Functional additive regression

Abstract

We suggest a new method, called Functional Additive Regression, or FAR, for efficiently performing high-dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, $X(t)$, and a scalar response, $Y$, in two key respects. First, FAR uses a penalized least squares optimization approach to efficiently deal with high-dimensional problems involving a large number of functional predictors. Second, FAR extends beyond the standard linear regression setting to fit general nonlinear additive models. We demonstrate that FAR can be implemented with a wide range of penalty functions using a highly efficient coordinate descent algorithm. Theoretical results are developed which provide motivation for the FAR optimization criterion. Finally, we show through simulations and two real data sets that FAR can significantly outperform competing methods.Comment: Published at http://dx.doi.org/10.1214/15-AOS1346 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Automatic Variable Selection for Partially Linear Functional Additive Model and Its Application to the Tecator Data Set

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Multivariate functional response regression, with application to fluorescence spectroscopy in a cervical pre-cancer study

“…In the scalar-on-function regression in which scalar responses are regressed on functional predictors (Morris, 2015), multiple functional variables can be easily incorporated as predictors, regardless of whether they have the same interpretation (Zhu and Cox, 2009; Zhu et al, 2010; Fan et al, 2015). However, since the scalar-on-function regression treats functional variables as covariates, it models the conditional expectation of a scalar given the functional variables, and thus does not directly characterize the inter-function correlations or complicated multi-level structures between the functional variables.…”

Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density

“…The first functional formulation of a linear model dates back to a discussion by Hastie and Mallows (1993), and it is later extended in detail by Ramsay and Silverman (2005). Since then, functional linear regression model has been further extended or modified to take into account possible nonlinear relationship, some of the regression models include the functional polynomial regression model (Yao and Müller, 2010;Horváth and Reeder, 2012), functional additive regression model (Müller and Yao, 2008;Febrero-Bande and González-Manteiga, 2013;Fan and James, 2013), and nonparametric functional regression model (Ferraty and Vieu, 2006;Ferraty, Van Keilegom, and Vieu, 2010). Due to the fast development in functional regression models, it has gained an increasing popularity in various fields of application, such as atmospheric radiation (Hlubinka and Prchal, 2007), chemometrics (Frank and Friedman, 1993;Ferraty and Vieu, 2002;Burba et al, 2009;Yao and Müller, 2010), climate variation forecasting (Shang and Hyndman, 2011), demographic modeling and forecasting (Hyndman and Ullah, 2007;Hyndman and Booth, 2008;Hyndman and Shang, 2009;Chiou and Müller, 2009), earthquake modeling , gene expression (Yao et al, 2005a;Chiou and Müller, 2007), health science (Harezlak, Coull, Laird, Magari, and Christiani, 2007), linguistics (Hastie et al, 1995;Malfait and Ramsay, 2003;Aston et al, 2010), medical research (Ratcliffe et al, 2002;Yao et al, 2005b;Erbas et al, 2007), ozone level prediction (Quintela-del-Río and Francisco-Fernández, 2011), and sulfur dioxide level prediction …”