A comparative study of nonparametric spatial models

Abstract

From the mid-twentieth century into the early 2000s, there have been significant developments and expansions in the world of nonparametric regression modeling. In particular, the advancement of nonparametric spatial regression models allows for the creation of models that better fit spatial data. In this report, we conducted a study to illustrate how the development of nonparametric spatial regression methods produced models that more closely fit spatial data. To do so, we compared the fits of select linear regression, nonparametric regression, and spatial regression models for spatial data. The following are the specific models investigated: multivariate linear regression, cubic B-spline, univariate kernel smoothing, geographically weighted regression (GWR), and cubic B-spline tensor product. First, a spatial simulation population data set was created using predefined coefficient functions, and samples of this population were used to fit each of the five models of interest. The mean integrated squared error (MISE) was calculated for the coefficient estimates of the models with functional coefficients. Across the 100 simulated samples, the B-spline tensor product model produced the smallest average mean squared error (MSE), yet larger MISE values for two of the four coefficients compared to the linear regression and GWR models. Second, an analysis of the 1970 Boston housing prices data set, which is provided in the package mlbench, was performed using 10-fold cross-validation for each of the models of interest. The mean squared prediction error (MSPE) was calculated for each combination of model and fold. The B-spline tensor product model had the smallest average MSPE, while the linear model had the largest average MSPE.