  We assume a nonparametric regression model with signals given by the sum of a
piecewise constant function and a smooth function. To detect the change-points
and estimate the regression functions, we propose PCpluS, a combination of the
fused Lasso and kernel smoothing. In contrast to existing approaches, it
explicitly uses the assumption that the signal can be decomposed into a
piecewise constant and a smooth function when detecting change-points. This is
motivated by several applications and by theoretical results about partial
linear model. Tuning parameters are selected by cross-validation. We argue that
in this setting minimizing the L1-loss is superior to minimizing the L2-loss.
We also highlight important consequences for cross-validation in piecewise
constant change-point regression. Simulations demonstrate that our approach has
a small average mean square error and detects change-points well, and we apply
the methodology to genome sequencing data to detect copy number variations.
Finally, we demonstrate its flexibility by combining it with smoothing splines
and by proposing extensions to multivariate and filtered data.
