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Mizanur Khondoker

2008-Jun-26 16:11 UTC

[R] constructing arbitrary (positive definite) covariance matrix

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

-- 
Mizanur Khondoker
Division of Pathway Medicine (DPM)
The University of Edinburgh Medical School
The Chancellor's Building
49 Little France Crescent
Edinburgh EH16 4SB
United Kingdom

Tel: +44 (0) 131 242 6287
Fax: +44 (0) 131 242 6244
http://www.pathwaymedicine.ed.ac.uk/

davidr at rhotrading.com

2008-Jun-26 17:09 UTC

head link

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

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

-- 
Mizanur Khondoker
Division of Pathway Medicine (DPM)
The University of Edinburgh Medical School
The Chancellor's Building
49 Little France Crescent
Edinburgh EH16 4SB
United Kingdom

Tel: +44 (0) 131 242 6287
Fax: +44 (0) 131 242 6244
http://www.pathwaymedicine.ed.ac.uk/

______________________________________________
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.

Gabor Grothendieck

2008-Jun-26 22:48 UTC

head link

[R] constructing arbitrary (positive definite) covariance matrix

Not sure if this is sufficient but nearcor in the sfsmisc package
will find the nearest correlation matrix to a given matrix.

On Thu, Jun 26, 2008 at 12:11 PM, Mizanur Khondoker
<Mizanur.Khondoker at ed.ac.uk> wrote:> 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
>
> --
> Mizanur Khondoker
> Division of Pathway Medicine (DPM)
> The University of Edinburgh Medical School
> The Chancellor's Building
> 49 Little France Crescent
> Edinburgh EH16 4SB
> United Kingdom
>
> Tel: +44 (0) 131 242 6287
> Fax: +44 (0) 131 242 6244
> http://www.pathwaymedicine.ed.ac.uk/
>
> ______________________________________________
> 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.
>

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