  This work focuses on the issue of variable selection in functional
regression. Unlike most work in this framework, our approach does not select
isolated points in the definition domain of the predictors, nor does it rely on
the expansion of the predictors in a given functional basis. It provides an
approach to select full intervals made of consecutive points. This feature
improves the interpretability of the estimated coefficients and is desirable in
the functional framework for which small shifts are frequent when comparing one
predictor (curve) to another. Our method is described in a semiparametric
framework based on Sliced Inverse Regression (SIR). SIR is an effective method
for dimension reduction of high-dimensional data which computes a linear
projection of the predictors in a low-dimensional space, without loss on
regression information. We extend the approaches of variable selection
developed for multidimensional SIR to select intervals rather than separated
evaluation points in the definition domain of the functional predictors.
Different and equivalent formulations of SIR are combined in a shrinkage
approach with a group-LASSO-like penalty. Finally, a fully automated iterative
procedure is also proposed to find the critical (interpretable) intervals. The
approach is proved efficient on simulated and real data.
