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#### -*- mode: R; kept-new-versions: 30; kept-old-versions: 20 -*-
#### MC-adjusted Outlyingness
#### ------------------------
###
### Original code from the web site from the Antwerpen Statistics Group :
### http://www.agoras.ua.ac.be/Robustn.htm
### which has a "MC" section and for the software links to
### ftp://ftp.win.ua.ac.be/pub/software/agoras/newfiles/mc.tar.gz
### and that contains mcrsoft/adjoutlyingness.R
##_NEW_ (2014): moved from Antwerpen to Leuwen,
## ===> http://wis.kuleuven.be/stat/robust/software
## has several links to 'robustbase', and S-plus code
## http://wis.kuleuven.be/stat/robust/Programs/adjusted-boxplot/adjusted-boxplot.ssc
## (copy in ../misc/Adjusted-boxplot.ssc
## MM [ FIXME ]:
## -----------
## 1) Use *transposed* B[] and A[] (now called 'E') matrices -- DONE
## 2) use IQR() instead of quantile(., .75) - quantile(., .25)
##--> but only *after* testing original code
## ^^^^^^^^^^^^^^^^^^^^^^^^
adjOutlyingness <- function(x, ndir = 250, p.samp = p, clower=4, cupper=3,
IQRtype = 7,
alpha.cutoff = 0.75, coef = 1.5,
qr.tol = 1e-12, keep.tol = 1e-12,
only.outlyingness = FALSE, maxit.mult = max(100, p),
trace.lev = 0,
## these are all passed to mc() {when applied to the projected data}
mcReflect = n <= 100, mcScale = TRUE, mcMaxit = 2*maxit.mult,
mcEps1 = 1e-12, mcEps2 = 1e-15,
mcTrace = max(0, trace.lev-1))
## Skewness-Adjusted Outlyingness
{
x <- data.matrix(x)
n <- nrow(x)
p <- ncol(x)
stopifnot(n >= 1, p >= 1, p.samp >= p, is.numeric(x))
if (p <= n) {
B <- matrix(0, p, ndir)
E <- matrix(1, p.samp, 1)
x. <- unname(x) # for speed in subsequent subsetting and solve
maxit <- as.integer(maxit.mult * ndir)
## ^^ original code had 'Inf', i.e. no iter.count check;
## often, maxit == ndir would suffice
if(trace.lev >= 2) p10 <- 10 ^ max(0, min(6 - trace.lev, floor(log10(maxit))))
i <- 1L
it <- 0L
while (i <= ndir && (it <- it+1L) < maxit) {
## we sort to get *identical* projections instead of "almost"
P <- x.[sort(sample.int(n, p.samp)), , drop=FALSE]
if ((qrP <- qr(P, tol = qr.tol))$rank == p) {
B[,i] <- solve(qrP, E, tol = qr.tol)
## if(trace.lev >= 2) cat(" it=",it,"; found direction # ", i,"\n", sep="")
i <- i+1L
} else if(trace.lev >= 2) {
if(it %% p10 == 0)
cat(" it=",it,": rank(qr(P ..)) = ", qrP$rank, " < p = ",p,"\n", sep="")
}
}
if(it == maxit) {
rnk.x <- qr(x, tol = qr.tol)$rank
if(rnk.x < p)
stop("Matrix 'x' is not of full rank: rankM(x) = ",rnk.x," < p = ",p,
"\n Use fullRank(x) instead")
## else
stop("** direction sampling iterations were not sufficient. Maybe try increasing 'maxit.mult'")
}
Bnorm <- sqrt(colSums(B^2))
Nx <- mean(abs(x.)) ## so the comparison is scale-equivariant:
keep <- Bnorm*Nx > keep.tol
if(!all(keep)) {
if(trace.lev)
cat("keeping ", sum(keep), "(out of", length(keep),") normalized directions\n")
Bnorm <- Bnorm[ keep ]
B <- B[, keep , drop=FALSE]
} else if(trace.lev) cat("keeping *all* ",length(keep)," normalized directions\n")
B <- B / rep(Bnorm, each = nrow(B)) # normalized B {called 'A' in orig.code}
}
else {
stop('More dimensions than observations: not yet implemented')
## MM: In LIBRA(matlab) they have it implemented:
## seed=0;
## nrich1=n*(n-1)/2;
## ndirect=min(250,nrich1);
## true = (ndirect == nrich1);
## B=extradir(x,ndir,seed,true); % n*ri
## ======== % Calculates ndirect directions through
## % two random choosen data points from data
## for i=1:size(B,1)
## Bnorm(i)=norm(B(i,:),2);
## end
## Bnorm=Bnorm(Bnorm > 1.e-12); %ndirect*1
## B=B(Bnorm > 1.e-12,:); %ndirect*n
## A=diag(1./Bnorm)*B; %ndirect*n
}
## NB: colSums( B^2 ) == 1
Y <- x %*% B # (n x p) %*% (p, nd') == (n x nd');
## nd' = ndir.final := ndir - {those not in 'keep'}
## Compute and sweep out the median
med <- colMedians(Y)
if(trace.lev) {
