\HeaderA{smooth.spline}{Fit a Smoothing Spline}{smooth.spline}
\keyword{smooth}{smooth.spline}
\begin{Description}\relax
Fits a cubic smoothing spline to the supplied data.
\end{Description}
\begin{Usage}
\begin{verbatim}
smooth.spline(x, y = NULL, w = NULL, df, spar = NULL,
              cv = FALSE, all.knots = FALSE, nknots = NULL,
              keep.data = TRUE, df.offset = 0, penalty = 1,
              control.spar = list())
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a vector giving the values of the predictor variable, or  a
list or a two-column matrix specifying x and y. 
\item[\code{y}] responses. If \code{y} is missing, the responses are assumed
to be specified by \code{x}.
\item[\code{w}] optional vector of weights of the same length as \code{x};
defaults to all 1.
\item[\code{df}] the desired equivalent number of degrees of freedom (trace of
the smoother matrix).
\item[\code{spar}] smoothing parameter, typically (but not necessarily) in
\eqn{(0,1]}{}.  The coefficient \eqn{\lambda}{} of the integral of the
squared second derivative in the fit (penalized log likelihood)
criterion is a monotone function of \code{spar}, see the details
below.
\item[\code{cv}] ordinary (\code{TRUE}) or \sQuote{generalized} cross-validation
(GCV) when \code{FALSE}.
\item[\code{all.knots}] if \code{TRUE}, all distinct points in \code{x} are used as
knots.  If \code{FALSE} (default), a subset of \code{x[]} is used,
specifically \code{x[j]} where the \code{nknots} indices are evenly
spaced in \code{1:n}, see also the next argument \code{nknots}.
\item[\code{nknots}] integer giving the number of knots to use when
\code{all.knots=FALSE}.  Per default, this is less than \eqn{n}{}, the
number of unique \code{x} values for \eqn{n > 49}{}.
\item[\code{keep.data}] logical specifying if the input data should be kept
in the result.  If \code{TRUE} (as per default), fitted values and
residuals are available from the result.
\item[\code{df.offset}] allows the degrees of freedom to be increased by
\code{df.offset} in the GCV criterion.
\item[\code{penalty}] the coefficient of the penalty for degrees of freedom
in the GCV criterion.
\item[\code{control.spar}] optional list with named components controlling the
root finding when the smoothing parameter \code{spar} is computed,
i.e., missing or \code{NULL}, see below.

\bold{Note} that this is partly \emph{experimental} and may change
with general spar computation improvements!

\describe{
\item[low:] lower bound for \code{spar}; defaults to -1.5 (used to
implicitly default to 0 in \R{} versions earlier than 1.4).
\item[high:] upper bound for \code{spar}; defaults to +1.5.
\item[tol:] the absolute precision (\bold{tol}erance) used; defaults
to 1e-4 (formerly 1e-3).
\item[eps:] the relative precision used; defaults to 2e-8 (formerly
0.00244).
\item[trace:] logical indicating if iterations should be traced.
\item[maxit:] integer giving the maximal number of iterations;
defaults to 500.
}
Note that \code{spar} is only searched for in the interval
\eqn{[low, high]}{}.

\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The \code{x} vector should contain at least four distinct values.
\emph{Distinct} here means \sQuote{distinct after rounding to 6 significant
digits}, i.e., \code{x} will be transformed to
\code{unique(sort(signif(x, 6)))}, and \code{y} and \code{w} are
pooled accordingly.

The computational \eqn{\lambda}{} used (as a function of
\eqn{s=spar}{\code{spar}}) is
\eqn{\lambda = r * 256^{3 s - 1}}{lambda = r * 256^(3*spar - 1)}
where
\eqn{r = tr(X' W X) / tr(\Sigma)}{},
\eqn{\Sigma}{} is the matrix given by
\eqn{\Sigma_{ij} = \int B_i''(t) B_j''(t) dt}{Sigma[i,j] = Integral B''[i](t) B''[j](t) dt},
\eqn{X}{} is given by \eqn{X_{ij} = B_j(x_i)}{X[i,j] = B[j](x[i])},
\eqn{W}{} is the diagonal matrix of weights (scaled such that
its trace is \eqn{n}{}, the original number of observations)
and \eqn{B_k(.)}{B[k](.)} is the \eqn{k}{}-th B-spline.

Note that with these definitions, \eqn{f_i = f(x_i)}{}, and the B-spline
basis representation \eqn{f = X c}{} (i.e., \eqn{c}{} is
the vector of spline coefficients), the penalized log likelihood is
\eqn{L = (y - f)' W (y - f) + \lambda c' \Sigma c}{}, and hence
\eqn{c}{} is the solution of the (ridge regression)
\eqn{(X' W X + \lambda \Sigma) c = X' W y}{}.

If \code{spar} is missing or \code{NULL}, the value of \code{df} is used to
determine the degree of smoothing.  If both are missing, leave-one-out
cross-validation (ordinary or \sQuote{generalized} as determined by
\code{cv}) is used to determine \eqn{\lambda}{}.
Note that from the above relation,
\code{spar} is \eqn{s = s0 + 0.0601 * \bold{\log}\lambda}{spar = s0 + 0.0601 * log(lambda)},
which is intentionally \emph{different} from the S-plus implementation
of \code{smooth.spline} (where \code{spar} is proportional to
\eqn{\lambda}{}).  In \R's (\eqn{\log \lambda}{}) scale, it makes more
sense to vary \code{spar} linearly.

Note however that currently the results may become very unreliable
for \code{spar} values smaller than about -1 or -2.  The same may
happen for values larger than 2 or so. Don't think of setting
\code{spar} or the controls \code{low} and \code{high} outside such a
safe range, unless you know what you are doing!

