b_mat: Construct B matrix

In dspline: Tools for Computations with Discrete Splines

Construct B matrix

Description

Constructs the extended discrete derivative matrix of a given order, with respect to given design points.

Usage

b_mat(k, xd, tf_weighting = FALSE, row_idx = NULL)

Arguments

k	Order for the extended discrete derivative matrix. Must be >= 0.
xd	Design points. Must be sorted in increasing order, and have length at least k+1.
tf_weighting	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.
row_idx	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.

Details

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 kth 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).

Value

Sparse matrix of dimension length(row_idx) by length(xd).

References

Tibshirani (2020), "Divided differences, falling factorials, and discrete splines: Another look at trend filtering and related problems", Section 6.2.

See Also

d_mat() for constructing the discrete derivative matrix, and b_mat_mult() for multiplying by the extended discrete derivative matrix.

Examples

b_mat(2, 1:10)
b_mat(2, 1:10 / 10)
b_mat(2, 1:10, row_idx = 4:7)

dspline documentation built on June 8, 2025, 9:40 p.m.