Additive Modeling of Functional Gradients

• Activity Number: 273
• Type: Invited
• Date/Time: Tuesday, August 2, 2011 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Fang Yao*+ and Hans-Georg Mueller
• Companies: University of Toronto and University of California at Davis
• Keywords: Functinal Derivative ; Functional Data ; Functional Regression ; Gradient Field ; Nonparametric Differentiation ; Principal Components

Abstract

We consider the problem of estimating functional derivatives and gradients in the framework of a functional regression setting where one observes functional predictors and scalar responses. Derivatives are then defined as functional directional derivatives which indicate how changes in the predictor function in a specified functional direction are associated with corresponding changes in the scalar response. Aiming at a model-free approach, navigating the curse of dimensionality requires the imposition of suitable structural constraints. Accordingly, we develop functional derivative estimation within an additive regression framework. This approach requires nothing more than estimating derivatives of one-dimensional nonparametric regressions, and thus is computationally very straightforward to implement, while it also provides substantial flexibility, fast computation and is consistent. We demonstrate the consistent estimation and interpretation of the resulting functional derivatives and functional gradient fields in a study of the dependence of lifetime fertility of flies on early life reproductive trajectories.