\HeaderA{lowess}{Scatter Plot Smoothing}{lowess}
\keyword{smooth}{lowess}
\begin{Description}\relax
This function performs the computations for the
\emph{LOWESS} smoother which uses locally-weighted polynomial
regression (see the references).
\end{Description}
\begin{Usage}
\begin{verbatim}
lowess(x, y = NULL, f = 2/3, iter = 3,
       delta = 0.01 * diff(range(xy$x[o])))
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, y}] vectors giving the coordinates of the points in the scatter plot.
Alternatively a single plotting structure can be specified.
\item[\code{f}] the smoother span. This gives the proportion of points in
the plot which influence the smooth at each value.
Larger values give more smoothness.
\item[\code{iter}] the number of \sQuote{robustifying} iterations which should be
performed.
Using smaller values of \code{iter} will make \code{lowess} run faster.
\item[\code{delta}] See Details.  Defaults to 1/100th of the range of \code{x}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
\code{lowess} is defined by a complex algorithm, the Ratfor original
of which (by W. S. Cleveland) can be found in the \R{} sources as file
\file{src/appl/lowess.doc}.  Normally a local linear polynomial fit is
used, but under some circumstances (see the file) a local constant fit
can be used.  \sQuote{Local} is defined by the distance to the
\code{floor(f*n)}th nearest neighbour, and tricubic weighting is used
for \code{x} which fall within the neighbourhood.

The initial fit is done using weighted least squares.  If
\code{iter > 0}, further weighted fits are done using the product of
the weights from the proximity of the \code{x} values and case weights
derived from the residuals at the previous iteration.  Specifically,
the case weight is Tukey's biweight, with cutoff 6 times the MAD of the
residuals.  (The current \R{} implementation differs from the original
in stopping iteration if the MAD is effectively zero since the
algorithm is highly unstable in that case.)

\code{delta} is used to speed up computation: instead of computing the
local polynomial fit at each data point it is not computed for points
within \code{delta} of the last computed point, and linear
interpolation is used to fill in the fitted values for the skipped
points.
\end{Details}
\begin{Value}
\code{lowess} returns a list containing components
\code{x} and \code{y} which give the coordinates of the smooth.
The smooth can be added to a plot of the original
points with the function \code{lines}: see the examples.
\end{Value}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Cleveland, W. S. (1979)
Robust locally weighted regression and smoothing scatterplots.
\emph{J. Amer. Statist. Assoc.} \bold{74}, 829--836.

Cleveland, W. S. (1981)
LOWESS: A program for smoothing scatterplots by robust locally weighted
regression. \emph{The American Statistician}, \bold{35}, 54.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{loess}{loess}}, a newer
formula based version of \code{lowess} (with different defaults!).
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
plot(cars, main = "lowess(cars)")
lines(lowess(cars), col = 2)
lines(lowess(cars, f=.2), col = 3)
legend(5, 120, c(paste("f = ", c("2/3", ".2"))), lty = 1, col = 2:3)
\end{ExampleCode}
\end{Examples}

