1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
|
\name{runmad}
\alias{runmad}
\title{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{
\item{x}{numeric vector of length n or matrix with n rows. If \code{x} is a
matrix than each column will be processed separately.}
\item{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 \code{center} function, since
\code{\link{runmed}} does not take even k's.}
\item{endrule}{character string indicating how the values at the beginning
and the end, of the data, should be treated. Only first and last \code{k2}
values at both ends are affected, where \code{k2} is the half-bandwidth
\code{k2 = k \%/\% 2}.
\itemize{
\item \code{"mad"} - applies the mad function to
smaller and smaller sections of the array. Equivalent to:
\code{for(i in 1:k2) out[i]=mad(x[1:(i+k2)])}.
\item \code{"trim"} - trim the ends; output array length is equal to
\code{length(x)-2*k2 (out = out[(k2+1):(n-k2)])}. This option mimics
output of \code{\link{apply}} \code{(\link{embed}(x,k),1,FUN)} and other
related functions.
\item \code{"keep"} - fill the ends with numbers from \code{x} vector
\code{(out[1:k2] = x[1:k2])}. This option makes more sense in case of
smoothing functions, kept here for consistency.
\item \code{"constant"} - fill the ends with first and last calculated
value in output array \code{(out[1:k2] = out[k2+1])}
\item \code{"NA"} - fill the ends with NA's \code{(out[1:k2] = NA)}
\item \code{"func"} - same as \code{"mad"} option except that implemented
in R for testing purposes. Avoid since it can be very slow for large windows.
}
Similar to \code{endrule} in \code{\link{runmed}} function which has the
following options: \dQuote{\code{c("median", "keep", "constant")}} .
}
\item{center}{moving window center. Defaults
to running median (\code{\link{runmed}} function). Similar to \code{center}
in \code{\link{mad}} function. For best acuracy at the edges use
\code{\link{runquantile}(x,k,0.5,type=2)}, which is slower than default
\code{\link{runmed}(x,k,endrule="med")}. If \code{x} is a 2D array (and
\code{endrule="mad"}) or if \code{endrule="func"} than array edges are
filled by repeated calls to
\dQuote{\code{\link{mad}(x, center=\link{median}(x), na.rm=TRUE)}} function.
Runmad's \code{center} parameter will be ignored for the beggining and the
end of output \code{y}. Please use
\code{center=\link{runquantile}(x,k,0.5,type=2)} for those cases.
}
\item{constant}{scale factor such that for Gaussian
distribution X, \code{\link{mad}}(X) is the same as \code{\link{sd}}(X).
Same as \code{constant} in \code{\link{mad}} function.}
\item{align}{specifies whether result should be centered (default),
left-aligned or right-aligned. If \code{endrule}="mad" then setting
\code{align} to "left" or "right" will fall back on slower implementation
equivalent to \code{endrule}="func". }
}
\details{
Apart from the end values, the result of y = runmad(x, k) is the same as
\dQuote{\code{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 \dQuote{\code{\link{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 \code{\link{runmed}}, a running window median function, all
functions listed in "see also" section are slower than very inefficient
\dQuote{\code{\link{apply}(\link{embed}(x,k),1,FUN)}} approach.
Functions \code{runquantile} and \code{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 \code{x}. Only in case of
\code{endrule="trim"} the output vectors will be shorter and output matrices
will have fewer rows.
}
\references{
About insertion sort used in \code{runmad} function see:
R. Sedgewick (1988): \emph{Algorithms}. Addison-Wesley (page 99)
}
\author{Jarek Tuszynski (SAIC) \email{jaroslaw.w.tuszynski@saic.com}}
\seealso{
Links related to:
\itemize{
\item \code{runmad} - \code{\link{mad}}, \code{\link[fSeries]{rollVar}} from
\pkg{fSeries} library
\item Other moving window functions from this package: \code{\link{runmin}},
\code{\link{runmax}}, \code{\link{runquantile}}, \code{\link{runmean}} and
\code{\link{runsd}}
\item generic running window functions: \code{\link{apply}}\code{
(\link{embed}(x,k), 1, FUN)} (fastest), \code{\link[fSeries]{rollFun}}
from \pkg{fSeries} (slow), \code{\link[gtools]{running}} from \pkg{gtools}
package (extremely slow for this purpose), \code{\link[zoo]{rapply}} from
\pkg{zoo} library, \code{\link[magic]{subsums}} from
\pkg{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)
y=runmad(x, k, center=m)
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])
lab = c("data", "runmed", "runmed-runmad/2", "runmed+runmad/2")
legend(0,0.9*n, lab, col=col, lty=1 )
# basic tests against apply/embed
eps = .Machine$double.eps ^ 0.5
k=25 # odd size window
a = runmad(x,k, center=runmed(x,k), endrule="trim")
b = apply(embed(x,k), 1, mad)
stopifnot(all(abs(a-b)<eps));
k=24 # even size window
a = runmad(x,k, center=runquantile(x,k,0.5,type=2), endrule="trim")
b = apply(embed(x,k), 1, mad)
stopifnot(all(abs(a-b)<eps));
# test against loop approach
# this test works fine at the R prompt but fails during package check - need to investigate
k=24; n=200;
x = rnorm(n,sd=30) + abs(seq(n)-n/4) # create random data
x = rep(1:5,40)
#x[seq(1,n,11)] = NaN; # commented out for time beeing - on to do list
#x[5] = NaN; # commented out for time beeing - on to do list
k2 = k%/%2
k1 = k-k2-1
ac = array(runquantile(x,k,0.5))
a = runmad(x, k, center=ac)
bc = array(0,n)
b = array(0,n)
for(j in 1:n) {
lo = max(1, j-k1)
hi = min(n, j+k2)
bc[j] = median(x[lo:hi], na.rm = TRUE)
b [j] = mad (x[lo:hi], na.rm = TRUE, center=bc[j])
}
eps = .Machine$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
a = runmad(x, k, center=c, endrule="mad" ) # fast C code
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;
a = runmad(x , k)
b = runmad(x[n:1], k)
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
\dontrun{
x=runif(1e5); k=51; # reduce vector and window sizes
system.time(runmad( x,k,endrule="trim"))
system.time(apply(embed(x,k), 1, mad))
}
}
\keyword{ts}
\keyword{array}
\keyword{utilities}
\concept{moving mad}
\concept{rolling mad}
\concept{running mad}
\concept{running window}
|
