# cSpline

##### C-Spline Basis for Polynomial Splines

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

##### Usage

```
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.

- df
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".- knots
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`

.- degree
Non-negative integer degree of the piecewise polynomial. The default value is 3 for cubic splines.

- intercept
If

`TRUE`

, an intercept is included in the basis; Default is`FALSE`

.- Boundary.knots
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`

.- scale
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.

##### 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`

.

##### 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.

##### References

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

##### See Also

`predict.cSpline`

for evaluation at given (new) values;
`deriv.cSpline`

for derivatives;
`iSpline`

for I-splines;
`mSpline`

for M-splines.

##### Examples

```
# NOT RUN {
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))
# }
```

*Documentation reproduced from package splines2, version 0.2.5, License: GPL (>= 3)*