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.

## References

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))