Constructs the extended discrete derivative matrix of a given order, with respect to given design points.
b_mat(k, xd, tf_weighting = FALSE, row_idx = NULL)Order for the extended discrete derivative matrix. Must be >= 0.
Design points. Must be sorted in increasing order, and have length
at least k+1.
Should "trend filtering weighting" be used? This is a
weighting of the discrete derivatives that is implicit in trend filtering;
see details for more information. The default is FALSE.
Vector of indices, a subset of 1:n where n = length(xd),
that indicates which rows of the constructed matrix should be returned. The
default is NULL, which is taken to mean 1:n.
Sparse matrix of dimension length(row_idx) by length(xd).
The extended discrete derivative matrix of order \(k\), with
respect to design points \(x_1 < \ldots < x_n\), is denoted \(B^k_n\).
It has dimension \(n \times n\), and is banded with bandwidth \(k+1\).
It can be constructed recursively, as follows. For \(k \geq 1\), we first
define the \(n \times n\) extended difference matrix \(\bar{B}_{n,k}\):
$$
\bar{B}_{n,k} =
\left[\begin{array}{rrrrrrrrr}
1 & 0 & \ldots & 0 & & & & \\
0 & 1 & \ldots & 0 & & & & \\
\vdots & & & & & & & \\
0 & 0 & \ldots & 1 & & & & \\
& & & -1 & 1 & 0 & \ldots & 0 & 0 \\
& & & 0 & -1 & 1 & \ldots & 0 & 0 \\
& & & \vdots & & & & & \\
& & & 0 & 0 & 0 & \ldots & -1 & 1
\end{array}\right]
\begin{array}{ll}
\left.\vphantom{\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}}
\right\} & \hspace{-5pt} \text{$k$ rows} \\
\left.\vphantom{\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}}
\right\} & \hspace{-5pt} \text{$n-k$ rows}
\end{array}.
$$
We also define the \(n \times n\) extended diagonal weight matrix
\(Z^k_n\) to have first \(k\) diagonal entries equal to 1 and last
\(n-k\) diagonal entries equal to \((x_{i+k} - x_i) / k\), \(i =
1,\ldots,n-k\). The \(k\)th order extended discrete derivative matrix
\(B^k_n\) is then given by the recursion:
$$
\begin{aligned}
B^1_n &= (Z^1_n)^{-1} \bar{B}_{n,1}, \\
B^k_n &= (Z^k_n)^{-1} \bar{B}_{n,k} \, B^{k-1}_n,
\quad \text{for $k \geq 2$}.
\end{aligned}
$$
We note that the discrete derivative matrix \(D^k_n\) from d_mat() is
simply given by the last \(n-k\) rows of the extended matrix \(B^k_n\).
The option tf_weighting = TRUE returns \(Z^k_n B^k_n\) where \(Z^k_n\)
is the \(n \times n\) diagonal matrix as described above. This weighting
is implicit in trend filtering, as explained in the help file for
d_mat_mult(). See also Sections 6.1 and 6.2 of Tibshirani (2020) for
further discussion.
Note: For multiplication of a given vector by \(B^k_n\), instead of
forming \(B^k_n\) with the current function and then carrying out the
multiplication, one should instead use b_mat_mult(), as this will be more
efficient (both will be linear time, but the latter saves the cost of
forming any matrix in the first place).
Tibshirani (2020), "Divided differences, falling factorials, and discrete splines: Another look at trend filtering and related problems", Section 6.2.
d_mat() for constructing the discrete derivative matrix, and
b_mat_mult() for multiplying by the extended discrete derivative matrix.