Functional projection pursuit regression

Abstract

In this paper we introduce a flexible approach to approximate the regression function in the case of a functional predictor and a scalar response. Following the Projection Pursuit Regression principle, we derive an additive decomposition which exploits the most interesting projections of the prediction variable to explain the response. On one hand, this approach allows us to avoid the well-known curse of dimensionality problem, and, on the other one, it can be used as an exploratory tool for the analysis of functional dataset. The terms of such decomposition are estimated with a procedure that combines a spline approximation and the one-dimensional NadarayaWatson approach. The good behavior of our procedure is illustrated from theoretical and practical points of view. Asymptotic results state that the terms in the additive decomposition can be estimated without suffering from the dimensionality problem, while some applications to real and simulated data show the high predictive performance of our method.

Rejoinder on: Probability enhanced effective dimension reduction for classifying sparse functional data

“…, β K } serves the purpose of dimension reduction to overcome the curse of dimensionality caused by exponentially decaying small ball probability. This is in line with the semiparametric ideas such as projection pursuit (Chen et al 2011;Ferraty et al 2003Ferraty et al , 2013Yao et al 2015) and partial linear modeling (Aneiros-Pérez and Vieu 2006; Lian 2011), among others. We would like to emphasize that the PEFCS based on sufficiency of linear projections includes (generalized) linear or additive regression as special cases, and is "link-free" compared to multiple index modeling methods.…”

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…”

A partitioned Single Functional Index Model

“…Such dataset has become a benchmark in functional regression studies: the aim is to predict the percentage of fat contained in each sample of meat from its near-infrared spectrum. Some empirical evidences on such case study emerge from literature: as pointed out in Ferraty et al (2013), the regression function exhibits a nonlinear nature; moreover the role of some specific points of the spectrometric curve in explaining the fat content has been emphasized in Ferraty et al (2010). Combining previous observations, it is reasonable to expect that the decomposition (1) can lead to a regression model with better prediction ability, and which can be able to provide a key to better understand the relationship between predictor and response.…”

“…The extensions to the functional framework of these ideas, such a functional semi-parametric methodology, have been intensively studied in the literature: conditions for identifiability for FSIM have been introduced in Ferraty et al (2003) and several estimation techniques are proposed in Ait-Saïdi et al (2008), Amato et al (2006) and . Moreover, this approach can be seen as the first step of the Functional Projection Pursuit regression developed in Ferraty et al (2013).…”