2.1 B-spline Basis Functions

The bSpline() function generates the basis matrix for B-splines and extends the function bs() of the package splines by providing 1) the piece-wise constant basis functions when degree = 0, 2) the derivatives of basis functions for a positive derivs, 3) the integrals of basis functions if integral = TRUE, 4) periodic basis functions based on B-splines if periodic = TRUE.

One example of linear B-splines with three internal knots is as follows:

knots <- c(0.3, 0.5, 0.6)
x <- seq(0, 1, 0.01)
bsMat <- bSpline(x, knots = knots, degree = 1, intercept = TRUE)
plot(bsMat, mark_knots = "all")

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

2.2 Integrals and Derivatives of B-splines

For convenience, the package also provides functions ibs() and dbs() for constructing the B-spline integrals and derivatives, respectively. Two toy examples are as follows:

ibsMat <- ibs(x, knots = knots, degree = 1, intercept = TRUE)
op <- par(mfrow = c(1, 2))
plot(bsMat, mark_knots = "internal")
plot(ibsMat, mark_knots = "internal")
abline(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)
plot(bsMat, mark_knots = "internal")
plot(dbsMat, mark_knots = "internal")

Cubic B-spline (left) and their first derivative (right).

We may also obtain the derivatives easily by the deriv() method as follows:

is_equivalent <- function(a, b) {
    all.equal(a, b, check.attributes = FALSE)
}
stopifnot(is_equivalent(dbsMat, deriv(bsMat)))

2.3 Periodic B-splines

The function bSpline() produces periodic spline basis functions following Piegl and Tiller (1997, chap. 12) when periodic = TRUE is specified. Different from the regular basis functions, the x is allowed to be placed outside the boundary and the Boundary.knots defines the cyclic interval. For instance, one may obtain the periodic cubic B-spline basis functions with cyclic interval (0, 1) as follows:

px <- seq(0, 3, 0.01)
pbsMat <- bSpline(px, knots = knots, Boundary.knots = c(0, 1),
                  intercept = TRUE, periodic = TRUE)
ipMat <- ibs(px, knots = knots, Boundary.knots = c(0, 1),
             intercept = TRUE, periodic = TRUE)
dp1Mat <- deriv(pbsMat)
dp2Mat <- deriv(pbsMat, derivs = 2)
par(mfrow = c(1, 2))
plot(pbsMat, ylab = "Periodic B-splines", mark_knots = "boundary")
plot(ipMat, ylab = "The integrals", mark_knots = "boundary")

plot(dp1Mat, ylab = "The 1st derivatives", mark_knots = "boundary")
plot(dp2Mat, ylab = "The 2nd derivatives", mark_knots = "boundary")

For reference, the corresponding integrals and derivatives are also visualized.

3.1 M-spline Basis Functions

M-splines (Ramsay 1988) can be considered the normalized version of B-splines with unit integral within boundary knots. An example given by Ramsay (1988) was a quadratic M-splines with three internal knots placed at 0.3, 0.5, and 0.6. The default boundary knots are the range of x, and thus 0 and 1 in this example.

msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)
par(op)
plot(msMat, mark_knots = "all")

Quadratic M-spline with three internal knots placed at 0.3, 0.5, and 0.6.

The derivative of the given order of M-splines can be obtained by specifying a positive integer to argument dervis of mSpline(). Similarly, for an existing mSpline object generated by mSpline(), one can use the deriv() method for derivaitives. For example, the first derivative of the M-splines given in the previous example can be obtained equivalently as follows:

dmsMat1 <- mSpline(x, knots = knots, degree = 2, intercept = TRUE, derivs = 1)
dmsMat2 <- deriv(msMat)
stopifnot(is_equivalent(dmsMat1, dmsMat2))

3.2 Periodic M-Splines

The mSpline() function produces periodic splines based on M-spline basis functions when periodic = TRUE is specified. The Boundary.knots defines the cyclic interval, which is the same with the periodic B-splines.

pmsMat <- mSpline(px, knots = knots, intercept = TRUE,
                  periodic = TRUE, Boundary.knots = c(0, 1))
plot(pmsMat, ylab = "Periodic Basis", mark_knots = "boundary")