B-Spline Basis for Polynomial SplinesSource:
Generates the spline basis matrix for B-splines representing the family of
piecewise polynomials with the specified interior knots, degree, and
boundary knots, evaluated at the values of
bSpline( x, df = NULL, knots = NULL, degree = 3L, intercept = FALSE, Boundary.knots = NULL, periodic = FALSE, derivs = 0L, integral = FALSE, warn.outside = getOption("splines2.warn.outside", TRUE), ... ) ibs( x, df = NULL, knots = NULL, degree = 3, intercept = FALSE, Boundary.knots = NULL, ... ) dbs( x, derivs = 1L, df = NULL, knots = NULL, degree = 3, intercept = FALSE, Boundary.knots = NULL, ... ) bsp( x, df = NULL, knots = NULL, degree = 3L, intercept = FALSE, Boundary.knots = NULL, periodic = FALSE, derivs = 0L, integral = FALSE, warn.outside = getOption("splines2.warn.outside", TRUE), ... )
The predictor variable. Missing values are allowed and will be returned as they are.
Degree of freedom that equals to the column number of the returned matrix. One can specify
knots, then the function chooses
df - degree - as.integer(intercept)internal knots at suitable quantiles of
xignoring missing values and those
xoutside of the boundary. For periodic splines,
df - as.integer(intercept)internal knots will be chosen at suitable quantiles of
xrelative to the beginning of the cyclic intervals they belong to (see Examples) and the number of internal knots must be greater or equal to the specified
degree - 1. If internal knots are specified via
knots, the specified
dfwill be ignored.
The internal breakpoints that define the splines. 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. For periodic splines, the number of knots must be greater or equal to the specified
degree - 1.
A nonnegative integer specifying the degree of the piecewise polynomial. The default value is
3for cubic splines. Zero degree is allowed for piecewise constant basis functions.
TRUE, the complete basis matrix will be returned. Otherwise, the first basis will be excluded from the output.
Boundary points at which to anchor the splines. By default, they are the range of
NA. If both
Boundary.knotsare supplied, the basis parameters do not depend on
x. Data can extend beyond
Boundary.knots. For periodic splines, the specified bounary knots define the cyclic interval.
A logical value. If
TRUE, the periodic splines will be returned. The default value is
A nonnegative integer specifying the order of derivatives of splines basis function. The default value is
A logical value. If
TRUE, the corresponding integrals of spline basis functions will be returned. The default value is
FALSE. For periodic splines, the integral of each basis is integrated from the left boundary knot.
A logical value indicating if a warning should be thrown out when any
xis outside the boundary. This option can also be set through
options("splines2.warn.outside")after the package is loaded.
Optional arguments that are not used.
A numeric matrix of
length(x) rows and
df columns if
df is specified. If
knots are specified instead, the
output matrix will consist of
length(knots) + degree +
as.integer(intercept) columns if
periodic = FALSE, or
length(knots) + as.integer(intercept) columns if
TRUE. Attributes that correspond to the arguments specified are
returned for usage of other functions in this package.
This function extends the
bs() function in the
for B-spline basis functions by allowing piecewise constant (left-closed and
right-open except on the right boundary) spline basis of degree zero. In
addition, the function provides derivatives or integrals of the B-spline
basis functions when one specifies the arguments
integral appropriately. The function constructs periodic B-splines
TRUE. All the implementations are based on
the closed-form recursion formula following De Boor (1978) and Wang and Yan
dbs() are provided for convenience.
The former provides the integrals of B-splines and is equivalent to
integral = TRUE. The latter produces the
derivatives of given order of B-splines and is equivalent to
bSpline() with default
derivs = 1. The function
is an alias of to encourage the use in a model formula.
De Boor, Carl. (1978). A practical guide to splines. Vol. 27. New York: Springer-Verlag.
Wang, W., & Yan, J. (2021). Shape-restricted regression splines with R package splines2. Journal of Data Science, 19(3),498--517.
knots for extracting internal and boundary knots.
library(splines2) set.seed(1) x <- runif(100) knots <- c(0.3, 0.5, 0.6) # internal knots ## cubic B-splines bsMat <- bSpline(x, knots = knots, degree = 3, intercept = TRUE) ibsMat <- update(bsMat, integral = TRUE) # the integrals d1Mat <- deriv(bsMat) # the 1st derivaitves d2Mat <- deriv(bsMat, 2) # the 2nd derivaitves op <- par(mfrow = c(2, 2), mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0)) plot(bsMat, ylab = "Cubic B-splines") plot(ibsMat, ylab = "The integrals") plot(d1Mat, ylab = "The 1st derivatives") plot(d2Mat, ylab = "The 2nd derivatives") ## evaluate at new values predict(bsMat, c(0.125, 0.801)) #> 1 2 3 4 5 6 7 #> [1,] 0.2276364 0.5596808 0.1956893 0.01699357 0.0000000 0.0000000 0.0000000 #> [2,] 0.0000000 0.0000000 0.0000000 0.05216125 0.3495278 0.4634015 0.1349094 ## periodic B-splines px <- seq(0, 3, 0.01) pbsMat <- bSpline(px, knots = knots, Boundary.knots = c(0, 1), intercept = TRUE, periodic = TRUE) ipMat <- update(pbsMat, integral = TRUE) dpMat <- deriv(pbsMat) dp2Mat <- deriv(pbsMat, 2) plot(pbsMat, ylab = "Periodic B-splines", mark_knots = "b") plot(ipMat, ylab = "The integrals", mark_knots = "b") plot(dpMat, ylab = "The 1st derivatives", mark_knots = "b") plot(dp2Mat, ylab = "The 2nd derivatives", mark_knots = "b") par(op) # reset to previous plotting settings