Generalized Spatially Varying Coefficient Models

In this article, we introduce a new class of nonparametric regression models, called generalized spatially varying coefficient models (GSVCMs), for data distributed over complex domains. For model estimation, we propose a nonparametric quasi-likelihood approach using the bivariate penalized spline approximation technique. We show that our estimation procedure is able to handle irregularly-shaped spatial domains with complex boundaries. Under some regularity conditions, the estimator for the coefficient function is proved to be consistent in the L2 sense and its convergence rate is established. We develop a numerically stable algorithm using penalized iteratively reweighted least squares method to estimate the coefficient functions in GSVCMs. To gain efficiency in the computation for large-scale data, we further propose a QR decomposition-based algorithm, which requires only sub-blocks of the design matrix to be computed at a time, so that it allows efficient estimation of GSVCMs for large datasets with modest computer hardware. The finite sample performance of the GSVCM and its estimation method is examined by simulations studies. The proposed method is also illustrated by an analysis of the crash data in Florida. Supplementary materials for this article are available online.

本文介绍了一类针对分布于复杂域上的数据的新型非参数回归模型——广义空间变系数模型（generalized spatially varying coefficient models, GSVCMs）。针对该模型的估计问题，本文提出了一种采用二元惩罚样条逼近技术的非参数拟似然方法。研究表明，所提估计流程能够处理带有复杂边界的不规则形状空间区域。在若干正则性条件下，本文证明了系数函数的估计量在L2范数意义下具有一致性，并推导得到其收敛速率。本文开发了一种基于惩罚迭代重加权最小二乘法的数值稳定算法，用于估计GSVCM中的系数函数。为提升大规模数据的计算效率，本文进一步提出了一种基于QR分解的算法，该算法仅需单次计算设计矩阵的子块，因此可在普通计算机硬件条件下高效完成大规模数据集下的GSVCM估计。通过模拟研究检验了GSVCM及其估计方法的有限样本性能，并以佛罗里达州的碰撞事故数据分析为例，对所提方法进行了实例验证。本文的补充材料可在线获取。