Fast robust additive models using gamma-divergence

Abstract

Abstract Additive models offer a flexible framework for modeling nonlinear relationships between predictors and a continuous response variable, where nonparametric terms are commonly modeled via penalized splines. However, such models can be sensitive to outliers, leading to potentially unreliable estimation and inferential results. Existing methods and software for robust additive models tend to be computationally burdensome, and can come with restrictions such as permitting only one nonparametric term or not offering uncertainty quantification. In this article, we propose a fast, robust approach to fitting additive models based on the gamma-divergence, which we refer to as gamma-divergence additive models or GDAMs. Specifically, we apply gamma-divergence to the restricted maximum likelihood function of the additive model, based on treating the smoothing coefficients as random effects. This leads to an efficient minorization-maximization algorithm that adaptively downweights the impact of outlying observations via a set of normalized power density weights. Simulation studies and an application to data concerning the distribution of federal grants across the United States confirm that GDAMs perform similarly to or better than many existing (non-)robust additive modeling methods under varying degrees of contamination, while also being computationally faster and more scalable.