rk function

Reproducing Kernel Basis

Generate a reproducing kernel basis matrix for a nominal, ordinal, or polynomial smoothing spline.

rk(x, df = NULL, knots = NULL, m = NULL, intercept = FALSE, 
   Boundary.knots = NULL, warn.outside = TRUE, 
   periodic = FALSE, xlev = levels(x))

Arguments

• x: the predictor vector of length n. Can be a factor, integer, or numeric, see Note.
• df: the degrees of freedom, i.e., number of knots to place at quantiles of x. Defaults to 5 but ignored if knots are provided.
• knots: the breakpoints (knots) defining the spline. If knots are provided, the df is defined as length(unique(c(knots, Boundary.knots))).
• m: the derivative penalty order: 0 = ordinal spline, 1 = linear spline, 2 = cubic spline, 3 = quintic spline
• intercept: should an intercept be included in the basis?
• Boundary.knots: the boundary points for spline basis. Defaults to range(x).
• warn.outside: if TRUE, a warning is provided when x values are outside of the Boundary.knots
• periodic: should the spline basis functions be constrained to be periodic with respect to the Boundary.knots?
• xlev: levels of x (only applicable if x is a factor)

Details

Given a vector of function realizations $f$, suppose that $f = X \beta$, where $X$ is the (unregularized) spline basis and $\beta$ is the coefficient vector. Let $Q$ denote the postive semi-definite penalty matrix, such that $\beta^\top Q \beta$ defines the roughness penalty for the spline. See Helwig (2017) for the form of $X$ and $Q$ for the various types of splines.

Consider the spectral parameterization of the form $f = Z \alpha$ where

is the regularized spline basis (that is returned by this function), and $\alpha = Q^{1/2} \beta$ are the reparameterized coefficients. Note that $X \beta = Z \alpha$ and $\beta^\top Q \beta = \alpha^\top \alpha$, so the spectral parameterization absorbs the penalty into the coefficients (see Helwig, 2021, 2024).

Syntax of this function is designed to mimic the syntax of the bs function.

Returns

Returns a basis function matrix of dimension n by df (plus 1 if an intercept is included) with the following attributes: - df: degrees of freedom

• knots: knots for spline basis

• m: derivative penalty order

• intercept: was an intercept included?

• Boundary.knots: boundary points of x

• periodic: is the basis periodic?

• xlev: factor levels (if applicable)

References

Helwig, N. E. (2017). Regression with ordered predictors via ordinal smoothing splines. Frontiers in Applied Mathematics and Statistics, 3(15), 1-13. tools:::Rd_expr_doi("10.3389/fams.2017.00015")

Helwig, N. E. (2021). Spectrally sparse nonparametric regression via elastic net regularized smoothers. Journal of Computational and Graphical Statistics, 30(1), 182-191. tools:::Rd_expr_doi("10.1080/10618600.2020.1806855")

Helwig, N. E. (2025). Versatile descent algorithms for group regularization and variable selection in generalized linear models. Journal of Computational and Graphical Statistics, 34(1), 239-252. tools:::Rd_expr_doi("10.1080/10618600.2024.2362232")

Author(s)

Nathaniel E. Helwig helwig@umn.edu

Note

The (default) type of spline basis depends on the class of the input x object:

• If x is an unordered factor, then a nominal spline basis is used

• If x is an ordered factor (and m = NULL), then an ordinal spline basis is used

• If x is an integer or numeric (and m = NULL), then a cubic spline basis is used

Note that you can override the default behavior by specifying the m argument.

Examples

######***######   NOMINAL SPLINE BASIS   ######***######

x <- as.factor(LETTERS[1:5])
basis <- rk(x)
plot(1:5, basis[,1], t = "l", ylim = extendrange(basis))
for(j in 2:ncol(basis)){
  lines(1:5, basis[,j], col = j)
}

######***######   ORDINAL SPLINE BASIS   ######***######

x <- as.ordered(LETTERS[1:5])
basis <- rk(x)
plot(1:5, basis[,1], t = "l", ylim = extendrange(basis))
for(j in 2:ncol(basis)){
  lines(1:5, basis[,j], col = j)
}

######***######   LINEAR SPLINE BASIS   ######***######

x <- seq(0, 1, length.out = 101)
basis <- rk(x, m = 1)
plot(x, basis[,1], t = "l", ylim = extendrange(basis))
for(j in 2:ncol(basis)){
  lines(x, basis[,j], col = j)
}

######***######   CUBIC SPLINE BASIS   ######***######

x <- seq(0, 1, length.out = 101)
basis <- rk(x)
basis <- scale(basis)  # for visualization only!
plot(x, basis[,1], t = "l", ylim = extendrange(basis))
for(j in 2:ncol(basis)){
  lines(x, basis[,j], col = j)
}

######***######   QUINTIC SPLINE BASIS   ######***######

x <- seq(0, 1, length.out = 101)
basis <- rk(x, m = 3)
basis <- scale(basis)  # for visualization only!
plot(x, basis[,1], t = "l", ylim = extendrange(basis))
for(j in 2:ncol(basis)){
  lines(x, basis[,j], col = j)
}