C-Spline Basis for Polynomial Splines

Generates the convex regression spline (called C-spline) basis matrix by integrating I-spline basis for a polynomial spline or the corresponding derivatives.

Usage

cSpline(
  x,
  df = NULL,
  knots = NULL,
  degree = 3L,
  intercept = TRUE,
  Boundary.knots = NULL,
  derivs = 0L,
  scale = TRUE,
  warn.outside = getOption("splines2.warn.outside", TRUE),
  ...
)

csp(
  x,
  df = NULL,
  knots = NULL,
  degree = 3L,
  intercept = TRUE,
  Boundary.knots = NULL,
  derivs = 0L,
  scale = TRUE,
  warn.outside = getOption("splines2.warn.outside", TRUE),
  ...
)

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. For periodic splines, df - as.integer(intercept) internal knots will be chosen at suitable quantiles of x relative to the beginning of the cyclic intervals they belong to (see Examples) and the number of internal knots must be greater or equal to the specified degree - 1. 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. For periodic splines, the number of knots must be greater or equal to the specified degree - 1. Duplicated internal knots are not allowed.
degree: The degree of C-spline defined to be the degree of the associated M-spline instead of actual polynomial degree. For example, C-spline basis of degree 2 is defined as the scaled double 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 C-Spline for shape-restricted regression, intercept = TRUE should be set even when an intercept term is considered additional to the spline basis in the model.
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. For periodic splines, the specified bounary knots define the cyclic interval.
derivs: A nonnegative integer specifying the order of derivatives of C-splines. The default value is 0L for C-spline basis functions.
scale: A logical value indicating if scaling C-splines is required. If TRUE by default, each C-spline basis is scaled to have unit height at right boundary knot. The corresponding I-spline and M-spline produced by deriv methods will be scaled to the same extent.
warn.outside: A logical value indicating if a warning should be thrown out when any x is outside the boundary. This option can also be set through options("splines2.warn.outside") after the package is loaded.
...: Optional arguments that are not used.

Value

A numeric matrix of length(x) rows and df columns if df is specified. If knots are specified instead, the output matrix will consist of length(knots) + degree + as.integer(intercept) columns. Attributes that correspond to the arguments specified are returned for usage of other functions in this package.

Details

It is an implementation of the closed-form C-spline basis derived from the recursion formula of I-splines and M-splines. The function csp() is an alias of to encourage the use in a model formula.

References

Meyer, M. C. (2008). Inference using shape-restricted regression splines. The Annals of Applied Statistics, 2(3), 1013–1033.

Examples

library(splines2)

x <- seq.int(0, 1, 0.01)
knots <- c(0.3, 0.5, 0.6)

### when 'scale = TRUE' (by default)
csMat <- cSpline(x, knots = knots, degree = 2)

plot(csMat, ylab = "C-spline basis", mark_knots = "internal")

isMat <- deriv(csMat)
msMat <- deriv(csMat, derivs = 2)
plot(isMat, ylab = "scaled I-spline basis")

plot(msMat, ylab = "scaled M-spline basis")


### when 'scale = FALSE'
csMat <- cSpline(x, knots = knots, degree = 2, scale = FALSE)

## the corresponding I-splines and M-splines (with same arguments)
isMat <- iSpline(x, knots = knots, degree = 2)
msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)

## or using deriv methods (more efficient)
isMat1 <- deriv(csMat)
msMat1 <- deriv(csMat, derivs = 2)

## equivalent
stopifnot(all.equal(isMat, isMat1, check.attributes = FALSE))
stopifnot(all.equal(msMat, msMat1, check.attributes = FALSE))