Estimation and inference in spatially varying coefficient models

Spatially varying coefficient models are a classical tool to explore the spatial nonstationarity of a regression relationship for spatial data. In this paper, we study the estimation and inference in spatially varying coefficient models for data distributed over complex domains. We use bivariate splines over triangulations to represent the coefficient functions. The estimators of the coefficient functions are consistent, and rates of convergence of the proposed estimators are established. A penalized bivariate spline estimation method is also introduced, in which a roughness penalty is incorporated to balance the goodness of fit and smoothness. In addition, we propose hypothesis tests to examine if the coefficient function is really varying over space or admits a certain parametric form. The proposed method is much more computationally efficient than the well‐known geographically weighted regression technique and thus usable for analyzing massive data sets. The performances of the estimators and the proposed tests are evaluated by simulation experiments. An environmental data example is used to illustrate the application of the proposed method.

[1]  Jianqing Fan,et al.  Statistical Estimation in Varying-Coefficient Models , 1999 .

[2]  Adrian Bowman,et al.  Multivariate varying‐coefficient models for an ecological system , 2009 .

[3]  Jianhua Z. Huang,et al.  Polynomial Spline Estimation and Inference for Varying Coefficient Models with Longitudinal Data , 2003 .

[4]  Lan Xue,et al.  ADDITIVE COEFFICIENT MODELING VIA POLYNOMIAL SPLINE , 2005 .

[5]  Heng Lian.  Variable selection for high-dimensional generalized varying-coefficient models , 2012 .

[6]  Eric A. Rosenberg,et al.  A Long-Term Hydrologically Based Dataset of Land Surface Fluxes and States for the Conterminous United States: Update and Extensions* , 2002 .

[7]  Andrew O. Finley,et al.  Comparing spatially‐varying coefficients models for analysis of ecological data with non‐stationary and anisotropic residual dependence , 2011 .

[8]  Ben Livneh,et al.  A Long-term hydrologically based dataset of land surface fluxes and states for the conterminous U.S.: Update and extensions , 2013 .

[9]  Eric R. Ziegel,et al.  Geographically Weighted Regression , 2006, Technometrics.

[10]  A S Fotheringham,et al.  Geographically weighted Poisson regression for disease association mapping , 2005, Statistics in medicine.

[11]  D. Tjøstheim,et al.  Nonparametric Estimation of Probability Density Functions for Irregularly Observed Spatial Data , 2014 .

[12]  James O. Ramsay,et al.  Spatial spline regression models , 2013 .

[13]  Yee Leung,et al.  Statistical Tests for Spatial Nonstationarity Based on the Geographically Weighted Regression Model , 2000 .

[14]  Dag Tjøstheim,et al.  Adaptively varying‐coefficient spatiotemporal models , 2009 .

[15]  Jianqing Fan,et al.  Profile likelihood inferences on semiparametric varying-coefficient partially linear models , 2005 .

[16]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[17]  Zudi Lu,et al.  Estimating spatial quantile regression with functional coefficients: A robust semiparametric framework , 2014, 1402.0958.

[18]  A. Lyapustin,et al.  Estimating Ground-Level PM2.5 Concentrations in the Southeastern U.S. using MAIAC AOD Retrievals , 2011 .

[19]  Dennis M. Gorman,et al.  Quantifying geographic variations in associations between alcohol distribution and violence: a comparison of geographically weighted regression and spatially varying coefficient models , 2007 .

[20]  Jun Zhu,et al.  Estimation for semiparametric nonlinear regression of irregularly located spatial time-series data , 2017 .

[21]  C. F. Sirmans,et al.  Spatial Modeling With Spatially Varying Coefficient Processes , 2003 .

[22]  A. Stewart Fotheringham,et al.  Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity , 2010 .

[23]  Xinwei Deng,et al.  Spatial modeling for refining and predicting surface potential mapping with enhanced resolution. , 2013, Nanoscale.

[24]  Zhongliang Cai,et al.  Coverage inequality and quality of volunteered geographic features in Chinese cities: Analyzing the associated local characteristics using geographically weighted regression , 2017 .

[25]  Yaping Huang,et al.  Spatially varying relationships between surface urban heat islands and driving factors across cities in China , 2019 .

[26]  A. Gelfand,et al.  Gaussian predictive process models for large spatial data sets , 2008, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[27]  Jstor.  Journal of the Royal Statistical Society. Series D, (The statistician) , 1993 .

[28]  Ming-Jun Lai,et al.  Bivariate Penalized Splines for Regression , 2013 .

[29]  Daniel A. Griffith,et al.  Spatially varying coefficient models in real estate: Eigenvector spatial filtering and alternative approaches , 2016, Comput. Environ. Urban Syst..

[30]  Qingguo Tang.  Robust estimation for functional coefficient regression models with spatial data , 2014 .

[31]  Wenyang Zhang,et al.  A semiparametric spatial dynamic model , 2014, 1405.6061.

[32]  S. Fotheringham,et al.  Some Notes on Parametric Significance Tests for Geographically Weighted Regression , 1999 .

[33]  D. Ruppert.  Selecting the Number of Knots for Penalized Splines , 2002 .

[34]  Renato M. Assunção,et al.  Space varying coefficient models for small area data , 2003 .

[35]  Michael H. Neumann,et al.  Bootstrap tests for simple structures in nonparametric time series regression , 2008 .

[36]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[37]  M. G. Estes,et al.  Estimating ground-level PM(2.5) concentrations in the southeastern U.S. using geographically weighted regression. , 2013, Environmental research.

[38]  Chang-Lin Mei,et al.  Spatially Varying Coefficient Models: Testing for Spatial Heteroscedasticity and Reweighting Estimation of the Coefficients , 2011 .

[39]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[40]  Simon N. Wood,et al.  Soap film smoothing , 2008 .

[41]  S. Koopman,et al.  Spline Smoothing Over Difficult Regions , 2008 .

[42]  Dewei Wang,et al.  Spatially modeling the effects of meteorological drivers of PM2.5 in the Eastern United States via a local linear penalized quantile regression estimator , 2017, Environmetrics.

[43]  Stanley R. Johnson,et al.  Varying Coefficient Models , 1984 .

[44]  S. Wood.  Thin plate regression splines , 2003 .

[45]  QingGuo Tang,et al.  B-spline estimation for varying coefficient regression with spatial data , 2009 .

[46]  Wen-Xiu Zhang,et al.  Testing the Importance of the Explanatory Variables in a Mixed Geographically Weighted Regression Model , 2006 .

[47]  Larry L. Schumaker,et al.  Spline functions on triangulations , 2007, Encyclopedia of mathematics and its applications.

[48]  Christopher Bitter,et al.  Incorporating spatial variation in housing attribute prices: a comparison of geographically weighted regression and the spatial expansion method , 2007, J. Geogr. Syst..

[49]  Daniel J. Jacob,et al.  Correlations between fine particulate matter (PM2.5) and meteorological variables in the United States: implications for the sensitivity of PM2.5 to climate change. , 2010 .

[50]  Jianqing Fan,et al.  Functional-Coefficient Regression Models for Nonlinear Time Series , 2000 .

[51]  David Wheeler,et al.  Multicollinearity and correlation among local regression coefficients in geographically weighted regression , 2005, J. Geogr. Syst..