This function generates the integral of B-spline basis matrix for a polynomial spline. The arguments are exactly the same with function bs in package splines.

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

Arguments

x

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

df

Degrees of freedom of the B-spline basis to be integrated. 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 B-spline basis to be integrated. 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 to be integrated. The default value is 3 for the integral of cubic B-splines.

intercept

If TRUE, an intercept is included in the basis; Default is FALSE.

Boundary.knots

Boundary points at which to anchor the B-spline basis to be integrated. 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.

...

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 integral of B-spline basis based on recursion relation. Internally, it calls 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.

References

De Boor, Carl. (1978). A practical guide to splines. Vol. 27. New York: Springer-Verlag.

See also

predict.ibs for evaluation at given (new) values; deriv.ibs for derivative method. bSpline for B-splines; dbs for derivatives of B-splines;

Examples

library(splines2) x <- seq.int(0, 1, 0.01) knots <- c(0.2, 0.4, 0.7, 0.9) ibsMat <- ibs(x, knots = knots, degree = 1, intercept = TRUE) ## the B-spline bases integrated by function bSpline (same arguments) bsMat0 <- bSpline(x, knots = knots, degree = 1, intercept = TRUE) ## or by function deriv (recommended) that directly extracts the existing ## result from the attribute of ibsMat and thus is much more efficient. bsMat <- deriv(ibsMat) stopifnot(all.equal(bsMat0, bsMat, check.attributes = FALSE)) # equivalent ## plot B-spline basis with their corresponding integrals library(graphics) par(mfrow = c(1, 2)) matplot(x, bsMat, type = "l", ylab = "B-spline basis") abline(v = knots, lty = 2, col = "gray") matplot(x, ibsMat, type = "l", ylab = "Integral of B-spline basis")
abline(v = knots, lty = 2, col = "gray")
par(mfrow = c(1, 1))