\HeaderA{runmed}{Running Medians -- Robust Scatter Plot Smoothing}{runmed}
\keyword{smooth}{runmed}
\keyword{robust}{runmed}
\begin{Description}\relax
Compute running medians of odd span.  This is the \dQuote{most robust}
scatter plot smoothing possible.  For efficiency (and historical
reason), you can use one of two different algorithms giving identical
results.
\end{Description}
\begin{Usage}
\begin{verbatim}
runmed(x, k, endrule = c("median", "keep", "constant"),
       algorithm = NULL, print.level = 0)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] numeric vector, the \dQuote{dependent} variable to be
smoothed.
\item[\code{k}] integer width of median window; must be odd.  Turlach had a
default of \code{k <- 1 + 2 * min((n-1)\%/\% 2, ceiling(0.1*n))}.
Use \code{k = 3} for \dQuote{minimal} robust smoothing eliminating
isolated outliers.
\item[\code{endrule}] character string indicating how the values at the
beginning and the end (of the data) should be treated.
\describe{
\item[\code{"keep"}] keeps the first and last \eqn{k_2}{k2} values
at both ends, where \eqn{k_2}{k2} is the half-bandwidth \code{k2
          = k \%/\% 2},
i.e., \code{y[j] = x[j]} for \eqn{j \in \{1,\ldots,k_2;
n-k_2+1,\ldots,n\}}{j = 1,..,k2 and (n-k2+1),..,n};
\item[\code{"constant"}] copies \code{median(y[1:k2])} to the first
values and analogously for the last ones making the smoothed ends
\emph{constant};
\item[\code{"median"}] the default, smoothes the ends by using
symmetrical medians of subsequently smaller bandwidth, but for
the very first and last value where Tukey's robust end-point
rule is applied, see \code{\LinkA{smoothEnds}{smoothEnds}}.
}

\item[\code{algorithm}] character string (partially matching \code{"Turlach"} or
\code{"Stuetzle"}) or the default \code{NULL}, specifying which algorithm
should be applied.  The default choice depends on \code{n = length(x)}
and \code{k} where \code{"Turlach"} will be used for larger problems.
\item[\code{print.level}] integer, indicating verboseness of algorithm;
should rarely be changed by average users.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
Apart from the end values, the result \code{y = runmed(x, k)} simply has
\code{y[j] = median(x[(j-k2):(j+k2)])} (k = 2*k2+1), computed very
efficiently.

The two algorithms are internally entirely different:
\describe{
\item["Turlach"] is the Härdle-Steiger
algorithm (see Ref.) as implemented by Berwin Turlach.
A tree algorithm is used, ensuring performance \eqn{O(n \log
k)}{O(n * log(k))} where \code{n <- length(x)} which is
asymptotically optimal.
\item["Stuetzle"] is the (older) Stuetzle-Friedman implementation
which makes use of median \emph{updating} when one observation
enters and one leaves the smoothing window.  While this performs as
\eqn{O(n \times k)}{O(n * k)} which is slower asymptotically, it is
considerably faster for small \eqn{k}{} or \eqn{n}{}.
}
\end{Details}
\begin{Value}
vector of smoothed values of the same length as \code{x} with an
\code{\LinkA{attr}{attr}}ibute \code{k} containing (the \sQuote{oddified})
\code{k}.
\end{Value}
\begin{Author}\relax
Martin Maechler \email{maechler@stat.math.ethz.ch},
based on Fortran code from Werner Stuetzle and S-plus and C code from
Berwin Turlach.
\end{Author}
\begin{References}\relax
Härdle, W. and Steiger, W. (1995)
[Algorithm AS 296] Optimal median smoothing,
\emph{Applied Statistics} \bold{44}, 258--264.

Jerome H. Friedman and Werner Stuetzle (1982)
\emph{Smoothing of Scatterplots};
Report, Dep. Statistics, Stanford U., Project Orion 003.

Martin Maechler (2003)
Fast Running Medians: Finite Sample and Asymptotic Optimality;
working paper available from the author.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{smoothEnds}{smoothEnds}} which implements Tukey's end point rule and
is called by default from \code{runmed(*, endrule = "median")}.
\code{\LinkA{smooth}{smooth}} uses running
medians of 3 for its compound smoothers.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
example(nhtemp)
myNHT <- as.vector(nhtemp)
myNHT[20] <- 2 * nhtemp[20]
plot(myNHT, type="b", ylim = c(48,60), main = "Running Medians Example")
lines(runmed(myNHT, 7), col = "red")

## special: multiple y values for one x
plot(cars, main = "'cars' data and runmed(dist, 3)")
lines(cars, col = "light gray", type = "c")
with(cars, lines(speed, runmed(dist, k = 3), col = 2))

## nice quadratic with a few outliers
y <- ys <- (-20:20)^2
y [c(1,10,21,41)] <- c(150, 30, 400, 450)
all(y == runmed(y, 1)) # 1-neigborhood <==> interpolation
plot(y) ## lines(y, lwd=.1, col="light gray")
lines(lowess(seq(y),y, f = .3), col = "brown")
lines(runmed(y, 7), lwd=2, col = "blue")
lines(runmed(y,11), lwd=2, col = "red")

## Lowess is not robust
y <- ys ; y[21] <- 6666 ; x <- seq(y)
col <- c("black", "brown","blue")
plot(y, col=col[1])
lines(lowess(x,y, f = .3), col = col[2])
lines(runmed(y, 7),      lwd=2, col = col[3])
legend(length(y),max(y), c("data", "lowess(y, f = 0.3)", "runmed(y, 7)"),
       xjust = 1, col = col, lty = c(0, 1,1), pch = c(1,NA,NA))
\end{ExampleCode}
\end{Examples}

