Spatial functional regression methods allow researchers to model spatially dependent functional random variables, often using a kriging-based interpolation method. However, less effort has been devoted towards the goal of modeling the relationship between a scalar response and a functional covariate throughout a spatial domain. Here, we introduce a method to characterize this association by estimating a coefficient function over points on a spatial lattice. Gridded data products are becoming commonplace in many fields, making this approach useful to researchers in many disciplines. As in our data application, we assume that the set of coefficient functions of interest are spatially similar; therefore, we aim to improve our estimates by penalizing dissimilarity between coefficient function estimates at adjacent points on the lattice. The results of our simulation study provide additional support for this approach. We perform an analysis of surface-level ozone in the Eastern US and consider three functional covariates: the vertical temperature profile (VTP), specific humidity, and omega. Our analysis suggests that the VTP and specific humidity may have stronger associations with surface-level O\(_3\) than omega, and that these relationships differ by region and by altitude. We also propose a permutation-based hypothesis test to determine whether it is reasonable to believe that coefficient functions truly differ from the zero function. Pointwise application of this test suggests that these atmospheric profile variables (APVs) may be useful predictor variables in much of the spatial domain.

空间函数回归方法允许研究人员对空间相关的函数随机变量进行建模，通常使用基于克里金法的插值方法。然而，对关系建模的目标投入较少在整个空间域中的标量响应和函数协变量之间。在这里，我们介绍了一种通过估计空间格子上点的系数函数来表征这种关联的方法。网格数据产品在许多领域变得司空见惯，这使得这种方法对许多学科的研究人员都很有用。与我们的数据应用程序一样，我们假设感兴趣的系数函数集在空间上是相似的；因此，我们的目标是通过惩罚格子上相邻点的系数函数估计之间的差异来改进我们的估计。我们的模拟研究结果为这种方法提供了额外的支持。我们对美国东部的地表臭氧进行了分析，并考虑了三个函数协变量：垂直温度分布 (VTP)、比湿度和欧米茄。\(_3\) 而不是 omega，并且这些关系因地区和海拔而异。我们还提出了一个基于置换的假设检验，以确定相信系数函数真正不同于零函数是否合理。该测试的逐点应用表明，这些大气廓线变量 (APV) 在大部分空间域中可能是有用的预测变量。