Introduction to splines2

M-splines using mSpline

M-splines (Ramsay 1988) can be considered as a 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 boundary knots by default are the range of the data x, thus 0 and 1 in this example.

library(splines2)
knots <- c(0.3, 0.5, 0.6)
x <- seq(0, 1, 0.01)
msOut <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)
library(graphics) # attach graphics just in case for plots
par(mar = c(2.5, 2.5, 0, 0), mgp = c(1.5, 0.5, 0))
matplot(x, msOut, type = "l", ylab = "y")
abline(v = knots, lty = 2, col = "gray") # mark internal knots

Quadratic M-splines with three internal knots.

I-splines using iSpline

I-splines (Ramsay 1988) are simply the integral of M-splines and thus monotonically non-decreasing with unit maximum value. A monotonically non-decreasing (non-increasing) function can be fitted by a linear combination of I-spline bases with non-negative (non-positive) coefficients, plus a constant function (where the coefficient of the constant function is unconstrained).

The example given by Ramsay (1988) was the I-splines corresponding to that quadratic M-splines with three internal knots placed at 0.3, 0.5, and 0.6. Note that the degree of I-splines is defined from the associated M-splines instead of their own polynomial degree.

isOut <- iSpline(x, knots = knots, degree = 2, intercept = TRUE)
par(mar = c(2.5, 2.5, 0, 0), mgp = c(1.5, 0.5, 0))
matplot(x, isOut, type = "l", ylab = "y")
abline(h = 1, v = knots, lty = 2, col = "gray")

I-splines of degree two with three internal knots.

In additional, the corresponding M-spline basis matrix is shipped in the attribute named msMat in the object returned by function iSpline. In other words, if we need both M-spline bases and their integral splines in model fitting, extra function call on mSpline should be avoided.

all.equal(msOut, attr(isOut, "msMat")) # 'msMat' attribute for M-splines

## [1] TRUE

C-splines using cSpline

Convex splines (Meyer 2008) called C-splines are a scaled version of I-splines’ integral with unit maximum value. Meyer (2008) applied C-splines to shape-restricted regression analysis. The monotone property of I-spines ensures the convexity of C-splines. A convex regression function can be estimated using linear combinations of the C-spline bases with non-negative coefficients, plus an unrestricted linear combination of the constant function and the identity function \(g(x)=x\). If the underlying regression function is both increasing and convex, the coefficient on the identity function is restricted to be nonnegative as well.

Function cSpline provides argument scale specifying whether scaling on C-spline bases is required. If scale = TRUE by default, each C-spline basis is scaled to have unit height at right boundary knot. The corresponding I-spline (M-spline) bases shipped in attributes isMat (msMat) are also scaled to the same extent and thus serve as the first (second) derivatives of the C-spline bases. If scale = FALSE, the actual integral of I-spline basis will be returned. The original I-splines integrated and the corresponding M-splines are available in attributes named isMat and msMat, respectively.

csOut1 <- cSpline(x, knots = knots, degree = 2, intercept = TRUE)
par(mar = c(2.5, 2.5, 0, 0), mgp = c(1.5, 0.5, 0))
matplot(x, csOut1, type = "l", ylab = "y")
abline(h = 1, v = knots, lty = 2, col = "gray")

C-splines of degree two with three internal knots.

Integral B-splines using ibs

A close-form recursive formula of B-spline integral given by De Boor (1978) is implemented. One toy example is given as follows:

ibsOut <- ibs(x, knots = knots, degree = 1, intercept = TRUE)
par(mar = c(2.5, 2.5, 0, 0), mgp = c(1.5, 0.5, 0), mfrow = c(1, 2))
matplot(x, attr(ibsOut, "bsMat"), type = "l", ylab = "y")
abline(v = knots, h = 1, lty = 2, col = "gray")
matplot(x, ibsOut, 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).

B-splines using bSpline

Function bSpline provides B-spline bases and allows degree = 0 for piecewise constant bases, which is one simple but handy extension to function bs in package splines. (For positive degree, bSpline internally call bs to do the hard work.) Step function or piecewise constant bases are often used in practice for a reasonable approximation without any assumption on the form of target function. One simple example of B-splines and M-splines of degree zero is given as follows:

bsOut0 <- bSpline(x, knots = knots, degree = 0, intercept = TRUE)
msOut0 <- mSpline(x, knots = knots, degree = 0, intercept = TRUE)
par(mar = c(2.5, 2.5, 0, 0), mgp = c(1.5, 0.5, 0), mfrow = c(1, 2))
matplot(x, bsOut0, type = "l", ylab = "y")
abline(v = knots, lty = 2, col = "gray")
matplot(x, msOut0, type = "l", ylab = "y")
abline(v = knots, lty = 2, col = "gray")

B-splines (left) and M-splines (right) of degree zero

Evaluation on New Values by predict

The methods for splines2 objects dispatched by generic function predict are useful if we want to evaluate the spline object at possibly new \(x\) values. For instance, if we want to evaluate the value of I-splines object in previous example at 0.275, 0.525, and 0.8, respectively, all we need is

predict(isOut, c(0.275, 0.525, 0.8))

##              1         2         3         4        5     6
## [1,] 0.9994213 0.7730556 0.2310764 0.0000000 0.000000 0.000
## [2,] 1.0000000 1.0000000 0.9765625 0.2696429 0.000625 0.000
## [3,] 1.0000000 1.0000000 1.0000000 0.9428571 0.580000 0.125

Technically speaking, the methods take all information needed, such as knots, degree, intercept, etc., from attributes of the original splines2 objects and call the corresponding function automatically for those new \(x\) values. Therefore, the predict methods will not be applicable if those attributes are somehow lost after certain operation.