Interpretable sparse sliced inverse regression for functional data

The focus is on the functional regression model in which a real random variable has to be predicted from functional predictors. We use the semiparametric framework of Sliced Inverse Regression (SIR). SIR is an effective method for dimension reduction of high dimensional data which computes a linear projection of thepredictors in a low dimensional space, without loss on regression information. We address the issue of variable selection in functional SIR in order to improve the interpretability of the components. We extend the approaches of variable selection developped for multidimensional SIR to select intervals rather than separated evaluation points in the definition domain of functional predictors. SIR is formulated in different and equivalent ways which are alternatively used to produce ridge estimates of the components which are shrinked afterwards with a group-LASSO like penalty. An iterative procedure which automatically defines the relevant intervals is also introduced to provide a priori nformation to the sparse model. The approach is proved efficient on simulated data