# runmed

##### Running Medians -- Robust Scatter Plot Smoothing

Compute running medians of odd span. This is the ‘most robust’ scatter plot smoothing possible. For efficiency (and historical reason), you can use one of two different algorithms giving identical results.

##### Usage

```
runmed(x, k, endrule = c("median", "keep", "constant"),
algorithm = NULL, print.level = 0)
```

##### Arguments

- x
- numeric vector, the ‘dependent’ variable to be smoothed.
- k
- integer width of median window; must be odd. Turlach had a
default of
`k <- 1 + 2 * min((n-1)%/% 2, ceiling(0.1*n))`

. Use`k = 3`

for ‘minimal’ robust smoothing eliminating isolated outliers. - endrule
- character string indicating how the values at the
beginning and the end (of the data) should be treated.
Can be abbreviated. Possible values are:
`"keep"`

- keeps the first and last \(k_2\) values
at both ends, where \(k_2\) is the half-bandwidth
`k2 = k %/% 2`

, i.e.,`y[j] = x[j]`

for \(j \in \{1,\ldots,k_2; n-k_2+1,\ldots,n\}\); `"constant"`

- copies
`median(y[1:k2])`

to the first values and analogously for the last ones making the smoothed ends*constant*; `"median"`

- the default, smooths 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
`smoothEnds`

.

- algorithm
- character string (partially matching
`"Turlach"`

or`"Stuetzle"`

) or the default`NULL`

, specifying which algorithm should be applied. The default choice depends on`n = length(x)`

and`k`

where`"Turlach"`

will be used for larger problems. - print.level
- integer, indicating verboseness of algorithm; should rarely be changed by average users.

##### Details

Apart from the end values, the result `y = runmed(x, k)`

simply has
`y[j] = median(x[(j-k2):(j+k2)])`

(`k = 2*k2+1`

), computed very
efficiently. The two algorithms are internally entirely different:

`"Turlach"`

- is the H<U+00E4>rdle--Steiger
algorithm (see Ref.) as implemented by Berwin Turlach.
A tree algorithm is used, ensuring performance \(O(n \log
k)\) where
`n = length(x)`

which is asymptotically optimal. `"Stuetzle"`

- is the (older) Stuetzle--Friedman implementation
which makes use of median
*updating*when one observation enters and one leaves the smoothing window. While this performs as \(O(n \times k)\) which is slower asymptotically, it is considerably faster for small \(k\) or \(n\).

`algorithm = "Steutzle"`

.
##### Value

vector of smoothed values of the same length as `x`

with an
`attr`

ibute `k`

containing (the ‘oddified’)
`k`

.

##### References

H<U+00E4>rdle, W. and Steiger, W. (1995)
[Algorithm AS 296] Optimal median smoothing,
*Applied Statistics* **44**, 258--264. Jerome H. Friedman and Werner Stuetzle (1982)
*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.

##### See Also

`smoothEnds`

which implements Tukey's end point rule and
is called by default from `runmed(*, endrule = "median")`

.
`smooth`

uses running
medians of 3 for its compound smoothers.

##### Examples

`library(stats)`

```
require(graphics)
utils::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))
<!-- %% FIXME: Show how to do it properly ! tapply(*, unique(.), median) -->
## 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-neighbourhood <==> interpolation
plot(y) ## lines(y, lwd = .1, col = "light gray")
lines(lowess(seq(y), y, f = 0.3), 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 = 0.3), col = col[2])
<!-- %% predict(loess(y ~ x, span = 0.3, degree=1, family = "symmetric")) -->
<!-- %% gives 6-line warning but does NOT break down -->
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))
```

*Documentation reproduced from package stats, version 3.3.3, License: Part of R 3.3.3*