P-splines with an l1 penalty for repeated measures

TL;DR

This paper explores the use of an  penalty in P-splines for semiparametric regression, especially for non-smooth functions, introducing an estimation method and demonstrating its effectiveness through simulations and real data.

Contribution

It introduces a new -penalized P-spline approach with an estimation procedure, degrees of freedom, and confidence bands, filling a gap in nonparametric regression methods.

Findings

Effective fitting of non-smooth functions demonstrated

Estimation procedure with ADMM and cross-validation developed

Application to stress study data shows practical utility

Abstract

P-splines are penalized B-splines, in which finite order differences in coefficients are typically penalized with an $\ell_2$ norm. P-splines can be used for semiparametric regression and can include random effects to account for within-subject variability. In addition to $\ell_2$ penalties, $\ell_1$-type penalties have been used in nonparametric and semiparametric regression to achieve greater flexibility, such as in locally adaptive regression splines, $\ell_1$ trend filtering, and the fused lasso additive model. However, there has been less focus on using $\ell_1$ penalties in P-splines, particularly for estimating conditional means. In this paper, we demonstrate the potential benefits of using an $\ell_1$ penalty in P-splines with an emphasis on fitting non-smooth functions. We propose an estimation procedure using the alternating direction method of multipliers and cross validation, and provide degrees of freedom and approximate confidence bands based on a ridge approximation to the $\ell_1$ penalized fit. We also demonstrate potential uses through simulations and an application to electrodermal activity data collected as part of a stress study.