Page 149 - Contributed Paper Session (CPS) - Volume 3

CPS1973 Matúš M. et al.
In general, for → 0 we expect changepoints to occur in each

observation and every level of smoothness while for → ∞ no

changepoints are expected and the final fit is fully determined by the vector
of parameters ∈ ℝ (the smooth function ).

0

The minimization problem (3) is a convex problem and it can be solved
using standard optimization tools. The smoothing parameter > 0 and the

sparsity parameter > 0 can be selected, for instance, by some Cross-

Validation technique. Alternatively, one can compute the whole solution paths
for > 0 using LARS algorithm (Efron at al., 2004) and to choose the final

model from a set of plausible models along the whole solution path.

2.1 Independent Changepoints: The type penalty term ( , . . . , −1 ).
0
1
1
in (3) can take various forms. Let us firstly mention the simplest scenario where
there is no hierarchical restriction imposed on the changepoint occurrences:
any discontinuity point in the function itself or its derivatives can occur on its
own. This property can be expressed by a specific penalty form, where

−1
(4) ( , . . . , ) = ∑ ∑ | () |
1 0 −1
=0 =1

Alternatively, one can consider a whole set of regularization parameters =

( , . . . , (−1) )
0

>
λJ = (λJ0,...,λJ(p−1)) to control for the sparsity in each smoothness level
∈ {0, . . . , − 1} separately.

2.2 Simultaneous Changepoints: Unlike the previous situation it can be
suitable for some scenarios to link the changepoint at some location across all
different levels of ∈ {0, . . . , − 1}. The motivation comes from some
practical examples where the shock processes in (2) are expected to become
all active at the same point. This quality can implemented into (3) by replacing
the standard LASSO penalty in (4) with the group LASSO penalty

(5)
,
which either selects the whole group of parameters (0) , . . . , (−1) for some
∈ {1, . . . , } to be nonzero, or all parameters within this group are set to zero
exactly.

2.3 Hierarchical Changepoints: An innovative approach to changepoints in
the nonparametric regression models can be obtained by using the overlap

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