cat("med <- colMedians(Y): ", length(med), " values; summary():\n")
print(summary(med))
}
Y <- Y - rep(med, each=n)
## central :<==> non-adjusted <==> "classical" outlyingness
central <- clower == 0 && cupper == 0
if(!central) {
## MM: mc() could be made faster if we could tell it that med(..) = 0
## vv
tmc <- apply(Y, 2L, mc, doReflect=mcReflect, doScale=mcScale,
maxit=mcMaxit, eps1=mcEps1, eps2=mcEps2, trace.lev = mcTrace)
## original Antwerpen *wrongly*: tmc <- mc(Y)
if(trace.lev) {
cat("Columnwise mc() got ", length(tmc), " values; summary():\n")
print(summary(tmc))
}
}
Q13 <- apply(Y, 2, quantile, c(.25, .75), names=FALSE, type = IQRtype)
Q1 <- Q13[1L,]; Q3 <- Q13[2L,]
IQR <- Q3 - Q1
## NOTA BENE(MM): simplified definition of tup/tlo here and below
## 2014-10-18: "flipped" sign (which Pieter Setaert (c/o Mia H) proposed, Jul.30,2014:
tup <- Q3 + coef*
(if(central) IQR else IQR*exp( cupper*tmc*(tmc >= 0) + clower*tmc*(tmc < 0)))
tlo <- Q1 - coef*
(if(central) IQR else IQR*exp(-clower*tmc*(tmc >= 0) - cupper*tmc*(tmc < 0)))
## Note: all(tlo < med & med < tup) # where med = 0
if(trace.lev >= 3) {
print( cbind(tlo, Q1, Q3, tup) )
}
## Instead of the loop:
## for (i in 1:ndir) {
## tup[i] <- max(Y[Y[,i] < tup[i], i])
## tlo[i] <- -min(Y[Y[,i] > tlo[i], i])
## ## MM: FIXED typo-bug : ^^^ this was missing!
## ## But after the fix, the function stops "working" for longley..
## ## because tlo[] becomes 0 too often, YZ[.,.] = c / 0 = Inf !
## }
Yup <- Ylo <- Y
Yup[Y >= rep(tup, each=n)] <- -Inf
Ylo[Y <= rep(tlo, each=n)] <- Inf
y.up <- apply(Yup, 2, max) # = max{ Y[i,] ; Y[i,] < tup[i] }
y.lo <- -apply(Ylo, 2, min) # = -min{ Y[i,] ; Y[i,] > tlo[i] }
if(trace.lev) {
cat(length(y.up), "lower & upper Y (:= X - med(.)) values:\n")
print(summary(y.lo))
print(summary(y.up))
}
tY <- t(Y)
## Note: column-wise medians are all 0 : "x_i > m" <==> y > 0
## Note: this loop is pretty fast
for (j in 1:n) { # when y = (X-med) = 0 ==> adjout = 0 rather than
## 0 / 0 --> NaN; e.g, in set.seed(3); adjOutlyingness(longley)
non0 <- 0 != (y <- tY[,j]); y <- y[non0]; I <- (y > 0)
D <- I*y.up[non0] + (1 - I)*y.lo[non0]
if(trace.lev >= 3) {
cat(sprintf("j=%2d: #{non0}= %2d; quantile(D)=\n", j, sum(non0)))
print(quantile(D), digits=3)
}
tY[non0, j] <- abs(y) / D
}
## We get +Inf above for "small n"; e.g. set.seed(11); adjOutlyingness(longley)
if(trace.lev) {
cat("outlyingnesses for all directions (of which max(.) will be chosen:\n")
print(quantile(tY, digits=3))
}
adjout <- apply(tY, 2, function(x) max(x[is.finite(x)]))
##---- ---
if(abs(trace.lev %% 1 - 0.7) < 1e-3) { ## really not for the end user ..
cat("Plotting outlyingnesses vs. observation i:\n")
matplot(t(tY), log="y"); axis(2, at=1); abline(h=1, lty=3)
Sys.sleep(2)
## somewhat revealing: 3 groups: very large | medium | very small (incl. 0 which are *not* plotted)
matplot(t(tY), log="y", type="b"); axis(2, at=1); abline(h=1, lty=3)
browser() ## <<<<<<<<<<<<<<<<<<<
}
if(only.outlyingness)
adjout
else {
Qadj <- quantile(adjout, probs = c(1 - alpha.cutoff, alpha.cutoff))
mcadjout <- if(cupper != 0) mc(adjout, doScale=mcScale, eps1=mcEps1, eps2=mcEps2) else 0
## ===
cutoff <- Qadj[2] + coef * (Qadj[2] - Qadj[1]) *
(if(mcadjout > 0) exp(cupper*mcadjout) else 1)
list(adjout = adjout, iter = it, ndir.final = sum(keep),
MCadjout = mcadjout, Qalph.adjout = Qadj, cutoff = cutoff,
nonOut = (adjout <= cutoff))
}
}
##' Compute a "full rank" version of matrix x,
##' by removing columns (or rows when nrow(x) < ncol(x)), using qr() and it's pivots
fullRank <- function(x, tol = 1e-7, qrx = qr(x, tol=tol)) {
d <- dim(x)
n <- d[[1L]]; p <- d[[2L]]
if(n < p)
return( t(fullRank(t(x), tol=tol)) )
## else n >= p >= rank(.)
rnk <- qrx$rank
if(rnk == p)
x
else
x[, qrx$pivot[seq_len(rnk)], drop=FALSE]
}
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