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I have an issue which I could not solve, although I tried and I got some help on R help forums too. I am trying to calculate Mahalanobis distances on a data frame, where I have several hundreds of groups and several hundreds of variables. Whatever I do, however I subset it I get the "system is computationally singular: reciprocal condition number" error.

It is clear that it is singular, but is there any way to get rid of it and run Maha? Should I forget Maha? Than what to use?

I have uploaded the data file to my ftp: http://mkk.szie.hu/dep/talt/lv/CentInpDuplNoHeader.txt It is a tab delimited txt file with no headers.

I was working with the (R) StatMatch Mahalanobis (also tried stats Mahalanobis) function. I have a deadline for this project (not a homework:)), and I could always use this funct, so I thought I will be able to quit the calculations short, but now I am just lost. link to previous question, sorry for crossposting, I have no idea how to migrate the previous one. http://stackoverflow.com/questions/13078909/system-is-computationally-singular-reciprocal-condition-number I would really appreciate any help.

Thanks for any help

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it's essentially a duplicate of this question where the OP wants to compute the Mahalanobis distance of a dataset with singular covariance. –  user603 Jul 28 '13 at 7:32

2 Answers 2

Why do you think there is no way that matrix could be singular?

A QR decomposition shows that the rank of this 380 x 372 matrix is just 300. In other words, it is highly singular:

url <- "http://mkk.szie.hu/dep/talt/lv/CentInpDuplNoHeader.txt"
df <- read.table(file = url, header = FALSE)
m <- as.matrix(df)

dim(m)
# [1] 380 372
qr(m)$rank
# [1] 300

Examining the matrix's singular values is another way to see the same thing:

head(table(svd(df)$d))

# 5.76661502353373e-13 2.57650568058543e-12  0.00929562094651422 
#                   71                    1                    1 
#   0.0277990885015625   0.0398152894712022   0.0469713341003743 
#                    1                    1                    1 
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Thanks @Josh for the fast reply. This is a dataset, for different physical and chemical properties of different geological layers and I would not expect it to be highly singular. Whatever, I also calculated svd and You are absolutely right. Anyhow I should still calculate de Maha distance. Is there any idea or solution how to do that if i have such a highly singular datasset? In my previous studies it never happened, although those were smaller and more specified ones. Thanks –  user1775772 Oct 26 '12 at 0:14
    
FYI As someone suggested, I added a small value (between 0.0001-0.0009) to every datavalue. It helped to get rid of the singularity. Thanks for your help –  user1775772 Oct 26 '12 at 13:08

A singular matrix means that some of the vectors are linear combinations of others. Thus, some vectors do not add any useful information to the Mahalanobis distance calculation. A generalized inverse or pseudoinverse effectively calculates an "inverse-like" matrix that ignores some of this noninformative information. This is superior to other methods that effectively add in a small amount of incorrect information (i.e. add a small constant to all data). Pseudoinverse covariance matrices have been used successfully with the Mahalanobis distance, see http://www.sciencedirect.com/science/article/pii/0146664X79900522.

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