rk: Reproducing Kernel Basis

In grpnet: Group Elastic Net Regularized GLMs and GAMs

Reproducing Kernel Basis

Description

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

Usage

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.

Value

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)

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.

Author(s)

Nathaniel E. Helwig <[e-mail]>

References

Helwig, N. E. (2017). Regression with ordered predictors via ordinal smoothing splines. Frontiers in Applied Mathematics and Statistics, 3(15), 1-13. \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10618600.2020.1806855")}

Helwig, N. E. (2024). Precise tensor product smoothing via spectral splines. Stats, 7(1), 34-53. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3390/stats7010003")}

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)
}

grpnet documentation built on Oct. 12, 2024, 1:07 a.m.