[RsR] Singular covariance in plot.lmrob

[Christian Hennig]

Hi Matias,

yes, I remember that something like this was discussed in Banff.
Actually I think that there are two different issues. One is with dummy/binary 
variables, which doesn't work well with a 50% breakdown covariance estimator 
generally (as I said, I think it may be an idea to offer to tune the breakdown 
point down as a plot.lmrob parameter... if 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%).
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.
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 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 using stats.ucl.ac.uk, www.homepages.ucl.ac.uk/~ucakche
>> 
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*** --- ***
Christian Hennig
University College London, Department of Statistical Science
Gower St., London WC1E 6BT, phone +44 207 679 1698
chrish using stats.ucl.ac.uk, www.homepages.ucl.ac.uk/~ucakche