Tukey's (Running Median) Smoothing

Tukey's smoothers, 3RS3R, 3RSS, 3R, etc.

robust, smooth
smooth(x, kind = c("3RS3R", "3RSS", "3RSR", "3R", "3", "S"),
       twiceit = FALSE, endrule = c("Tukey", "copy"), do.ends = FALSE)

a vector or time series


a character string indicating the kind of smoother required; defaults to "3RS3R".


logical, indicating if the result should be ‘twiced’. Twicing a smoother \(S(y)\) means \(S(y) + S(y - S(y))\), i.e., adding smoothed residuals to the smoothed values. This decreases bias (increasing variance).


a character string indicating the rule for smoothing at the boundary. Either "Tukey" (default) or "copy".


logical, indicating if the 3-splitting of ties should also happen at the boundaries (ends). This is only used for kind = "S".


3 is Tukey's short notation for running medians of length 3,

3R stands for Repeated 3 until convergence, and

S for Splitting of horizontal stretches of length 2 or 3.

Hence, 3RS3R is a concatenation of 3R, S and 3R, 3RSS similarly, whereas 3RSR means first 3R and then (S and 3) Repeated until convergence -- which can be bad.


An object of class "tukeysmooth" (which has print and summary methods) and is a vector or time series containing the smoothed values with additional attributes.


S and S-PLUS use a different (somewhat better) Tukey smoother in smooth(*). Note that there are other smoothing methods which provide rather better results. These were designed for hand calculations and may be used mainly for didactical purposes.

Since R version 1.2, smooth does really implement Tukey's end-point rule correctly (see argument endrule).

kind = "3RSR" has been the default till R-1.1, but it can have very bad properties, see the examples.

Note that repeated application of smooth(*) does smooth more, for the "3RS*" kinds.


Tukey, J. W. (1977). Exploratory Data Analysis, Reading Massachusetts: Addison-Wesley.

See Also

runmed for running medians; lowess and loess; supsmu and smooth.spline.

  • smooth
library(stats) # NOT RUN { require(graphics) ## see also demo(smooth) ! x1 <- c(4, 1, 3, 6, 6, 4, 1, 6, 2, 4, 2) # very artificial (x3R <- smooth(x1, "3R")) # 2 iterations of "3" smooth(x3R, kind = "S") sm.3RS <- function(x, ...) smooth(smooth(x, "3R", ...), "S", ...) y <- c(1, 1, 19:1) plot(y, main = "misbehaviour of \"3RSR\"", col.main = 3) lines(sm.3RS(y)) lines(smooth(y)) lines(smooth(y, "3RSR"), col = 3, lwd = 2) # the horror x <- c(8:10, 10, 0, 0, 9, 9) plot(x, main = "breakdown of 3R and S and hence 3RSS") matlines(cbind(smooth(x, "3R"), smooth(x, "S"), smooth(x, "3RSS"), smooth(x))) presidents[] <- 0 # silly summary(sm3 <- smooth(presidents, "3R")) summary(sm2 <- smooth(presidents,"3RSS")) summary(sm <- smooth(presidents)) all.equal(c(sm2), c(smooth(smooth(sm3, "S"), "S"))) # 3RSS === 3R S S all.equal(c(sm), c(smooth(smooth(sm3, "S"), "3R"))) # 3RS3R === 3R S 3R plot(presidents, main = "smooth(presidents0, *) : 3R and default 3RS3R") lines(sm3, col = 3, lwd = 1.5) lines(sm, col = 2, lwd = 1.25) # }
Documentation reproduced from package stats, version 3.6.2, License: Part of R 3.6.2

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