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

Thu Dec 10 17:24:20 CET 2015

Dear Peter,

> -----Original Message-----
> From: peter dalgaard [mailto:pdalgd at gmail.com]
> Sent: Thursday, December 10, 2015 11:09 AM
> To: Stefano Sofia
> Cc: Fox, John; r-help at r-project.org
> Subject: Re: [R] matrix which results singular but at the same time
> positive definite
> 
> Looks like the ill-conditioning is almost entirely due to scaling, e.g.

Yes, that's my point. Sorry I didn't make it clearer.

Best,
 John

> 
> > eigen(cov2cor(m))
> $values
> [1] 1.7234899 1.3380701 0.6619299 0.2765101
> ...
> 
> This is an annoyance in several parts of numerical linear algebra:
> Routines assume that R^n has all coordinates on a similar scale and
> therefore think that anything on the order of 1e-7 or so is effectively
> zero.
> 
> Condition numbers do this too:
> 
> > kappa(m)
> [1] 1.066582e+13
> > kappa(cov2cor(m))
> [1] 5.489243
> 
> 
> -pd
> 
> On 10 Dec 2015, at 16:41 , Stefano Sofia
> <stefano.sofia at regione.marche.it> wrote:
> 
> > Dear John,
> > thank you for your considerations.
> > This matrix (which is a variance matrix) is part of an algorithm for
> forward-filtering and backward-sampling of Dynamic Linear Models (West
> and Harrison, 1997), applied to DLM representation of ARIMA processes
> (Petris, Petrone, Campagnoli).  It is therefore very difficult to
> explain why this variance matrix becomes so ill conditioned. This
> already happens at the first iteration of the algorithm. I will try to
> work on initial conditions and some fixed parameters.
> >
> > Thank you again
> > Stefano
> >
> >
> > ________________________________________
> > Da: Fox, John [jfox at mcmaster.ca]
> > Inviato: giovedì 10 dicembre 2015 14.41
> > A: Stefano Sofia; r-help at r-project.org
> > Oggetto: RE: matrix which results singular but at the same time
> positive        definite
> >
> > Dear Stefano,
> >
> > You've already had a couple of informative responses directly
> addressing your question, but are you aware how ill-conditioned the
> matrix is (one of the responses alluded to this)?
> >
> >> kappa(X, exact=TRUE)
> > [1] 7.313338e+12
> >
> >> eigen(X)$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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> > ______________________________________________
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> 
> --
> Peter Dalgaard, Professor,
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Office: A 4.23
> Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com
> 
> 
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