# Reproducing Kernel Basis Generate a reproducing kernel basis matrix for a nominal, ordinal, or polynomial smoothing spline. ```r 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 $$ Z = X Q^{-1/2} $$ 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 ## 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 ```r ######***###### 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) } ```

rk function