Generates the I-spline (integral of M-spline) basis matrix for a polynomial spline or the corresponding derivatives of given order.

  df = NULL,
  knots = NULL,
  degree = 3L,
  intercept = TRUE,
  Boundary.knots = NULL,
  derivs = 0L,



The predictor variable. Missing values are allowed and will be returned as they are.


Degree of freedom that equals to the column number of 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. If internal knots are specified via knots, the specified df will be ignored.


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.


The degree of I-spline defined to be the degree of the associated M-spline instead of actual polynomial degree. For example, I-spline basis of degree 2 is defined as the integral of associated M-spline basis of degree 2.


If TRUE by default, all spline bases are included. Notice that when using I-Spline for monotonic regression, intercept = TRUE should be set even when an intercept term is considered additional to the spline bases in the model.


Boundary points at which to anchor the 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 that are not used.


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


It is an implementation of the close form I-spline basis based on the recursion formula given by Ramsay (1988).


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

See also

mSpline for M-splines; cSpline for C-splines;


library(splines2) ## Example given in the reference paper by Ramsay (1988) x <-, 1, by = 0.01) knots <- c(0.3, 0.5, 0.6) isMat <- iSpline(x, knots = knots, degree = 2) par(mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0)) 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, intercept = TRUE) stopifnot(all.equal(msMat1, msMat2))