P-splines with an $\ell_{1}$ penalty for repeated measures

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 correlations. 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.