Constructs the discrete B-spline basis matrix of a given order, with respect to given design points and given knot points.
n_mat(k, xd, normalized = TRUE, knot_idx = NULL)Order for the discrete B-spline basis matrix. Must be >= 0.
Design points. Must be sorted in increasing order, and have length
at least k+1.
Should the discrete B-spline basis vectors be normalized to
attain a maximum value of 1 over the design points? The default is TRUE.
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.
Sparse matrix of dimension length(xd) by length(knot_idx) + k + 1.
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.
Tibshirani (2020), "Divided differences, falling factorials, and discrete splines: Another look at trend filtering and related problems", Sections 7, 8.2, and 8.3.
h_eval() for constructing evaluations of the discrete B-spline
basis at arbitrary query points.