n_mat: Construct N matrix

In dspline: Tools for Computations with Discrete Splines

Construct N matrix

Description

Constructs the discrete B-spline basis matrix of a given order, with respect to given design points and given knot points.

Usage

n_mat(k, xd, normalized = TRUE, knot_idx = NULL)

Arguments

k	Order for the discrete B-spline basis matrix. Must be >= 0.
xd	Design points. Must be sorted in increasing order, and have length at least k+1.
normalized	Should the discrete B-spline basis vectors be normalized to attain a maximum value of 1 over the design points? The default is TRUE.
knot_idx	Vector of indices, a subset of (k+1):(n-1) where n = length(xd), that indicates which design points should be used as knot points for the discrete B-splines. Must be sorted in increasing order. The default is NULL, which is taken to mean (k+1):(n-1) as in the "canonical" discrete spline space (though in this case the returned N matrix will be trivial: it will be the identity matrix). See details.

Details

The discrete B-spline basis matrix of order k, with respect to design points x_1 < \ldots < x_n, and knot set T \subseteq x_{(k+1):(n-1)} is denoted N^k_T. It has dimension (|T|+k+1) \times n, and its entries are given by:

(N^k_T)_{ij} = \eta^k_j(x_i),

where \eta^k_1, \ldots, \eta^k_m are discrete B-spline (DB-spline) basis functions and m = |T|+k+1. As is suggested by their name, the DB-spline functions are linearly independent and span the space of discrete splines with knots at T. Each DB-spline \eta^k_j has a key local support property: it is supported on an interval containing at most k+2 adjacent knots.

The functions \eta^k_1, \ldots, \eta^k_m are, in general, not available in closed-form, and are defined by setting up and solving a sequence of locally-defined linear systems. For any knot set T, computation of the evaluations of all DB-splines at the design points can be done in O(nk^2) operations; see Sections 7, 8.2, and 8.3 of Tibshirani (2020) for details. The current function uses a sparse QR decomposition from the Eigen::SparseQR module in C++ in order to solve the local linear systems.

When T = x_{(k+1):(n-1)}, the knot set corresponding to the "canonical" discrete spline space (spanned by the falling factorial basis functions h^k_1, \ldots, h^k_n whose evaluations make up H^k_n; see the help file for h_mat()), the DB-spline basis matrix, which we denote by N^k_n, is trivial: it equals the n \times n identity matrix, N^k_n = I_n. Therefore DB-splines are really only useful for knot sets T that are proper subsets of x_{(k+1):(n-1)}. Specification of the knot set T is done via the argument knot_idx.

Value

Sparse matrix of dimension length(xd) by length(knot_idx) + k + 1.

References

Tibshirani (2020), "Divided differences, falling factorials, and discrete splines: Another look at trend filtering and related problems", Sections 7, 8.2, and 8.3.

See Also

h_eval() for constructing evaluations of the discrete B-spline basis at arbitrary query points.

Examples

n_mat(2, 1:10, knot_idx = c(3, 5, 7))
n_mat(2, 1:10, knot_idx = c(4, 6, 8))

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