Introduction to splines2

Wenjie Wang

2023-01-13

The R package splines2 is intended to be a comprehensive, efficient supplement to the base package splines. It provides functions constructing a variety of regression spline basis functions that are not available from splines. Most functions have a very similar user interface with the function splines::bs(). To be more specific, it provides functions to construct basis matrices of

• B-splines
• M-splines
• I-splines
• C-splines
• periodic M-splines
• natural cubic splines
• generalized Bernstein polynomials

along with their integrals (except C-splines) and derivatives of given order by closed-form recursive formulas.

Compared to the package splines, the package splines2 allows piecewise constant basis functions for B-splines and provides a more user-friendly interface for their derivatives with consistent handling on NA’s. Most of the implementations had been (re)written in C++ with the help of Rcpp and RcppArmadillo since v0.3.0, which boosted the computational performance.

In the remaining of this vignette, we illustrated the basic usage of most functions in the package through examples. See the package manual for the details of function usage.

B-splines with their integrals and derivatives

Function bSpline() provides B-spline basis matrix and allows degree = 0 for piece-wise constant basis function, which extends the bs() function in package splines with a better computational performance. One example of linear B-splines with three internal knots is given as follows:

library(splines2)
knots <- c(0.3, 0.5, 0.6)
x <- seq(0, 1, 0.01)
bsMat <- bSpline(x, knots = knots, degree = 1, intercept = TRUE)
matplot(x, bsMat, type = "l", ylab = "y")
abline(v = knots, lty = 2, col = "gray")

B-splines of degree one with three internal knots placed at 0.3, 0.5, and 0.6.

The closed-form recursive formula of B-spline integrals and derivatives given by De Boor (1978) is implemented in function ibs() and dbs(), respectively. Two toy examples are given as follows:

ibsMat <- ibs(x, knots = knots, degree = 1, intercept = TRUE)
par(mfrow = c(1, 2))
matplot(x, bsMat, type = "l", ylab = "y")
abline(v = knots, h = 1, lty = 2, col = "gray")
matplot(x, ibsMat, type = "l", ylab = "y")
abline(v = knots, h = c(0.15, 0.2, 0.25), lty = 2, col = "gray")

Piecewise linear B-splines (left) and their integrals (right).

bsMat <- bSpline(x, knots = knots, intercept = TRUE)
dbsMat <- dbs(x, knots = knots, intercept = TRUE)
par(mfrow = c(1, 2))
matplot(x, bsMat, type = "l", ylab = "y")
abline(v = knots, lty = 2, col = "gray")
matplot(x, dbsMat, type = "l", ylab = "y")
abline(v = knots, lty = 2, col = "gray")