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

peter dalgaard pdalgd at gmail.com
Thu Dec 10 17:09:12 CET 2015 

Looks like the ill-conditioning is almost entirely due to scaling, e.g.

> 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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> ________________________________
> 
> AVVISO IMPORTANTE: Questo messaggio di posta elettronica può contenere informazioni confidenziali, pertanto è destinato solo a persone autorizzate alla ricezione. I messaggi di posta elettronica per i client di Regione Marche possono contenere informazioni confidenziali e con privilegi legali. Se non si è il destinatario specificato, non leggere, copiare, inoltrare o archiviare questo messaggio. Se si è ricevuto questo messaggio per errore, inoltrarlo al mittente ed eliminarlo completamente dal sistema del proprio computer. Ai sensi dell’art. 6 della DGR n. 1394/2008 si segnala che, in caso di necessità ed urgenza, la risposta al presente messaggio di posta elettronica può essere visionata da persone estranee al destinatario.
> IMPORTANT NOTICE: This e-mail message is intended to be received only by persons entitled to receive the confidential information it may contain. E-mail messages to clients of Regione Marche may contain information that is confidential and legally privileged. Please do not read, copy, forward, or store this message unless you are an intended recipient of it. If you have received this message in error, please forward it to the sender and delete it completely from your computer system.
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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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