Constructs the discrete B-spline basis matrix of a given order, with respect to given design points and given knot points.
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)wheren = 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 isNULL, 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.
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))
#> 10 x 6 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1 . . . . .
#> [2,] . 1.0000000 . . . .
#> [3,] . 0.3333333 0.8888889 . . .
#> [4,] . . 1.0000000 0.3125000 . .
#> [5,] . . 0.3333333 0.9375000 . .
#> [6,] . . . 1.0000000 0.2857143 .
#> [7,] . . . 0.5000000 0.8571429 .
#> [8,] . . . 0.1666667 1.0000000 0.1666667
#> [9,] . . . . 0.7142857 0.5000000
#> [10,] . . . . . 1.0000000
n_mat(2, 1:10, knot_idx = c(4, 6, 8))
#> 10 x 6 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1.0000000 . . . . .
#> [2,] 0.3333333 0.8888889 . . . .
#> [3,] . 1.0000000 0.3333333 . . .
#> [4,] . 0.3333333 1.0000000 . . .
#> [5,] . . 1.0000000 0.3333333 . .
#> [6,] . . 0.3333333 1.0000000 . .
#> [7,] . . . 1.0000000 0.3333333 .
#> [8,] . . . 0.3333333 1.0000000 .
#> [9,] . . . . 0.8888889 0.3333333
#> [10,] . . . . . 1.0000000