[R] always about positive definite matrix

[Mike Miller]

(Apologies to the cc-list: I'm resending from a different address because 
I didn't realize it was going to r-help.)

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
>> 
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>
> -- 
> Spencer Graves, PE, PhD
> President and Chief Operating Officer
> Structure Inspection and Monitoring, Inc.
> 751 Emerson Ct.
> San José, CA 95126
> ph:  408-655-4567
>