Derivatives of B-Splines

Produces the derivatives of given order of B-splines.

Usage

dbs(
  x,
  derivs = 1L,
  df = NULL,
  knots = NULL,
  degree = 3L,
  intercept = FALSE,
  Boundary.knots = NULL,
  ...
)

Arguments

x: The predictor variable. Missing values are allowed and will be returned as they are.
derivs: A positive integer specifying the order of derivative. The default value is 1L for the first derivative.
df: Degree of freedom that equals to the column number of the returned matrix. One can specify df rather than knots, then the function chooses df - degree - as.integer(intercept) internal knots at suitable quantiles of x ignoring missing values and those x outside of the boundary. If internal knots are specified via knots, the specified df will be ignored.
knots: The internal breakpoints that define the splines. The default is NULL, which results in a basis for ordinary polynomial regression. Typical values are the mean or median for one knot, quantiles for more knots.
degree: A nonnegative integer specifying the degree of the piecewise polynomial. The default value is 3 for cubic splines. Zero degree is allowed for piecewise constant basis functions.
intercept: If TRUE, the complete basis matrix will be returned. Otherwise, the first basis will be excluded from the output.
Boundary.knots: Boundary points at which to anchor the splines. By default, they are the range of x excluding NA. If both knots and Boundary.knots are supplied, the basis parameters do not depend on x. Data can extend beyond Boundary.knots.
...: Optional arguments that are not used.

Value

A numeric matrix of length(x) rows and df columns if df is specified or length(knots) + degree + as.integer(intercept) columns if knots are specified instead. Attributes that correspond to the arguments specified are returned mainly for other functions in this package.

Details

This function provides a more user-friendly interface and a more consistent handling for NA's than splines::splineDesign() for derivatives of B-splines. The implementation is based on the closed-form recursion formula. At knots, the derivative is defined to be the right derivative except at the right boundary knot.

References

De Boor, Carl. (1978). A practical guide to splines. Vol. 27. New York: Springer-Verlag.

Examples

library(splines2)
x <- seq.int(0, 1, 0.01)
knots <- c(0.2, 0.4, 0.7)
## the second derivative of cubic B-splines with three internal knots
dMat <- dbs(x, derivs = 2L, knots = knots, intercept = TRUE)

## compare with the results from splineDesign
ord <- attr(dMat, "degree") + 1L
bKnots <- attr(dMat, "Boundary.knots")
aKnots <- c(rep(bKnots[1L], ord), knots, rep(bKnots[2L], ord))
res <- splines::splineDesign(aKnots, x = x, derivs = 2L)
stopifnot(all.equal(res, dMat, check.attributes = FALSE))