models (GSVCMs) to extend a class of SVCMs to investigate the effects of local features on various types of responses across locations. The GSVCM assumes that the conditional expectation of response in the exponential dispersion family is connected to a combination of linear predictors via a link function. The functional coefficients in the systematic component are approximated by using the bivariate spline over a triangulation (BPST) technique. We propose the procedures for estimation and test of nonstationarity for coefficient functions in the GSVCM. Simulation and application to the crash data in Florida clarify the benefits of the GSVCM by describing the spatial nonstationarity in associations between outcomes of interest and regional characteristics over complex domains.
In practice, there are situations where coefficient functions are assumed to be in a certain subspace. For example, they should be either non-negative or non-positive on a domain by nature. However, optimization on a global space of coefficient functions does not ensure that estimates preserve meaningful features in their signs. The second project illustrates SVCMs under the sign preservation on coefficient functions. We reformulate the constrained optimization problem concerning approximation to functional coefficients via the BPST, and describe an algorithm that produces efficient and sensible estimated smooth coefficient surfaces by preserving their sign over a domain. Inferences on coefficient functions in SVCMs under sign preservation conditions are also proposed.
Despite the fact that mean regression is useful when modeling the conditional mean response, its estimates are sensitive to the violation of model assumptions without describing the full distributional properties of the response. Although many studies on SVCMs have been conducted in the context of mean regression, little attention has been paid to studying quantile regression with multiple spatially varying coefficients, especially over complex domains. In the last project, we study the estimation and inference of quantile spatially varying coefficient models over complicated domains. For model estimation, we propose a quantile regression method adopting the BPST to approximate the unknown functional coefficients. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the optimization problem. Numerical studies have confirmed the excellent performance of the proposed approach. The proposed method is also illustrated by the analysis of mortality data in the U.S.
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Type
dissertation