This function generates the convex regression spline (called C-spline) basis matrix by integrating I-spline basis for a polynomial spline.

cSpline(x, df = NULL, knots = NULL, degree = 3L, intercept = FALSE,
Boundary.knots = range(x, na.rm = TRUE), scale = TRUE, ...)

## Arguments

x The predictor variable. Missing values are allowed and will be returned as they were. 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". 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. Non-negative integer degree of the piecewise polynomial. The default value is 3 for cubic splines. If TRUE, an intercept is included in the basis; Default is FALSE. Boundary points at which to anchor the C-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. Logical value (TRUE by default) indicating whether scaling on C-spline basis is required. If TRUE, C-spline basis is scaled to have unit height at right boundary knot; the corresponding I-spline and M-spline basis matrices shipped in attributes are also scaled to the same extent. Optional arguments for future usage.

## Value

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

## Details

It is an implementation of the close form C-spline basis derived from the recursion formula of I-spline and M-spline. Internally, it calls iSpline and generates a basis matrix for representing the family of piecewise polynomials and their corresponding integrals with the specified interior knots and degree, evaluated at the values of x.

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

predict.cSpline for evaluation at given (new) values; deriv.cSpline for derivatives; iSpline for I-splines; mSpline for M-splines.

## 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, intercept = TRUE)

library(graphics)
matplot(x, csMat, type = "l", ylab = "C-spline basis")abline(v = knots, lty = 2, col = "gray") isMat <- deriv(csMat)
msMat <- deriv(csMat, derivs = 2)
matplot(x, isMat, type = "l", ylab = "scaled I-spline basis") matplot(x, msMat, type = "l", ylab = "scaled M-spline basis") ### when 'scale = FALSE'
csMat <- cSpline(x, knots = knots, degree = 2,
intercept = TRUE, scale = FALSE)
## the corresponding I-splines and M-splines (with same arguments)
isMat <- iSpline(x, knots = knots, degree = 2, intercept = TRUE)
msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)
## or using deriv methods (much 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))