b_mat_mult: Multiply by B matrix

In dspline: Tools for Computations with Discrete Splines

Multiply by B matrix

Description

Multiplies a given vector by B, the extended discrete derivative matrix of a given order, with respect to given design points.

Usage

b_mat_mult(v, k, xd, tf_weighting = FALSE, transpose = FALSE, inverse = FALSE)

Arguments

v	Vector to be multiplied by B, the extended discrete derivative matrix.
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.
transpose	Multiply by the transpose of B? The default is FALSE.
inverse	Multiply by the inverse of B? The default is FALSE.

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 is square, having dimension n \times n. Acting on a vector v of function evaluations at the design points, denoted v = f(x_{1:n}), it gives the discrete derivatives of f at the points x_{1:n}:

B^k_n v = (\Delta^k_n f) (x_{1:n}).

The matrix B^k_n can be constructed recursively as the product of a diagonally-weighted first difference matrix and B^{k-1}_n; see the help file for b_mat(), or Section 6.2 of Tibshirani (2020). Therefore, multiplication by B^k_n or by its transpose can be performed in O(nk) operations based on iterated weighted differences. See Appendix D of Tibshirani (2020) for details.

The option tf_weighting = TRUE performs multiplication by Z^k_n B^k_n where Z^k_n is an n \times n diagonal matrix whose top left k \times k block equals the identity matrix and bottom right (n-k) \times (n-k) block equals W^k_n, the latter being a diagonal weight matrix that is implicit in trend filtering, as explained in the help file for d_mat_mult().

Lastly, the matrix B^k_n has a special inverse relationship to the falling factorial basis matrix H^{k-1}_n of degree k-1 with knots in x_{k:(n-1)}; it satisfies:

Z^k_n B^k_n H^{k-1}_n = I_n,

where Z^k_n is the n \times n diagonal matrix as described above, and I_n is the n \times n identity matrix. This, combined with the fact that the falling factorial basis matrix has an efficient recursive representation in terms of weighted cumulative sums, means that multiplying by (B^k_n)^{-1} or its transpose can be performed in O(nk) operations. See Section 6.3 and Appendix D of Tibshirani (2020) for details.

Value

Product of the extended discrete derivative matrix B and the input vector v.

References

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

See Also

discrete_deriv() for discrete differentiation at arbitrary query points, d_mat_mult() for multiplying by the discrete derivative matrix, and b_mat() for constructing the extended discrete derivative matrix.

Examples

v = sort(runif(10))
as.vector(b_mat(2, 1:10) %*% v)
b_mat_mult(v, 2, 1:10) 

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