Good evening Christian,
yes, I remember that something like this was discussed in Banff.
Actually I think that there are two different issues. (snip)
Yes, I agree.
there is a binary variable with 80% of the data on one value, then I'd
rather go for a 10% breakdown point
and not 50%).
This will indeed prevent you from finding a singular covariance matrix. But
I would go a bit further and question what information (regarding leverage)
can be expected to be gained by keeping this variable in the design matrix.
I am inclined to remove categorical variables from the design matrix when
looking for high-leverage points. Having said this, however, I would like to
hear other opinions.
But this is not the issue with the dataset cited below, where all
variables are quantitative but two of them are discret with a much lower
number of distinct values than observations so that there may be many, but
much fewer than 50% observations with the same x-value.
The problem with such data is not with the definition, but with the
algorithm.
I agree as well. And will wait for the experts' comments on this.
It should be pretty straightforward to prevent plot.lmrob from producing
an error in this case with the try command and just make it giving out the
other plots and a warning. However, I don't know whether there is a
Yes, I hope I find time for this in the coming weeks after our teaching
Term ends. This will also be partially solved if we do not use binary
variables (particularly when they result from coding categorical
covariates).
Matias
reliable method to construct a better initialization of the covariance
estimator.
Christian
On Fri, 28 Nov 2008, Matias Salibian-Barrera wrote:
Thanks Christian for reminding me of this issue. It was discussed in
Banff last year, and it may in principle happen any time you have a
categorical explanatory variable in your model, as the design matrix becomes
sparse and sub-sampling search algorithms tend to produce too many singular
subsamples of size p+1.
I am not sure that this can be fixed by lowering the BP in the current
MCD algorithm. Note how your example fails with a message that 14 (out of
392) obs. are on a lower-dimensional hyperplane. Shouldn't we be considering
samples of size ~ 200? I believe this error message may be more related to
the random subsampling search than the BP of the target estimator. Maybe
Valentin can help me understand what is happening here.
For the linear regression case, I would argue the following: since
Mahalanobis distances can be hard to interpret for categorical variables,
one possibility would be to simply remove these "factor" variables when
calculating the distances for the plot. Sometimes, however, the user may
have already "coded" the factors into rows of 0's and 1's (instead of using
proper factor variables in the formula), which would be a more difficult
case to protect against.
For the more general multivariate location/scatter problem, I believe the
default "failing" behaviour of the MCD algorithm may need to be revisited,
since, as you mention, one may still want to get a (singular) covariance
matrix estimator when half the data are lying on a lower-dimensional
hyperplane. While we've had this conversation in the past, we never reached
much of an consensus. Maybe it is time to try again.
Matias
Christian Hennig wrote:
Dear list,
I have come across several situations in which the robust Mahalanobis
distance vs. residuals plot, the first default plot in plot.lmrob, gave an
error like this:
# recomputing robust Mahalanobis distances
# The covariance matrix has become singular during
# the iterations of the MCD algorithm.
# There are 14 observations (in the entire dataset of 392 obs.) lying on
# the hyperplane with equation a_1*(x_i1 - m_1) + ... + a_p*(x_ip - m_p)
# = 0 with (m_1,...,m_p) the mean of these observations and coefficients
# a_i from the vector a <- c(-0.0102123, 0, 0, 0, 0, -0.9999479)
# Error in solve.default(cov, ...) :
# system is computationally singular: reciprocal condition number =
2.33304e-3
This particular error has been produced with the Auto-mpg dataset from
http://archive.ics.uci.edu/ml/datasets.html
autod <- read.table("auto-mpg.data",col.names=c("mpg","cylinders",
"displacement","horsepower","weight","acceleration",
"modelyear","origin","carname"),na.strings="?")
autoc <- autod[complete.cases(autod),]
auto17 <- autoc[,1:7]
rautolm <-
lmrob(mpg~cylinders+displacement+horsepower+weight+acceleration+
modelyear,data=auto17)
plot(rautolm)
(I don't claim that this is the most reasonable thing to do with these
data because of nonlinearity, anyway...)
This problem happens easily if at least one of the variables is discrete
and there are several observations with the same value.
Such a situation is by no means atypical and therefore I think that it's
worthwhile that something is done about this, for example checking
singularity
internally and in that case trying a different initial sample. It may
also make sense to give the option that the robust covariance matrix is
tuned down to 25% breakdown, say, because one may still want to see a bit if
half of the data lie on a lower dimensional hyperplane (in case of a binary
x-variable) but regression still makes sense.
Best regards,
Christian
*** --- ***
Christian Hennig
University College London, Department of Statistical Science
Gower St., London WC1E 6BT, phone +44 207 679 1698
chrish at stats.ucl.ac.uk, www.homepages.ucl.ac.uk/~ucakche<http://www.homepages.ucl.ac.uk/%7Eucakche>