P-splines with an l1 penalty for repeated measures

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.