Ridge regression with adaptive additive rectangles and other piecewise functional templates
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We propose an L_2-based penalization algorithm for functional linear regression models, where the coefficient function is shrunk towards a data-driven shape template γ, which is constrained to belong to a class of piecewise functions by restricting its basis expansion. In particular, we focus on the case where γ can be 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, the proposed parametrization 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.
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