1. Martin Maechler's comments should be taken as replacements
for anything I wrote where appropriate. Any apparent conflict is a
result of his superior knowledge.
2. 'eigen' returns the eigenvalue decomposition assuming the
matrix is symmetric, ignoring anything in m[upper.tri(m)].
3. The basic idea behind both posdefify and nearPD is to compute
the eigenenvalues and vectors, then replace any eigenvalues that are
small or negative with some suitable small positive number and
reconstruct the matrix from this modified eigenvalue decomposition.
posdefify and nearPD implement modifications of this basic idea.
4. I recommend in the summary you mention nearPD but not
posdefify, because nearPD was written more recently using the results of
research not available to the authors when posdefify was written.
MARTIN: There is a typo in the first line of the documentation for
"symmpart". It currently reads, "symmpart(x) computes the
symmetric
part (x + t(x))/2 and the skew symmetric part (x - t(x))/2 of a square
matrix x.". It should read, "symmpart(x) computes the symmetric part
(x
+ t(x))/2 and skewpart the skew symmetric part (x - t(x))/2 of a square
matrix x."
Hope this helps.
Spencer
On 2/4/2011 6:26 AM, Stefano Sofia wrote:> Dear R-users,
> I followed with high interest the thread about positive definite matrix.
> I tracked all the messages of the discussion and I am trying to make a
summary of all the correlated problems that arose from the discussion and the
best solutions to overcome them.
> As far as I understood, the main problems are two: assessing the symmetry
of the given matrix and dealing with eigenvalues very close to zero.
> Do I miss some important points?
>
> The functions that have been mentioned are eigen (I think in particualr the
isSymmetric.matrix function), the function posdefify of the sfmisc package and
the function nearPD of the Matrix package. I believe that some conversations
have not been shared with the mailing list and therefore I find difficult to
trace everything.
>
> I understood very well the summary in four points given by Dr.Spencer
Graves (message 53 of ISSUE 30, VOL 95), and parts of the comments added by
Dr.Martin Maechler (message 71 of the same issue).
> I am not able to understand the improvement given by posdefify with respect
to eigen and why nearPD is even better.
>
> Any final help?
> thank you for your attention
>
> Stefano Sofia PhD
> Weather Department of Civil Protection Marche Region
>
> AVVISO IMPORTANTE: Questo messaggio di posta elettronica pu? contenere
informazioni confidenziali, pertanto ? 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 ?
il destinatario specificato, non leggere, copiare, inoltrare o archiviare questo
messaggio. Se si ? 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 necessit? ed urgenza, la
risposta al presente messaggio di posta elettronica pu? 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.
--
Spencer Graves, PE, PhD
President and Chief Operating Officer
Structure Inspection and Monitoring, Inc.
751 Emerson Ct.
San Jos?, CA 95126
ph: 408-655-4567
1. Martin Maechler's comments should be taken as replacements
for anything I wrote where appropriate. Any apparent conflict is a
result of his superior knowledge.
2. 'eigen' returns the eigenvalue decomposition assuming the
matrix is symmetric, ignoring anything in m[upper.tri(m)].
3. The basic idea behind both posdefify and nearPD is to compute
the eigenenvalues and vectors, then replace any eigenvalues that are
small or negative with some suitable small positive number and
reconstruct the matrix from this modified eigenvalue decomposition.
posdefify and nearPD implement modifications of this basic idea.
4. I recommend in the summary you mention nearPD but not
posdefify, because nearPD was written more recently using the results of
research not available to the authors when posdefify was written.
MARTIN: There is a typo in the first line of the documentation for
"symmpart". It currently reads, "symmpart(x) computes the
symmetric
part (x + t(x))/2 and the skew symmetric part (x - t(x))/2 of a square
matrix x.". It should read, "symmpart(x) computes the symmetric part
(x
+ t(x))/2 and skewpart the skew symmetric part (x - t(x))/2 of a square
matrix x."
Hope this helps.
Spencer
On 2/4/2011 6:26 AM, Stefano Sofia wrote:> Dear R-users,
> I followed with high interest the thread about positive definite matrix.
> I tracked all the messages of the discussion and I am trying to make a
summary of all the correlated problems that arose from the discussion and the
best solutions to overcome them.
> As far as I understood, the main problems are two: assessing the symmetry
of the given matrix and dealing with eigenvalues very close to zero.
> Do I miss some important points?
>
> The functions that have been mentioned are eigen (I think in particualr the
isSymmetric.matrix function), the function posdefify of the sfmisc package and
the function nearPD of the Matrix package. I believe that some conversations
have not been shared with the mailing list and therefore I find difficult to
trace everything.
>
> I understood very well the summary in four points given by Dr.Spencer
Graves (message 53 of ISSUE 30, VOL 95), and parts of the comments added by
Dr.Martin Maechler (message 71 of the same issue).
