d_mat: Construct D matrix

In dspline: Tools for Computations with Discrete Splines

Construct D matrix

Description

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

Usage

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

Arguments

k	Order for the 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-k) 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-k).

Details

The discrete derivative matrix of order k, with respect to design points x_1 < \ldots < x_n, is denoted D^k_n. It has dimension (n-k) \times n, and is banded with bandwidth k+1. It can be constructed recursively, as follows. We first define the (n-1) \times n first difference matrix \bar{D}_n:

\bar{D}_n = \left[\begin{array}{rrrrrr} -1 & 1 & 0 & \ldots & 0 & 0 \\ 0 & -1 & 1 & \ldots & 0 & 0 \\ \vdots & & & & & \\ 0 & 0 & 0 & \ldots & -1 & 1 \end{array}\right],

and for k \geq 1, define the (n-k) \times (n-k) diagonal weight matrix W^k_n to have diagonal entries (x_{i+k} - x_i) / k, i = 1,\ldots,n-k. The kth order discrete derivative matrix D^k_n is then given by the recursion:

\begin{aligned} D^1_n &= (W^1_n)^{-1} \bar{D}_n, \\ D^k_n &= (W^k_n)^{-1} \bar{D}_{n-k+1} \, D^{k-1}_n, \quad \text{for $k \geq 2$}. \end{aligned}

We note that \bar{D}_{n-k+1} above denotes the (n-k) \times (n-k+1) version of the first difference matrix that is defined in the second-to-last display.

The option tf_weighting = TRUE returns W^k_n D^k_n where W^k_n is the (n-k) \times (n-k) diagonal matrix as described above. This weighting is implicit in trend filtering, as explained in the help file for d_mat_mult(). See also Section 6.1 of Tibshirani (2020) for further discussion.

Note: For multiplication of a given vector by D^k_n, instead of forming D^k_n with the current function and then carrying out the multiplication, one should instead use d_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 6.1.

See Also

b_mat() for constructing the extended discrete derivative matrix, and d_mat_mult() for multiplying by the discrete derivative matrix.

Examples

d_mat(2, 1:10)
d_mat(2, 1:10 / 10)
d_mat(2, 1:10, row_idx = 2:5)

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