\Header{bs}{Generate a Basis for Polynomial Splines}
\keyword{smooth}{bs}
\begin{Description}\relax
Generate the B-spline basis matrix for a cubic spline.\end{Description}
\begin{Usage}
\begin{verbatim}
bs(x, df, knots, degree=3, intercept=FALSE, Boundary.knots)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] the predictor variable.
\item[\code{df}] degrees of freedom; one can specify \code{df} rather than
\code{knots}; \code{bs()} then chooses \code{df-degree-1} knots at
suitable quantiles of \code{x}.
\item[\code{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[\code{degree}] degree of the piecewise polynomial---default is 3 for
cubic splines.
\item[\code{intercept}] if \code{TRUE}, an intercept is included in the
basis; default is \code{FALSE}.
\item[\code{Boundary.knots}] boundary points at which to anchor the B-spline
basis (default the range of the 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}
\end{ldescription}
\end{Arguments}
\begin{Value}
A matrix of dimension \code{length(x) * df}, where either \code{df}
was supplied or if \code{knots} were supplied,
\code{df = length(knots) + 3 + 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()}.

\code{bs()} is based on the function \code{spline.des()} written by
Douglas Bates. 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. 

Beware of making predictions with new \code{x} values when \code{df}
is used as an argument. Either use \code{safe.predict.gam()}, or else
specify \code{knots} and \code{Boundary.knots}.\end{Value}
\begin{SeeAlso}\relax
\code{ns}, \code{poly}, \code{smooth.spline},
\code{predict.bs}.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
data(women)
bs(women$height, df = 5)
summary(fm1 <- lm(weight ~ bs(height, df = 5), data = women))
\end{ExampleCode}
\end{Examples}

