Evaluates the discrete B-spline basis of a given order, with respect to given design points, evaluated at arbitrary query points.
Arguments
- k
Order for the discrete B-spline basis. Must be >= 0.
- xd
Design points. Must be sorted in increasing order, and have length at least
k+1.- x
Query points. Must be sorted in increasing order.
- 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).- N
Matrix of discrete B-spline evaluations at the design points. The default is
NULL, which means that this is precomputed before constructing the matrix of discrete B-spline evaluations at the query points. IfNis non-NULL, then the argumentnormalizedwill be ignored (as this would have only been used to construct N at the design points).
Details
The discrete B-spline basis functions of order \(k\), defined with
respect to design points \(x_1 < \ldots < x_n\), are denoted
\(\eta^k_1, \ldots, \eta^k_n\). For a discussion of their properties and
further references, see the help file for n_mat(). The current function
produces a matrix of evaluations of the discrete B-spline basis at an
arbitrary sequence of query points. For each query point \(x\), this
matrix has a corresponding row with entries:
$$
\eta^k_j(x), \; j = 1, \ldots, n.
$$
Unlike the falling factorial basis, the discrete B-spline basis is not
generally available in closed-form. Therefore, the current function (unlike
h_eval()) will first check if it should precompute the evaluations of the
discrete B-spline basis at the design points. If the argument N is
non-NULL, then it will use this as the matrix of evaluations at the
design points; if N is NULL, then it will call n_mat() to produce
such a matrix, and will pass to this function the arguments normalized
and knot_idx accordingly.
After obtaining the matrix of discrete B-spline evaluations at the design
points, the fast interpolation scheme from dspline_interp() is used to
produce evaluations at the query points.
See also
n_mat() for constructing evaluations of the discrete B-spline
basis at the design points.
Examples
xd = 1:10 / 10
x = 1:9 / 10 + 0.05
n_mat(2, xd, 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_eval(2, xd, x, knot_idx = c(3, 5, 7))
#> 9 x 6 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 0.375 0.70833333 -0.11111111 . . .
#> [2,] -0.125 0.87500000 0.33333333 . . .
#> [3,] . 0.12500000 1.04166667 0.11718750 . .
#> [4,] . -0.04166667 0.76388889 0.58593750 . .
#> [5,] . . 0.12500000 1.03906250 0.1071429 .
#> [6,] . . -0.04166667 0.82031250 0.5357143 .
#> [7,] . . . 0.31250000 0.9821429 0.0625000
#> [8,] . . . 0.06250000 0.9107143 0.3125000
#> [9,] . . . -0.02083333 0.4107143 0.7291667