  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.
