Functional Generalized Additive Models

<div><p>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 <i>t</i> of <i>F</i>{<i>X</i>(<i>t</i>), <i>t</i>} where <i>F</i>( ·, ·) is an unknown regression function and <i>X</i>(<i>t</i>) is a functional covariate. Rather than having an additive model in a finite number of principal components as by Müller and Yao (<a href="#cit0023" target="_blank">2008</a>), 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 <i>F</i>( ·, ·) 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 <i>X</i>(<i>t</i>) is a signal from diffusion tensor imaging at position, <i>t</i>, 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. The FGAM is implemented in R in the refund package.  There are additional supplementary materials available online.</p></div>