Ridge regression with adaptive additive rectangles and other piecewise functional templates

We propose a penalization algorithm for functional linear regression models, where the coefficient function fl is shrunk towards a data-driven shape template y. To the best of our knowledge, we employ the nonzero centered L-2 penalty in a novel manner, as the center of the penalty y is also optimized while being constrained to belong to a class of piecewise functions gamma, by restricting its basis expansion. This indirect penalization allows the user to control the overall shape of beta, by imposing his prior knowledge on y through the definition of gamma, without limiting the flexibility of the estimated model. In particular, we focus on the case where y is expressed as a sum of q rectangles that are adaptively positioned with respect to the regression error. As the problem of finding the optimal knot placement of a piecewise function is nonconvex, we also propose a novel parametrization that allows to reduce the number of variables in the global optimization scheme, resulting in a fitting algorithm that alternates between approximating a suitable template and solving a convex ridge-like problem. The predictive power and interpretability of our method is shown on multiple simulations and two real world case studies.& nbsp;