Page 148 - Contributed Paper Session (CPS) - Volume 3

CPS1973 Matúš M. et al.
does not play any role when estimating discontinuities in the function
itself, however, it effects the situation when estimating jumps in the
corresponding derivatives (see Figure 1 for an illustration).
Let’s assume that , where is the set of

standard B-spline basis functions of the order ∈ ℕ and = ( , . . . , ) is

1
the vector of unknown parameters which needs to be estimated to obtain the
estimate of . Similarly, for the shock processes{ , . . . , −1 } we define
0
0
, for = 0, . . . , − 1, where () denotes the
+
positive part of ∈ ℝ . It is easy to see that the set of functions
defines a truncated power spline basis of order
∈ {0, . . . , − 1} over a set of knot points , which are, without any loss
of generality, assumed to be unique.
Finally, the sparsity principle is introduced with respect to () parameters,
()
such that = 0 holds for almost all indexes = 1, . . . , and = 0, . . . , −
1 (thus, there is no structural break at the location ), but some few

exceptions—existing changepoints. Moreover, for some identifiability reasons,
it is also assumed that all shock processes are inactive at the beginning (for
instance, ( ) = 0, and thus, , for all = 0, . . . , − 1).
1

The estimate for the underlying regression function in (1) can be now
obtained as a solution of the minimization problem

(3) Minimize

( , 0 , . . . , p−1 ) ∈ ℝ

where = ( , . . . , ) is the response vector, = ( , . . . , ) , and

1
1

, for = 0, . . . , − 1 are the unknown vectors of
parameters to be estimated Parameters > 0 and > 0 are some tuning

parameters, which controls for the overall amount of smoothness in the final
estimate (parameter ) and the amount of sparsity in the parameter vectors

, for = 0, . . . , − 1 (tuning parameter ) which are

included in the type penalty ( , . . . , −1 ). The penalty can take various
1
0
1
,
, () ()
forms (see below). Matrices = (ℎ ( )) and = (ℎ ℓ ( ))

ℓ

,ℓ=1
,ℓ=1
stand for a classical smoothing spline design matrix and a j-order jump
generating bases with sparse vectors of parameters , for
= 0, . . . , − 1 . The − norm penalty in (3) controls for the overall
2
smoothness of the estimate of and is such that = , where is a

0
matrix of mutual products of second derivatives of the basis functions
.
137 | I S I W S C 2 0 1 9