[R] matrix which results singular but at the same time positive definite

[Fox, John]

Dear Stefano,

I don't really know anything about your application, but my point is about the scaling of the variables. Can't you rescale some or all of the variables so their variances aren't so different? For example, the correlation matrix among these four variables isn't ill-conditioned.

Best,
 John

> -----Original Message-----
> From: Stefano Sofia [mailto:stefano.sofia at regione.marche.it]
> Sent: Thursday, December 10, 2015 10:42 AM
> To: Fox, John; r-help at r-project.org
> Subject: RE: matrix which results singular but at the same time positive
> definite
> 
> Dear John,
> thank you for your considerations.
> This matrix (which is a variance matrix) is part of an algorithm for
> forward-filtering and backward-sampling of Dynamic Linear Models (West
> and Harrison, 1997), applied to DLM representation of ARIMA processes
> (Petris, Petrone, Campagnoli).  It is therefore very difficult to
> explain why this variance matrix becomes so ill conditioned. This
> already happens at the first iteration of the algorithm. I will try to
> work on initial conditions and some fixed parameters.
> 
> Thank you again
> Stefano
> 
> 
> ________________________________________
> Da: Fox, John [jfox at mcmaster.ca]
> Inviato: giovedì 10 dicembre 2015 14.41
> A: Stefano Sofia; r-help at r-project.org
> Oggetto: RE: matrix which results singular but at the same time positive
> definite
> 
> Dear Stefano,
> 
> You've already had a couple of informative responses directly addressing
> your question, but are you aware how ill-conditioned the matrix is (one
> of the responses alluded to this)?
> 
> > kappa(X, exact=TRUE)
> [1] 7.313338e+12
> 
> > eigen(X)$values
> [1] 4.964711e+00 9.356881e-01 4.863392e-12 6.788344e-13
> 
> Two of the variables have variances around 10^0 and the other two around
> 10^-12. Of course, you haven't said anything about the context, but does
> it really make sense to analyze the data on these scales?
> 
> Best,
>  John
> 
> -----------------------------
> John Fox, Professor
> McMaster University
> Hamilton, Ontario
> Canada L8S 4M4
> Web: socserv.mcmaster.ca/jfox
> 
> 
> 
> 
> > -----Original Message-----
> > From: R-help [mailto:r-help-bounces at r-project.org] On Behalf Of
> Stefano Sofia
> > Sent: December 10, 2015 5:08 AM
> > To: r-help at r-project.org
> > Subject: [R] matrix which results singular but at the same time
> positive definite
> >
> > Dear list users,
> > through the "matrixcalc" package I am performing some checks of
> variance
> > matrices (which must be positive definite).
> > In this example, it happens that the matrix A here reported is
> singular but
> > positive definite. Is it possible?
> >
> >               [,1]          [,2]          [,3]          [,4]
> > [1,]  1.904255e-12 -1.904255e-12 -8.238960e-13 -1.240294e-12 [2,] -
> > 1.904255e-12  3.637979e-12  1.364242e-12  1.818989e-12 [3,] -
> 8.238960e-13
> > 1.364242e-12  4.809988e+00  7.742369e-01 [4,] -1.240294e-12
> 1.818989e-12
> > 7.742369e-01  1.090411e+00
> >
> > print(is.non.singular.matrix(A, tol = 1e-18)) FALSE
> print(is.positive.definite(A,
> > tol=1e-18)) TRUE
> >
> > Is there something wrong with this matrix?
> > Any comment will be appreciated.
> > Stefano
> >
> >
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