Functional Generalized Additive Models

Abstract

We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the link-transformed mean response as the integral with respect to t of F{X(t), t} where F(·,·) is an unknown regression function and X(t) is a functional covariate. Rather than having an additive model in a finite number of principal components as in Müller and Yao (2008), our model incorporates the functional predictor directly and thus our model can be viewed as the natural functional extension of generalized additive models. We estimate F(·,·) using tensor-product B-splines with roughness penalties. A pointwise quantile transformation of the functional predictor is also considered to ensure each tensor-product B-spline has observed data on its support. The methods are evaluated using simulated data and their predictive performance is compared with other competing scalar-on-function regression alternatives. We illustrate the usefulness of our approach through an application to brain tractography, where X(t) is a signal from diffusion tensor imaging at position, t, along a tract in the brain. In one example, the response is disease-status (case or control) and in a second example, it is the score on a cognitive test. R code for performing the simulations and fitting the FGAM can be found in supplemental materials available online.

Cox regression models with functional covariates for survival data

“…Functional data regression is under intense methodological development (Marx and Eilers, 1999; Cardot et al, 1999, 2003; James, 2002; Müller and Stadtmüller, 2005; Cardot and Sarda, 2005; Ferraty and Vieu, 2006; Reiss and Ogden, 2007; Ramsay et al, 2009; James et al, 2009; Goldsmith et al, 2011; Harezlak and Randolph, 2011; Ferraty, 2011; McLean et al, 2012), though there are only a few modeling attempts in the case when outcome is time-to-event data. Probably the first paper to consider this topic was James (2002), who used a functional generalized linear model to model right-censored life expectancy, where censored outcomes are handled using the procedure of Schmee and Hahn (1979).…”

Function-on-Function Linear Regression by Signal Compression

“…Many methods have been developed for models with scalar responses and functional predictors (scalar-on-function regression) or functional responses and scalar predictors (function-on-scalar regression). A sampling of papers includes Ramsay and Dalzell (1991), Cardot et al (1999), Brown et al (2001), Ratcliffe et al (2002), Ramsay and Silverman (2005), Reiss and Ogden (2007), Marx and Eilers (1999), James (2002), Müller and Stadtmüller (2005), Goldsmith et al (2012) for linear or generalized linear scalar-on-function models, James and Silverman (2005), Li and Marx (2008), Yao and Müller (2010), McLean et al (2014) for non-linear scalar-on-function models, and Hart and Wehrly (1986), Faraway (1997), Guo et al (2003), Lin et al (2004), Morris and Carroll (2006), Reiss et al (2010) on function-on-scalar models. But there has been comparatively less work on function-on-function regression done to date, especially when there are multiple functional predictors.…”

Nonlinear function on function additive model with multiple predictor curves

“…The single-index model has been extended to multiple-index model (James and Silverman, 2005;Chen et al, 2011;Ferraty et al, 2013) with multiple linear functionals of the single predictor: Müller et al (2013) and McLean et al (2014) proposed the continuously additive model y = µ + ∫ I F (X(s), s)ds + ε, where…”