Séance Séminaire

Séminaire de Probabilités et Statistique

Bootstrap and inference in nonparametric regression when both response and predictor are functional data

Technologies progress in terms of computational tools or memory capicities allow to collect and handle Functional Data which are random variables observed in some continuous way (for instance, a functional dataset may content a collection of curves/surfaces, etc). Functional Data Analysis (FDA) is a very exciting part of modern Statistics. This is certainly due to their numerous potentialities in terms of applications. New theoretical and computational challenges arise which are extremely motivating for the statistical community. From the 1990's, lots of investigations involving functional data focused on linear modelling and successful functional statistical methods have been achieved (see the monographies of Ramsay and Silverman, 2002 and 2005, or Bosq, 2000). More recently, from the 2000's, nonparametric approaches have been widely investigated (for an overview see Ferraty and Vieu, 2006), especially in regression model in order to take into account nonlinear relationship between scalar response and functional predictors.

This talk focuses on the situation when the response is also a functional variable. For instance, in the heating-district data used in this presentation, the problematic is to predict a daily load-demand curves (hourly consumption of energy = functional response) by taking into account a corresponding daily temperature curve (functional predictor). We propose to regress nonparametrically the functional response on the functional predictor by using a standard functional kernel estimator. Asymptotic results as well as practical aspects will be presented. However, getting confidence areas from asymptotic distribution seems to be unrealistic in this functional data situation. An alternative standard tool for doing that is the so-called bootstrap method. We will see how the bootstrap methodology can be extended to the setting when one regresses a functional response on a functional predictor. Theoretical properties will be stated whereas simulations and real datasets will illustrate the good practical behaviour of the boostrap method in this functional situation. As a by-product, we show how functional pseudo-confidence areas can be built.

References:
Bosq, D. (2000). Linear processes in fuction spaces, theory and applications. Lecture Notes in Statistics, 149, Springer-Verlag, New York.
Ferraty, F., Vieu, P. (2006). NonParametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics. Springer, New York.
Ramsay, J.O., Silverman, B.W. (2002) Applied functional data analysis; methods and case studies. Springer-Verlag, New York.
Ramsay, J.O., Silverman, B.W. (2005) Functional data analysis. Second edition. Springer-Verlag, New York.