Page 147 - Contributed Paper Session (CPS) - Volume 3

CPS1973 Matúš M. et al.
for all ∈ , where ⊂ ℝ is some compact domain of interest (for instance,
interval [0,1]). Function is a smooth function of the order ∈ ℕ
0
at least (for instance, = 3), and functions ’s are so called background

shock processes of the lower orders ∈ {0, . . . , − 1}, having all the same
domain as , or respectively. In other words, function is a piece-wise
0
0
constant function generating jumps in m, function is a continuous piece-
1
wise linear function generating jumps in the first derivative of m, etc. For every
location where j-th order polynomial pieces of functions “join“ together,

there is a j-th order structural break in the underlying regression function
(a jump in the j-th order derivative). The idea is to develop a direct estimation
approach that can fully reconstruct the underlying function , while also
estimating all locations where the changepoints occur (thus, defining also the
orders and magnitudes of all jumps, or breaks respectively).

(a) Piece-wise constant trend (b) Piece-wise linear trend

FIGURE 1. An illustration of the optimal changepoint location gained from
the underlying data: two piece-wise constant models in (a) are equivalent if
there are no other data points available in between observations and +1 .

On the other hand, two models in (b) are clearly different even though there
are no more data points between and +1 . Thus, there is no information

about the optimal changepoint location given within the data for a jump in
the model but there is sufficient information obtained implicitly in the data
for the optimal changepoint location for higher order derivatives.

From the computational point of view, splines are used to estimate the
underlying function and its components—the smooth part and the
0
shock processes , . . . , −1 . However, in order to stay within the convex
0
optimization framework, the shock processes are only allowed to be active at
the observational points ’s and, moreover, they are assumed to be active

only at some very few points—changepoints. Therefore, the sparsity principle
is employed when estimating the shock processes to achieve this property.
Restricting changepoint occurrences onto the set of observational points

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