## Median Absolute Deviation of Moving Windows

### Description

Moving (aka running, rolling) Window MAD (Median Absolute Deviation) calculated over a vector

### Usage

   runmad(x, k, center = runmed(x,k), constant = 1.4826,
endrule=c("mad", "NA", "trim", "keep", "constant", "func"),
align = c("center", "left", "right"))


### Arguments

 x numeric vector of length n or matrix with n rows. If x is a matrix than each column will be processed separately. k width of moving window; must be an integer between one and n. In case of even k's one will have to provide different center function, since runmed does not take even k's. endrule character string indicating how the values at the beginning and the end, of the data, should be treated. Only first and last k2 values at both ends are affected, where k2 is the half-bandwidth k2 = k %/% 2. "mad" - applies the mad function to smaller and smaller sections of the array. Equivalent to: for(i in 1:k2) out[i]=mad(x[1:(i+k2)]). "trim" - trim the ends; output array length is equal to length(x)-2*k2 (out = out[(k2+1):(n-k2)]). This option mimics output of apply (embed(x,k),1,FUN) and other related functions. "keep" - fill the ends with numbers from x vector (out[1:k2] = x[1:k2]). This option makes more sense in case of smoothing functions, kept here for consistency. "constant" - fill the ends with first and last calculated value in output array (out[1:k2] = out[k2+1]) "NA" - fill the ends with NA's (out[1:k2] = NA) "func" - same as "mad" option except that implemented in R for testing purposes. Avoid since it can be very slow for large windows. Similar to endrule in runmed function which has the following options: “c("median", "keep", "constant")” . center moving window center. Defaults to running median (runmed function). Similar to center in mad function. For best acuracy at the edges use runquantile(x,k,0.5,type=2), which is slower than default runmed(x,k,endrule="med"). If x is a 2D array (and endrule="mad") or if endrule="func" than array edges are filled by repeated calls to “mad(x, center=median(x), na.rm=TRUE)” function. Runmad's center parameter will be ignored for the beggining and the end of output y. Please use center=runquantile(x,k,0.5,type=2) for those cases. constant scale factor such that for Gaussian distribution X, mad(X) is the same as sd(X). Same as constant in mad function. align specifies whether result should be centered (default), left-aligned or right-aligned. If endrule="mad" then setting align to "left" or "right" will fall back on slower implementation equivalent to endrule="func".

### Details

Apart from the end values, the result of y = runmad(x, k) is the same as “for(j=(1+k2):(n-k2)) y[j]=mad(x[(j-k2):(j+k2)], na.rm = TRUE)”. It can handle non-finite numbers like NaN's and Inf's (like “mad(x, na.rm = TRUE)”).

The main incentive to write this set of functions was relative slowness of majority of moving window functions available in R and its packages. With the exception of runmed, a running window median function, all functions listed in "see also" section are slower than very inefficient “apply(embed(x,k),1,FUN)” approach.

Functions runquantile and runmad are using insertion sort to sort the moving window, but gain speed by remembering results of the previous sort. Since each time the window is moved, only one point changes, all but one points in the window are already sorted. Insertion sort can fix that in O(k) time.

### Value

Returns a numeric vector or matrix of the same size as x. Only in case of endrule="trim" the output vectors will be shorter and output matrices will have fewer rows.

### Author(s)

Jarek Tuszynski (SAIC) jaroslaw.w.tuszynski@saic.com

### References

About insertion sort used in runmad function see: R. Sedgewick (1988): Algorithms. Addison-Wesley (page 99)

• runmad - mad

• Other moving window functions from this package: runmin, runmax, runquantile, runmean and runsd

• generic running window functions: apply (embed(x,k), 1, FUN) (fastest), running from gtools package (extremely slow for this purpose), subsums from magic library can perform running window operations on data with any dimensions.

### Examples

  # show runmed function
k=25; n=200;
x = rnorm(n,sd=30) + abs(seq(n)-n/4)
col = c("black", "red", "green")
m=runmed(x, k)
plot(x, col=col[1], main = "Moving Window Analysis Functions")
lines(m    , col=col[2])
lines(m-y/2, col=col[3])
lines(m+y/2, col=col[3])
legend(0,0.9*n, lab, col=col, lty=1 )

# basic tests against apply/embed
eps = .Machine$double.eps ^ 0.5 k=25 # odd size window a = runmad(x,k, center=runmed(x,k), endrule="trim") b = apply(embed(x,k), 1, mad) stopifnot(all(abs(a-b)<eps)); k=24 # even size window a = runmad(x,k, center=runquantile(x,k,0.5,type=2), endrule="trim") b = apply(embed(x,k), 1, mad) stopifnot(all(abs(a-b)<eps)); # test against loop approach # this test works fine at the R prompt but fails during package check - need to investigate k=24; n=200; x = rnorm(n,sd=30) + abs(seq(n)-n/4) # create random data x = rep(1:5,40) #x[seq(1,n,11)] = NaN; # commented out for time beeing - on to do list #x[5] = NaN; # commented out for time beeing - on to do list k2 = k k1 = k-k2-1 ac = array(runquantile(x,k,0.5)) a = runmad(x, k, center=ac) bc = array(0,n) b = array(0,n) for(j in 1:n) { lo = max(1, j-k1) hi = min(n, j+k2) bc[j] = median(x[lo:hi], na.rm = TRUE) b [j] = mad (x[lo:hi], na.rm = TRUE, center=bc[j]) } eps = .Machine$double.eps ^ 0.5
#stopifnot(all(abs(ac-bc)<eps)); # commented out for time beeing - on to do list
#stopifnot(all(abs(a-b)<eps));   # commented out for time beeing - on to do list

# compare calculation at array ends
k=25; n=200;
x = rnorm(n,sd=30) + abs(seq(n)-n/4)
c = runquantile(x,k,0.5,type=2)             # find the center
b = runmad(x, k, center=c, endrule="func")  # slow R code
stopifnot(all(abs(a-b)<eps));

# test if moving windows forward and backward gives the same results
k=51;
stopifnot(all(a[n:1]==b, na.rm=TRUE));

# test vector vs. matrix inputs, especially for the edge handling
nRow=200; k=25; nCol=10
x = rnorm(nRow,sd=30) + abs(seq(nRow)-n/4)
X = matrix(rep(x, nCol ), nRow, nCol) # replicate x in columns of X
a = runmad(x, k, center = runquantile(x,k,0.5,type=2))
b = runmad(X, k, center = runquantile(X,k,0.5,type=2))
stopifnot(all(abs(a-b[,1])<eps));        # vector vs. 2D array
stopifnot(all(abs(b[,1]-b[,nCol])<eps)); # compare rows within 2D array

# speed comparison
## Not run:
x=runif(1e5); k=51;                       # reduce vector and window sizes