M-Spline Basis for Polynomial Splines and its Derivatives

This function generates the monotone regression spline (or simply called M-spline) basis matrix for a polynomial spline or its derivatives of given order.

mSpline(x, df = NULL, knots = NULL, degree = 3L, intercept = FALSE,
        Boundary.knots = range(x, na.rm = TRUE), derivs = 0L, ...)

Arguments

x	The predictor variable. Missing values are allowed and will be returned as they were.
df	Degrees of freedom. One can specify `df` rather than `knots`, then the function chooses "df - degree" (minus one if there is an intercept) knots at suitable quantiles of `x` (which will ignore missing values). The default, `NULL`, corresponds to no inner knots, i.e., "degree - intercept".
knots	The internal breakpoints that define the spline. 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. See also `Boundary.knots`.
degree	Non-negative integer degree of the piecewise polynomial. The default value is 3 for cubic splines. Zero degree is allowed for piecewise constant basis.
intercept	If `TRUE`, an intercept is included in the basis; Default is `FALSE`.
Boundary.knots	Boundary points at which to anchor the M-spline basis. By default, they are the range of the non-`NA` data. If both `knots` and `Boundary.knots` are supplied, the basis parameters do not depend on `x`. Data can extend beyond `Boundary.knots`.
derivs	A non-negative integer specifying the order of derivatives of M-splines. The default value is `0L` for M-spline bases.
...	Optional arguments for future usage.

Value

A matrix of dimension length(x) by df = degree + length(knots) (plus one if intercept is included). Attributes that correspond to the arguments specified are returned for usage of other functions in this package.

Details

It is an implementation of the close form M-spline basis based on relationship between M-spline basis and B-spline basis. In fact, M-spline basis is a rescaled version of B-spline basis. Internally, it calls function bSpline and generates a basis matrix for representing the family of piecewise polynomials with the specified interior knots and degree, evaluated at the values of x.

References

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

Examples

## Example given in the reference paper by Ramsay (1988)
library(splines2)
x <- seq.int(0, 1, 0.01)
knots <- c(0.3, 0.5, 0.6)
msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)

library(graphics)
matplot(x, msMat, type = "l", ylab = "M-spline basis")
abline(v = knots, lty = 2, col = "gray")

## derivatives of M-splines
dmsMat <- mSpline(x, knots = knots, degree = 2,
                  intercept = TRUE, derivs = 1)
## or using the 'deriv' method
dmsMat1 <- deriv(msMat)
stopifnot(all.equal(dmsMat, dmsMat1, check.attributes = FALSE))