[R] [SPAM] - constructing arbitrary (positive definite) covariance matrix - Found word(s) list error in the Text body

[Moshe Olshansky]

If the main diagonal element of matrix A is 1 and the off diagonal element is a then for any vector x we get that t(x)*A*x = (1-a)*sum(x^2) +a*(sum(x))^2 . If we want A to be positive (semi)definite we need this expression to be positive (non-negative) for any x!= 0. Since sum(x)^2/sum(x*2) <= n where n is the dimension of the matrix and equality is possible we get that A is positive (semi)definite if and only if -1/(n-1) <= a <= 1 (sharp inequalities for positive definiteness).
Since any symmetric (semi)positive definite matrix can be a covariance matrix this describes all the matrices which satisfy the requirement.


--- On Fri, 27/6/08, Patrick Burns <pburns at pburns.seanet.com> wrote:

> From: Patrick Burns <pburns at pburns.seanet.com>
> Subject: Re: [R] [SPAM] - constructing arbitrary (positive definite) covariance matrix - Found word(s) list error in the Text body
> To: davidr at rhotrading.com
> Cc: "Mizanur Khondoker" <Mizanur.Khondoker at ed.ac.uk>, r-help at r-project.org
> Received: Friday, 27 June, 2008, 3:15 AM
> To make David's approach a little more concrete:
> You can always have correlations all equal to 1 --
> the variables are all the same, except for the names
> you've given them.  You can have two variables
> with correlation -1, but you can't get a third variable
> that has -1 correlation to both of the first two.
> 
> 
> Patrick Burns
> patrick at burns-stat.com
> +44 (0)20 8525 0696
> http://www.burns-stat.com
> (home of S Poetry and "A Guide for the Unwilling S
> User")
> 
> davidr at rhotrading.com wrote:
> > Well, if you think about the geometry, all
> correlations equal usually
> > won't work. Think of the SDs as the sides of a
> simplex and the
> > correlations as the cosines of the angles between the
> sides (pick one
> > variable as the 'origin'.) Only certain values
> will give a valid
> > covariance or correlation matrix.
> > HTH,
> > David L. Reiner, PhD
> > Head Quant
> > Rho Trading Securities, LLC
> > -----Original Message-----
> > From: r-help-bounces at r-project.org
> [mailto:r-help-bounces at r-project.org]
> > On Behalf Of Mizanur Khondoker
> > Sent: Thursday, June 26, 2008 11:11 AM
> > To: r-help at r-project.org
> > Subject: [SPAM] - [R] constructing arbitrary (positive
> definite)
> > covariance matrix - Found word(s) list error in the
> Text body
> >
> > Dear list,
> >
> > I am trying to use the  'mvrnorm'  function
> from the MASS package for
> > simulating multivariate Gaussian data with given
> covariance matrix.
> > The diagonal elements of my covariance matrix should
> be the same,
> > i.e., all variables have the same marginal variance.
> Also all
> > correlations between all pair of variables should be
> identical, but
> > could be any value in [-1,1]. The problem I am having
> is that the
> > matrix I create is not always positive definite (and
> hence mvrnorm
> > fails).
> >
> > Is there any simple way of constructing covariance
> matrix of the above
> > structure (equal variance, same pairwise correlation
> from [-1, 1])
> > that will always be positive definite?
> > I have noticed that covraince matrices created using
> the following COV
> > function are positive definite for  -0.5 < r <1.
> However, for  r <
> > -0.5, the matrix is not positive definite.
> > Does anyone have any idea why this is the case?  For
> my simualtion, I
> > need to generate multivariate data for the whole range
> of r,  [-1, 1]
> > for a give value of sd.
> >
> > Any help/ suggestion would be greatly appreciated.
> >
> > Examples
> > ########
> > COV<-function (p = 3, sd = 1, r= 0.5){
> >     cov <- diag(sd^2, ncol=p, nrow=p)
> >     for (i in 1:p) {
> >         for (j in 1:p) {
> >             if (i != j) {
> >                 cov[i, j] <- r * sd*sd
> >             }
> >         }
> >     }
> >    cov
> > }
> >
> >   
> >> library(MASS)
> >> ### Simualte multivarite gaussin data (works OK)
> >> Sigma<-COV(p = 3, sd = 2, r= 0.5)
> >> mu<-1:3
> >> mvrnorm(5, mu=mu, Sigma=Sigma)
> >>     
> >           [,1]     [,2]     [,3]
> > [1,] 1.2979984 1.843248 4.460891
> > [2,] 2.1061054 1.457201 3.774833
> > [3,] 2.1578538 2.761939 4.589977
> > [4,] 0.8775056 4.240710 2.203712
> > [5,] 0.2698180 2.075759 2.869573
> >   
> >> ### Simualte multivarite gaussin data ( gives
> Error)
> >> Sigma<-COV(p = 3, sd = 2, r= -0.6)
> >> mu<-1:3
> >> mvrnorm(5, mu=mu, Sigma=Sigma)
> >>     
> > Error in mvrnorm(5, mu = mu, Sigma = Sigma) :
> >   'Sigma' is not positive definite
> >
> >
> 
> ______________________________________________
> R-help at r-project.org mailing list
> 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.