Spatially Varying Coefficient Models with Sign Preservation of the Coefficient Functions

This paper considers the estimation and inference of spatially varying coefficient models, while preserving the sign of the coefficient functions. In practice, there are various situations where coefficient functions are assumed to be in a certain subspace. For example, they should be either nonnegative or nonpositive on a domain by their nature. However, optimization on a global space of coefficient functions does not ensure that estimates preserve meaningful features in their signs. In this paper, we propose sign-preserved and efficient estimators of the coefficient functions using novel bivariate spline estimators under their smoothness conditions. Our algorithm, based on the alternating direction method of multipliers, yields estimated coefficient functions that are nonnegative or nonpositive, consistent, and efficient. Simulation studies are conducted to address the advantages of the sign preservation method for a specific situation, where coefficient functions have sign constraints. Furthermore, we propose residual bootstrap-based confidence intervals for sign preserving coefficient functions over the domain of interest after adjusting the inherent bias of penalized smoothing spline techniques. Finally, we evaluate our method in a case study using air temperature, land surface temperature, and elevation in the USA. Supplementary materials accompanying this paper appear online.