Looks like the ill-conditioning is almost entirely due to scaling, e.g. > eigen(cov2cor(m)) $values [1] 1.7234899 1.3380701 0.6619299 0.2765101 ...
This is an annoyance in several parts of numerical linear algebra: Routines assume that R^n has all coordinates on a similar scale and therefore think that anything on the order of 1e-7 or so is effectively zero. Condition numbers do this too: > kappa(m) [1] 1.066582e+13 > kappa(cov2cor(m)) [1] 5.489243 -pd On 10 Dec 2015, at 16:41 , Stefano Sofia <stefano.so...@regione.marche.it> wrote: > Dear John, > thank you for your considerations. > This matrix (which is a variance matrix) is part of an algorithm for > forward-filtering and backward-sampling of Dynamic Linear Models (West and > Harrison, 1997), applied to DLM representation of ARIMA processes (Petris, > Petrone, Campagnoli). It is therefore very difficult to explain why this > variance matrix becomes so ill conditioned. This already happens at the first > iteration of the algorithm. I will try to work on initial conditions and some > fixed parameters. > > Thank you again > Stefano > > > ________________________________________ > Da: Fox, John [j...@mcmaster.ca] > Inviato: giovedì 10 dicembre 2015 14.41 > A: Stefano Sofia; r-help@r-project.org > Oggetto: RE: matrix which results singular but at the same time positive > definite > > Dear Stefano, > > You've already had a couple of informative responses directly addressing your > question, but are you aware how ill-conditioned the matrix is (one of the > responses alluded to this)? > >> kappa(X, exact=TRUE) > [1] 7.313338e+12 > >> eigen(X)$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-boun...@r-project.org] On Behalf Of Stefano Sofia >> Sent: December 10, 2015 5:08 AM >> To: r-help@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 >> >> >> ________________________________ >> >> 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. >> >> [[alternative HTML version deleted]] >> >> ______________________________________________ >> R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see >> https://stat.ethz.ch/mailman/listinfo/r-help >> PLEASE do read the posting guide http://www.R-project.org/posting- >> guide.html >> and provide commented, minimal, self-contained, reproducible code. > > ________________________________ > > 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. > > ______________________________________________ > R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. -- Peter Dalgaard, Professor, Center for Statistics, Copenhagen Business School Solbjerg Plads 3, 2000 Frederiksberg, Denmark Phone: (+45)38153501 Office: A 4.23 Email: pd....@cbs.dk Priv: pda...@gmail.com ______________________________________________ R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
