GAMs with integrated model selection using penalized regression splines and applications to environmental modelling

Abstract

Generalized additive models (GAMs) have been popularized by the work of Hastie and Tibshirani ( Generalized Additive Models (1990)) and the availability of user friendly gam software in Splus. However, whilst it is flexible and efficient, the gam framework based on backfitting with linear smoothers presents some difficulties when it comes to model selection and inference. On the other hand, the mathematically elegant work of Wahba ( Spline Models for Observational Data (1990)) and co-workers on Generalized Spline Smoothing (GSS) provides a rigorous framework for model selection (SIAM J. Sci. Statist. Comput. 12 (1991) 383) and inference with GAMs constructed from smoothing splines: but unfortunately these models are computationally very expensive with operations counts that are of cubic order in the number of data. A ‘middle way’ between these approaches is to construct GAMs using penalized regression splines (e.g. Marx and Eilers, Comput. Statist. Data Anal. (1998)). In this paper, we develop this idea further and show how GAMs constructed using penalized regression splines can be used to get most of the practical benefits of GSS models, including well founded model selection and multi-dimensional smooth terms, with the ease of use and low computational cost of backfit GAMs. Inference with the resulting methods also requires slightly fewer approximations than are employed in the GAM modelling software provided in Splus. This paper presents the basic mathematical and numerical approach to GAMs implemented in the R package mgcv, and includes two environmental examples using the methods as implemented in the package.