The \sQuote{generalized} cross-validation method will work correctly when
there are duplicated points in \code{x}.  However, it is ambiguous what
leave-one-out cross-validation means with duplicated points, and the
internal code uses an approximation that involves leaving out groups
of duplicated points.  \code{cv=TRUE} is best avoided in that case.
\end{Details}
\begin{Value}
An object of class \code{"smooth.spline"} with components
\begin{ldescription}
\item[\code{x}] the \emph{distinct} \code{x} values in increasing order, see
the \sQuote{Details} above.
\item[\code{y}] the fitted values corresponding to \code{x}.
\item[\code{w}] the weights used at the unique values of \code{x}.
\item[\code{yin}] the y values used at the unique \code{y} values.
\item[\code{data}] only if \code{keep.data = TRUE}: itself a
\code{\LinkA{list}{list}} with components \code{x}, \code{y} and \code{w}
of the same length.  These are the original \eqn{(x_i,y_i,w_i),
i=1,\dots,n}{}, values where \code{data\$x} may have repeated values and
hence be longer than the above \code{x} component; see details.

\item[\code{lev}] leverages, the diagonal values of the smoother matrix.
\item[\code{cv.crit}] cross-validation score, \sQuote{generalized} or true, depending
on \code{cv}.
\item[\code{pen.crit}] penalized criterion
\item[\code{crit}] the criterion value minimized in the underlying
\code{.Fortran} routine \file{sslvrg}.
\item[\code{df}] equivalent degrees of freedom used.  Note that (currently)
this value may become quite imprecise when the true \code{df} is
between and 1 and 2.

\item[\code{spar}] the value of \code{spar} computed or given.
\item[\code{lambda}] the value of \eqn{\lambda}{} corresponding to \code{spar},
see the details above.
\item[\code{iparms}] named integer(3) vector where \code{..\$ipars["iter"]}
gives number of spar computing iterations used.
\item[\code{fit}] list for use by \code{\LinkA{predict.smooth.spline}{predict.smooth.spline}}, with
components
\describe{
\item[knot:] the knot sequence (including the repeated boundary
knots).
\item[nk:] number of coefficients or number of \sQuote{proper}
knots plus 2.
\item[coef:] coefficients for the spline basis used.
\item[min, range:] numbers giving the corresponding quantities of
\code{x}.
}

\item[\code{call}] the matched call.
\end{ldescription}
\end{Value}
\begin{Note}\relax
The default \code{all.knots = FALSE} and \code{nknots = NULL} entails
using only \eqn{O(n^{0.2})}{}
knots instead of \eqn{n}{} for \eqn{n > 49}{}.  This cuts speed and memory
requirements, but not drastically anymore since \R{} version 1.5.1 where
it is only \eqn{O(n_k) + O(n)}{O(nk) + O(n)} where \eqn{n_k}{nk} is
the number of knots.
In this case where not all unique \code{x} values are
used as knots, the result is not a smoothing spline in the strict
sense, but very close unless a small smoothing parameter (or large
\code{df}) is used.
\end{Note}
\begin{Author}\relax
\R{} implementation by B. D. Ripley and Martin Maechler
(\code{spar/lambda}, etc).

This function is based on code in the \code{GAMFIT} Fortran program by
T. Hastie and R. Tibshirani (\url{http://lib.stat.cmu.edu/general/}),
which makes use of spline code by Finbarr O'Sullivan.  Its design
parallels the \code{smooth.spline} function of Chambers \& Hastie (1992).
\end{Author}
\begin{References}\relax
Chambers, J. M. and Hastie, T. J. (1992)
\emph{Statistical Models in S}, Wadsworth \& Brooks/Cole.

Green, P. J. and Silverman, B. W. (1994)
\emph{Nonparametric Regression and Generalized Linear Models:
A Roughness Penalty Approach.} Chapman and Hall.

Hastie, T. J. and Tibshirani, R. J. (1990)
\emph{Generalized Additive Models.}  Chapman and Hall.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{predict.smooth.spline}{predict.smooth.spline}} for evaluating the spline
and its derivatives.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
require(graphics)

attach(cars)
plot(speed, dist, main = "data(cars)  &  smoothing splines")
cars.spl <- smooth.spline(speed, dist)
(cars.spl)
## This example has duplicate points, so avoid cv=TRUE

lines(cars.spl, col = "blue")
lines(smooth.spline(speed, dist, df=10), lty=2, col = "red")
legend(5,120,c(paste("default [C.V.] => df =",round(cars.spl$df,1)),
               "s( * , df = 10)"), col = c("blue","red"), lty = 1:2,
       bg='bisque')
detach()

## Residual (Tukey Anscombe) plot:
plot(residuals(cars.spl) ~ fitted(cars.spl))
abline(h = 0, col="gray")

## consistency check:
stopifnot(all.equal(cars$dist,
                    fitted(cars.spl) + residuals(cars.spl)))

##-- artificial example
y18 <- c(1:3,5,4,7:3,2*(2:5),rep(10,4))
xx  <- seq(1,length(y18), len=201)
(s2  <- smooth.spline(y18)) # GCV
(s02 <- smooth.spline(y18, spar = 0.2))
plot(y18, main=deparse(s2$call), col.main=2)
lines(s2, col = "gray"); lines(predict(s2, xx), col = 2)
lines(predict(s02, xx), col = 3); mtext(deparse(s02$call), col = 3)

## The following shows the problematic behavior of 'spar' searching:
(s2  <- smooth.spline(y18, control = list(trace=TRUE,tol=1e-6, low= -1.5)))
(s2m <- smooth.spline(y18, cv = TRUE,
                      control = list(trace=TRUE,tol=1e-6, low= -1.5)))
## both above do quite similarly (Df = 8.5 +- 0.2)


\end{ExampleCode}
\end{Examples}

