Running Medians – Robust Scatter Plot Smoothing

Description

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

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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 k2 values at both ends, where k2 is the half-bandwidth k2 = k %/% 2, i.e., y[j] = x[j] for j = 1, …, k2 and (n-k2+1), …, 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 Härdle–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 * k) which is slower asymptotically, it is considerably faster for small k or n.

Currently long vectors are only supported for algorithm = "Steutzle".

Value

vector of smoothed values of the same length as x with an attribute k containing (the ‘oddified’) k.

Author(s)

Martin Maechler maechler@stat.math.ethz.ch, based on Fortran code from Werner Stuetzle and S-PLUS and C code from Berwin Turlach.

References

Härdle, 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

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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))


## 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])


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))

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