Introduction to splines2

Wenjie Wang

2021-01-06

The splines2 package (version 0.4.1) provides functions constructing a variety of regression spline bases that are not available from the splines package shipped with base R. Most functions have a very similar user interface with the function splines::bs(). To be more specific, it provides functions to construct basis matrix of

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

and their integrals (except C-spliness) and derivatives of given order by close-form recursive formulas. Compared to the splines package, splines2 allows piecewise constant basis 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 introduce the basic usage of most functions in the package through examples. The details of function syntax are available in the package manual and thus will not be discussed.

B-splines with their integrals and derivatives

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

The close-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-spline bases (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")