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% File src/library/splines/man/bs.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2025 R Core Team
% Distributed under GPL 2 or later

\name{bs}
\alias{bs}
\title{B-Spline Basis for Polynomial Splines}
\description{
  Generate the B-spline basis matrix for a polynomial spline.
}
\usage{
bs(x, df = NULL, knots = NULL, degree = 3, intercept = FALSE,
   Boundary.knots = range(x), warn.outside = TRUE)
}
\arguments{
  \item{x}{the predictor variable.  Missing values are allowed.}
  \item{df}{degrees of freedom; one can specify \code{df} rather than
    \code{knots}; \code{bs()} then chooses \code{df-degree} (minus one
    if there is an intercept) knots at suitable quantiles of \code{x}
    (which will ignore missing values).  The default, \code{NULL},
    takes the number of inner knots as \code{length(knots)}.  If that is
    zero as per default, that corresponds to \code{df = degree - intercept}.}
  \item{knots}{the \emph{internal} breakpoints that define the
    spline.  The default is \code{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
    \code{Boundary.knots}.}
  \item{degree}{degree of the piecewise polynomial---default is \code{3} for
    cubic splines.}
  \item{intercept}{if \code{TRUE}, an intercept is included in the
    basis; default is \code{FALSE}.}
  \item{Boundary.knots}{boundary points at which to anchor the B-spline
    basis (default the range of the non-\code{\link{NA}} data).  If both
    \code{knots} and \code{Boundary.knots} are supplied, the basis
    parameters do not depend on \code{x}.  Data can extend beyond
    \code{Boundary.knots}.}
  \item{warn.outside}{\code{\link{logical}} indicating if a
    \code{\link{warning}} should be signalled in case some \code{x} values
    are outside the boundary knots.}
}
\details{
  \code{bs} is based on the function \code{\link{splineDesign}}.
  It generates a basis matrix for
  representing the family of piecewise polynomials with the specified
  interior knots and degree, evaluated at the values of \code{x}.  A
  primary use is in modeling formulas to directly specify a piecewise
  polynomial term in a model.

  When \code{Boundary.knots} are set \emph{inside} \code{range(x)},
  \code{bs()} now uses a \sQuote{pivot} inside the respective boundary
  knot which is important for derivative evaluation.  In \R versions
  \eqn{\le} 3.2.2, the boundary knot itself had been used as
  pivot, which lead to somewhat wrong extrapolations.

  When used in a model \code{\link{formula}} containing a \code{subset = *}
  part, the \code{x} in \code{bs(x)} is evaluated before subsetting which
  may lead to unexpected results.  For more information see the
  \sQuote{Details} section of the \code{\link{model.frame}}.
}
\value{
  A matrix of dimension \code{c(length(x), df)}, where either \code{df}
  was supplied or if \code{knots} were supplied, \code{df =
  length(knots) + degree} plus one if there is an intercept.  Attributes
  are returned that correspond to the arguments to \code{bs}, and
  explicitly give the \code{knots}, \code{Boundary.knots} etc for use by
  \code{predict.bs()}.
}
\seealso{
  \code{\link{ns}}, \code{\link{poly}}, \code{\link{smooth.spline}},
  \code{\link{predict.bs}}, \code{\link{SafePrediction}}
}
\author{Douglas Bates and Bill Venables.  Tweaks by R Core, and a patch
  fixing extrapolation \dQuote{outside} \code{Boundary.knots} by Trevor
  Hastie.
}
\references{
  \bibshow{R:Hastie:1992}
}
\examples{
require(stats); require(graphics)
bs(women$height, df = 5)
summary(fm1 <- lm(weight ~ bs(height, df = 5), data = women))

## example of safe prediction
plot(women, xlab = "Height (in)", ylab = "Weight (lb)")
ht <- seq(57, 73, length.out = 200)
lines(ht, predict(fm1, data.frame(height = ht)))
\dontshow{
## Consistency:
x <- c(1:3, 5:6)
stopifnot(identical(bs(x), bs(x, df = 3)),
          identical(bs(x, df = 4), bs(x, df = 4, knots = NULL)), # not true till 2.15.2
          !is.null(kk <- attr(bs(x), "knots")), # not true till 1.5.1
          length(kk) == 0)
}
}
\keyword{smooth}