> I am not able to understand the improvement given by posdefify with respect
to eigen and why nearPD is even better.
>
> Any final help?
> thank you for your attention
>
> Stefano Sofia PhD
> Weather Department of Civil Protection Marche Region
>
> AVVISO IMPORTANTE: Questo messaggio di posta elettronica pu? contenere
informazioni confidenziali, pertanto ? 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 ?
il destinatario specificato, non leggere, copiare, inoltrare o archiviare questo
messaggio. Se si ? 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 necessit? ed urgenza, la
risposta al presente messaggio di posta elettronica pu? 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.
--
Spencer Graves, PE, PhD
President and Chief Operating Officer
Structure Inspection and Monitoring, Inc.
751 Emerson Ct.
San Jos?, CA 95126
ph: 408-655-4567
I'm also not an expert on this topic. I just wanted to list a couple of ways that non-PD matrices might arise. I'll just add now a couple of pointers: First, I believe the term "semipositive definite" is considered ambiguous because in some literature it means that the matrix the smallest eigenvalue is zero and in other literature it means that no eigenvalue is negative (but there might be no zero eigenvalues). I think I might have read about this ambiguity in Searle's excellent "Matrix Algebra Useful for Statistics": http://www.amazon.com/Matrix-Algebra-Useful-Statistics-Probability/dp/0471866814 Second, people in certain stat areas often recommend this chapter by Werner Worthke, formerly of SAS Institute, Inc.: Wothke, Werner (1995), "Nonpositive Definite Matrices in Structural Equation Modeling", in "Testing Structural Equation Models", by Kenneth A. Bollen and J. Scott Long (eds.), Sage Publications, Newbury Park pp. 256-293. http://www.amazon.com/Testing-Structural-Equation-Models-Editions/dp/0803945078 I think you'll find basic definitions and explanations there along with a lot of information about how non-positive definite matrices may arise in real-world applications. Best, Mike On Fri, 4 Feb 2011, Spencer Graves wrote:> 1. Martin Maechler's comments should be taken as replacements for > anything I wrote where appropriate. Any apparent conflict is a result of his > superior knowledge. > > > 2. 'eigen' returns the eigenvalue decomposition assuming the matrix is > symmetric, ignoring anything in m[upper.tri(m)]. > > > 3. The basic idea behind both posdefify and nearPD is to compute the > eigenenvalues and vectors, then replace any eigenvalues that are small or > negative with some suitable small positive number and reconstruct the matrix > from this modified eigenvalue decomposition. posdefify and nearPD implement > modifications of this basic idea. > > > 4. I recommend in the summary you mention nearPD but not posdefify, > because nearPD was written more recently using the results of research not > available to the authors when posdefify was written. > > > MARTIN: There is a typo in the first line of the documentation for > "symmpart". It currently reads, "symmpart(x) computes the symmetric part (x > + t(x))/2 and the skew symmetric part (x - t(x))/2 of a square matrix x.". > It should read, "symmpart(x) computes the symmetric part (x + t(x))/2 and > skewpart the skew symmetric part (x - t(x))/2 of a square matrix x." > > > Hope this helps. > Spencer > > > On 2/4/2011 6:26 AM, Stefano Sofia wrote: >> Dear R-users, >> I followed with high interest the thread about positive definite matrix. >> I tracked all the messages of the discussion and I am trying to make a >> summary of all the correlated problems that arose from the discussion and >> the best solutions to overcome them. >> As far as I understood, the main problems are two: assessing the symmetry >> of the given matrix and dealing with eigenvalues very close to zero. >> Do I miss some important points? >> >> The functions that have been mentioned are eigen (I think in particualr the >> isSymmetric.matrix function), the function posdefify of the sfmisc package >> and the function nearPD of the Matrix package. I believe that some >> conversations have not been shared with the mailing list and therefore I >> find difficult to trace everything. >> >> I understood very well the summary in four points given by Dr.Spencer >> Graves (message 53 of ISSUE 30, VOL 95), and parts of the comments added by >> Dr.Martin Maechler (message 71 of the same issue). >> I am not able to understand the improvement given by posdefify with respect >> to eigen and why nearPD is even better. >> >> Any final help? >> thank you for your attention >> >> Stefano Sofia PhD >> Weather Department of Civil Protection Marche Region >> >> AVVISO IMPORTANTE: Questo messaggio di posta elettronica pu? contenere >> informazioni confidenziali, pertanto ? 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 ? il destinatario specificato, non leggere, copiare, inoltrare o >> archiviare questo messaggio. Se si ? 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 necessit? ed urgenza, la risposta al presente messaggio di posta >> elettronica pu? 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. > > > -- > Spencer Graves, PE, PhD > President and Chief Operating Officer > Structure Inspection and Monitoring, Inc. > 751 Emerson Ct. > San Jos?, CA 95126 > ph: 408-655-4567 >
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