I-Spline Basis for Polynomial Splines

Generates the I-spline (integral of M-spline) basis matrix for a polynomial spline or the corresponding derivatives of given order.

Usage

iSpline(
  x,
  df = NULL,
  knots = NULL,
  degree = 3L,
  intercept = TRUE,
  Boundary.knots = NULL,
  derivs = 0L,
  ...
)

Arguments

x: The predictor variable. Missing values are allowed and will be returned as they are.
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: The degree of I-spline defined to be the degree of the associated M-spline instead of actual polynomial degree. For example, I-spline basis of degree 2 is defined as the integral of associated M-spline basis of degree 2.
intercept: If TRUE by default, all of the spline basis functions are returned. Notice that when using I-Spline for monotonic regression, intercept = TRUE should be set even when an intercept term is considered additional to the spline basis functions.
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.
derivs: A nonnegative integer specifying the order of derivatives of I-splines.
...: 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

It is an implementation of the closed-form I-spline basis based on the recursion formula given by Ramsay (1988).

References

Ramsay, J. O. (1988). Monotone regression splines in action. Statistical Science, 3(4), 425--441.

Examples

library(splines2)

## Example given in the reference paper by Ramsay (1988)
x <- seq.int(0, 1, by = 0.01)
knots <- c(0.3, 0.5, 0.6)
isMat <- iSpline(x, knots = knots, degree = 2)

op <- par(mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0))
matplot(x, isMat, type = "l", ylab = "I-spline basis")
abline(v = knots, lty = 2, col = "gray")


## reset to previous plotting settings
par(op)

## the derivative of I-splines is M-spline
msMat1 <- iSpline(x, knots = knots, degree = 2, derivs = 1)
msMat2 <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)
stopifnot(all.equal(msMat1, msMat2))