User's Guide to lqa - LMU

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initialized using the function structure(), where all its elements are listed. Note that
the membership of the class penalty must also be given here.
For the fused lasso penalty, things are a little bit more complicated. Since a j , j =
p + 1, . . . , 2p − 1 consists of two non-zero elements we cannot apply getpenmat(). The
first derivatives of the penalty terms are
p ′ λ,j(|ξ j |) =
{
λ 1 1 {ξj ≠0}, j = 1, . . . , p,
λ 2 1 {ξj ≠0}, j = p + 1, . . . , 2p − 1.
This will be returned by the first.derivative() function, see below. A summarized
version of the coefficients is
⎡
⎢
⎣
a 1 a 2 . . . a p a p+1 a p+2 . . . a 2p−1
1 0 . . . 0 −1 0 . . . 0
0 1 . . . 0 1 −1 . . . 0
0 0 . . . 0 0 1 . . . 0
. .
.. . . . . . .. .
0 0 . . . 0 0 0 . . . 0
0 0 . . . 0 0 0 . . . −1
0 0 . . . 1 0 0 . . . 1
⎤
.
⎥
⎦
This (p × (2p − 1))-dimensional matrix will be returned by a.coefs(). So the complete
source code of the fused.lasso object is:
R> fused.lasso