Construct D matrix — d

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

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).

Value

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

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 $k$th 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).

References

Tibshirani (2020), "Divided differences, falling factorials, and discrete splines: Another look at trend filtering and related problems", Section 6.1.