Functional Adaptive Model Estimation

In this article we are interested in modeling the relationship between a scalar, Y , and a functional predictor, X(t). We introduce a highly flexible approach called ”Functional Adaptive Model Estimation” (FAME) which extends generalized linear models (GLM), generalized additive models (GAM) and projection pursuit regression (PPR) to handle functional predictors. The FAME approach can model any of the standard exponential family of response distributions that are assumed for GLM or GAM while maintaining the flexibility of PPR. For example standard linear or logistic regression with functional predictors, as well as far more complicated models, can easily be applied using this approach. A functional principal components decomposition of the predictor functions is used to aid visualization of the relationship between X(t) and Y . We also show how the FAME procedure can be extended to deal with multiple functional and standard finite dimensional predictors, possibly with missing data. The FAME approach is illustrated on simulated data as well as on the prediction of arthritis based on bone shape. We end with a discussion of the relationships between standard regression approaches, their extensions to functional data and FAME. Some key words: Functional predictor; Functional principal components; Generalized linear models; Generalized additive models; Projection pursuit regression.