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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.

Fractal dimension, functional data, kernel estimator, mixing processes, nonparametric regression, semi-normed linear space, 28A80, 60G25, 62G08, 62G05, 62G20, 62G99,

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Given a functional regression model with scalar response, the aim is to present a methodology in order to approximate in a semi-parametric way the unknown regression operator through a single index approach, but taking possible structural changes into account. Our paper presents this methodology and illustrates its behaviour both on simulated and real curves datasets. It appears, from an example of interest in spectrometry, that the method provides a nice exploratory tool both for analyzing structural changes in the spectrum and for visualizing the most informative directions, still keeping good predictive power. Even if the main objective of this work is to discuss applied issues of the method, asymptotic behaviour is shortly described.