With the proliferation of gridded data products, modeling the relationship between a scalar response and a functional covariate over the points on a regular lattice is becoming increasingly important. In this work, our overall aim is to better understand the relationship between high quantiles of surface-level ozone and the vertical temperature profile (VTP), a functional covariate, over the US Southwest in the summer. We develop our penalized functional quantile regression based approach within the framework provided by functional data analysis. As we assume that coefficient functions at points on the lattice exhibit spatial similarity, we obtain improved estimates by penalizing dissimilarity between nearby coefficient function estimates. In order to better account for the high degree of diversity of this region, we more strongly weight differences between coefficient function estimates at cells that exhibit a higher degree of geographical and climatological similarity. Our analysis suggests that the VTP is associated with acute surface-level ozone events during the summertime over this region, and the nature of this relationship differs spatially.

Traditionally, actuaries make their predictions based on simple, robust methods. Stochastic models become increasingly popular because they can enrich the point estimates with error estimates or even provide the whole probability distribution. Here, we construct such a model for German inpatient health expenses per age using the functional data approach. This allows us to see in which age groups the expenses change the most and where predictions are most uncertain. Jumps in the derived model parameters indicate that 3 years might be outliers. In fact, they can be explained by changes in the reimbursement system and must be dealt with. As an application, we compute the probability distribution of the total health expenses in the upcoming years.