P-splines with an l 1 penalty for repeated measures

Abstract

P-splines are penalized B-splines, in which finite order differences in coefficients are typically penalized with an l 2 norm. P-splines can be used for semiparametric regression and can include random effects to account for within-subject correlations. In addition to l 2 penalties, l 1 -type penalties have been used in nonparametric and semiparametric regression to achieve greater flexibility, such as in locally adaptive regression splines, l 1 trend filtering, and the fused lasso additive model. However, there has been less focus on using l 1 penalties in P-splines, particularly for estimating conditional means. In this paper, we demonstrate the potential benefits of using an l 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 l 1 penalized fit. We also demonstrate potential uses through simulations and an application to electrodermal activity data collected as part of a stress study.