Adding structure to generalized additive models, with applications in ecology

Abstract: Generalized additive models (GAMs) that connect a set of scalar covariates that map 1-1 to a response variable are commonly employed in ecological and other scientific disciplines. However, covariates are often inherently non-scalar, taking on multiple values for each observation of the response. They can sometimes have a temporal structure, e.g., a time series of temperature and precipitation measurements, or a spatial structure, e.g., multiple soil pH measurements made at nearby locations. While aggregating or selectively summarizing such covariates to yield a scalar covariate allows the use of standard GAM fitting procedures, exactly how to do so can be problematic, e.g., using a mean or median value for some subsequence of a time series, and information is necessarily lost as well. On the other hand naively including all $p$ components of a vector-valued covariate as $p$ separate covariates, say, without recognizing the structure, can lead to problems of multicollinearity, data sets that are excessively wide given the sample size, and difficulty extracting the primary signal provided by the covariate. In this paper we introduce three useful extensions to GAMs that provide means of efficiently and effectively handling vector-valued covariates without requiring one to choose problematic aggregations or selective summarizations. These extensions are varying-coefficient, scalar-on-function and distributed lag models. While these models have existed for some time they remain relatively underused in ecology. This article aims to show when these models can be useful and how to fit them with the popular R package \texttt{mgcv}.