R solve:system is exactly singular

Question

I am solving simple optimization problem. The data set has 26 columns and over 3000 rows. The source code looks like

Means   <- colMeans(Returns)
Sigma   <- cov(Returns)
invSigma1 <- solve(Sigma)

And everything works perfect- but then I want to do the same for shorter period (only 261 rows) and the solve function writes the following error:

solve(Sigma)
Error in solve.default(Sigma) : 
  Lapack routine dgesv: system is exactly singular 

Its weird because when I do the same with some random numbers:

Returns<-matrix(runif(6786,-1,1), nrow=261)
Means   <- colMeans(Returns)
Sigma   <- cov(Returns)
invSigma <- solve(Sigma)

no error occurs at all. Could someone explain me where could be the problem and how to treat it. Thank you very much, Alex

Jezabelle

Answer 1

Using solve with a single parameter is a request to invert a matrix. The error message is telling you that your matrix is singular and cannot be inverted.

Rexer

The following page, near the bottom, helped me understand the concepts of inversion and singularity issue: link – Sixpence

Answer 2

I guess your code uses somewhere in the second case a singular matrix (i.e. not invertible), and the solve function needs to invert it. This has nothing to do with the size but with the fact that some of your vectors are (probably) colinear.

Bois

The data set contains daily returns of 26 asset returns, it should be invertible, no? I am confused why there is no problem when I have the whole data set while shorten produces error. Any ideas how to deal with it? – Jezabelle

@Jezabelle Check det(Sigma): If it is zero then it is impossible to invert. – Incompetent

Indeed, it is 0. What is the problem when I shorten the data set? – Jezabelle

The only thing that I dont get is why there is a problem with smaller sample size, respectively how to deal with it cos I need the invert it – Jezabelle

@Jezabelle Either you have just been unlucky with the numbers, or you have reduced the data too much to be able to ensure linear independence. The following links may be of use: m-hikari.com/imf-2010/45-48-2010/kaimakamisIMF45-48-2010.pdf arxiv.org/abs/1010.0601 – Incompetent

Answer 3

Lapack is a Linear Algebra package which is used by R (actually it's used everywhere) underneath solve(), dgesv spits this kind of error when the matrix you passed as a parameter is singular.

As an addendum: dgesv performs LU decomposition, which, when using your matrix, forces a division by 0, since this is ill-defined, it throws this error. This only happens when matrix is singular or when it's singular on your machine (due to approximation you can have a really small number be considered 0)

I'd suggest you check its determinant if the matrix you're using contains mostly integers and is not big. If it's big, then take a look at this link.

Choreodrama

Answer 4

I can understand your question. The problem is that your matrix is perpendicular. You can see your first number and the last number of your matrix is same.

Yezd