Multivariate Barrow Wheel Distribution Random Vectors
Generate p-dimensional random vectors according to Stahel's Barrow Wheel Distribution.
rbwheel(n, p, frac = 1/p, sig1 = 0.05, sig2 = 1/10,
rGood = rnorm,
rOut = function(n) sqrt(rchisq(n, p - 1)) * sign(runif(n, -1, 1)),
U1 = rep(1, p),
scaleAfter = TRUE, scaleBefore = FALSE, spherize = FALSE,
fullResult = FALSE)n |
integer, specifying the sample size. |
p |
integer, specifying the dimension (aka number of variables). |
frac |
numeric, the proportion of outliers. The default, 1/p, corresponds to the (asymptotic) breakdown point of M-estimators. |
sig1 |
thickness of the “wheel”, (= σ
|
sig2 |
thickness of the “axis” (compared to 1). |
rGood |
function; the generator for “good” observations. |
rOut |
function, generating the outlier observations. |
U1 |
p-vector to which (1,0,…,0) is rotated. |
scaleAfter |
logical indicating if the matrix is re-scaled after
rotation (via |
scaleBefore |
logical indicating if the matrix is re-scaled before
rotation (via |
spherize |
logical indicating if the matrix is to be “spherized”, i.e., rotated and scaled to have empirical covariance I_p. This means that the principal components are used (before rotation). |
fullResult |
logical indicating if in addition to the n x p matrix, some intermediate quantities are returned as well. |
....
By default (when fullResult is FALSE), an
n x p matrix of n sample vectors of the
p dimensional barrow wheel distribution, with an attribute,
n1 specifying the exact number of “good” observations,
n1 ~= (1-f)*n, f = frac.
If fullResult is TRUE, a list with components
X |
the n x p matrix of above,
|
X0 |
......... |
A |
the p x p rotation matrix, see above. |
n1 |
the number of “good” observations, see above. |
n2 |
the number of “outlying” observations, n2 = n - n1. |
Werner Stahel and Martin Maechler
Stahel, W.~A. and Mächler, M. (2009). Comment on “invariant co-ordinate selection”, Journal of the Royal Statistical Society B 71, 584–586.
set.seed(17)
rX8 <- rbwheel(1000,8, fullResult = TRUE, scaleAfter=FALSE)
with(rX8, stopifnot(all.equal(X, X0 %*% A, tol = 1e-15),
all.equal(X0, X %*% t(A), tol = 1e-15)))
##--> here, don't need to keep X0 (nor A, since that is Qrot(p))
## for n = 100, you don't see "it", but may guess .. :
n <- 100
pairs(r <- rbwheel(n,6))
n1 <- attr(r,"n1") ; pairs(r, col=1+((1:n) > n1))
## for n = 500, you *do* see it :
n <- 500
pairs(r <- rbwheel(n,6))
## show explicitly
n1 <- attr(r,"n1") ; pairs(r, col=1+((1:n) > n1))
## but increasing sig2 does help:
pairs(r <- rbwheel(n,6, sig2 = .2))
## show explicitly
n1 <- attr(r,"n1") ; pairs(r, col=1+((1:n) > n1))
set.seed(12)
pairs(X <- rbwheel(n, 7, spherize=TRUE))
colSums(X) # already centered
if(require("ICS") && require("robustbase")) {
# ICS: Compare M-estimate [Max.Lik. of t_{df = 2}] with high-breakdown :
stopifnot(require("MASS"))
X.paM <- ics(X, S1 = cov, S2 = function(.) cov.trob(., nu=2)$cov, stdKurt = FALSE)
X.paM.<- ics(X, S1 = cov, S2 = function(.) tM(., df=2)$V, stdKurt = FALSE)
X.paR <- ics(X, S1 = cov, S2 = function(.) covMcd(.)$cov, stdKurt = FALSE)
plot(X.paM) # not at all clear
plot(X.paM.)# ditto
plot(X.paR)# very clear
}
## Similar such experiments ---> demo(rbwheel_d) and demo(rbwheel_ics)
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