This function generates the I-spline (integral of M-spline) basis matrix for a polynomial spline or its derivatives of given order..

iSpline(x, df = NULL, knots = NULL, degree = 3L, intercept = FALSE,
Boundary.knots = range(x, na.rm = TRUE), derivs = 0L, ...)

## 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. Note that the degree of I-spline is defined to be the degree of the associated M-spline instead of actual polynomial degree. In other words, I-spline basis of degree 2 is defined as the integral of associated M-spline basis of degree 2. If TRUE, an intercept is included in the basis; Default is FALSE. Boundary points at which to anchor the I-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. A non-negative integer specifying the order of derivatives of I-splines. Optional arguments for future usage.

## Value

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

## Details

It is an implementation of the close form I-spline basis based on the recursion formula of B-spline basis. Internally, it calls mSpline and bSpline, 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.

Ramsay, J. O. (1988). Monotone regression splines in action. Statistical science, 3(4), 425--441.

predict.iSpline for evaluation at given (new) values; deriv.iSpline for derivative method; mSpline for M-splines; cSpline for C-splines;

## Examples

## Example given in the reference paper by Ramsay (1988)
library(splines2)
x <- seq.int(0, 1, by = 0.01)
knots <- c(0.3, 0.5, 0.6)
isMat <- iSpline(x, knots = knots, degree = 2, intercept = TRUE)

library(graphics)
matplot(x, isMat, type = "l", ylab = "I-spline basis")abline(v = knots, lty = 2, col = "gray") ## the derivative of I-splines is M-spline
msMat1 <- iSpline(x, knots = knots, degree = 2, derivs = 1)
msMat2 <- mSpline(x, knots = knots, degree = 2)
stopifnot(all.equal(msMat1, msMat2))