Estimation and inference in spatially varying coefficient models

Abstract

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.

Spatial autoregressive partially linear varying coefficient models

“…According to Assumption (A2), the eigenvalues of n −1 Z Z is bounded below and above except on an event with probability going to zero. By Lemma C.5 in Mu, Wang, and Wang (2018), the the eigenvalues of n −1 K n X B X B is bounded below and above except on an event with probability going to zero. Therefore, the eigenvalues of nU 11 and nK −1 n U 22 are bounded below and above except on an event with probability going to zero.…”

Spatiotemporal Autoregressive Partially Linear Varying Coefficient Models

“…For the spatial dimension, we consider triangulation of a polygonal domain Ω, which is an effective tool to handle data distributed on irregular 2D regions with complex boundaries and/or interior holes. Lindgren et al (2015), Mu et al (2018) and Yu et al (2020a) used triangulation to partition the spatial domain into triangles. In the following we use τ to denote a triangle, which is a convex hull of three points not located in one line.…”

“…The STAR-PLVCM encompasses many existing models as special cases, such as the spatiotemporal autoregressive (STAR) model when all the β 0k 's are assumed to be constant (Pace et al, 1998); the binary treatment model with spatial interactions, when X ik consists of a constant term only; the semiparametric SAR model when X ik consists of a constant term only and its coefficient-effect is assumed to be spatially dependent only (Su and Jin, 2010); the partially linear varying coefficient model (Li and Liang, 2008) when there is no neighbor effect in the model, i.e., α 0 = 0; the TVCM (Fan and Zhang, 2008;Park et al, 2015;Yang et al, 2006) when only the time-index is included in the coefficient functions; and the SVCM in Fotheringham et al (2002); Gelfand et al (2003); Mu et al (2018) when only the spatial-index is included and neighbor effects are not considered.…”

Semiparametric model selection for identification of environmental covariates related to adult groundfish catches and weights

“…Spatiotemporal variation among values is incorporated through three-dimensional functions based on spline smoothing 22 – 27 , accounting for spatial and temporal autocorrelation and improving prediction and inference 28 , 29 . Similar structures have been widely used in the related spatially-varying coefficient models 30 , 31 and temporally-varying coefficient models 32 , including applications in forestry 33 , ecology 34 , and economics 35 , 36 . Spline-based methods, which are often used to estimate in SIVCMs 37 , 38 , are more robust for spatially correlated data and do not require the spatial variation to be specified by a functional form 39 , 40 .…”