[RsR] Singular covariance in plot.lmrob

[Matias Salibian-Barrera]

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 using stats.ucl.ac.uk, www.homepages.ucl.ac.uk/~ucakche
>>>
>>> _______________________________________________
>>> R-SIG-Robust using r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-robust
>>
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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