Re: [R] [SPAM] - constructing arbitrary (positive definite) covariance matrix - Found word(s) list error in the Text body
Thanks to everyone who responded to my email.
Moshe's email explains clearly why my matrices were not positive
definite for certain negative correlations.
I now have better understanding of the problem.
Thanks
Mizan
2008/6/27 Moshe Olshansky [EMAIL PROTECTED]:
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 [EMAIL PROTECTED] wrote:
From: Patrick Burns [EMAIL PROTECTED]
Subject: Re: [R] [SPAM] - constructing arbitrary (positive definite)
covariance matrix - Found word(s) list error in the Text body
To: [EMAIL PROTECTED]
Cc: Mizanur Khondoker [EMAIL PROTECTED], r-help@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
[EMAIL PROTECTED]
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and A Guide for the Unwilling S
User)
[EMAIL PROTECTED] 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: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED]
On Behalf Of Mizanur Khondoker
Sent: Thursday, June 26, 2008 11:11 AM
To: r-help@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@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.
--
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
Re: [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: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
On Behalf Of Mizanur Khondoker
Sent: Thursday, June 26, 2008 11:11 AM
To: r-help@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@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.
__
R-help@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.
Re: [R] [SPAM] - constructing arbitrary (positive definite) covariance matrix - Found word(s) list error in the Text body
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
[EMAIL PROTECTED]
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and A Guide for the Unwilling S User)
[EMAIL PROTECTED] 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: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
On Behalf Of Mizanur Khondoker
Sent: Thursday, June 26, 2008 11:11 AM
To: r-help@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@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.
Re: [R] [SPAM] - constructing arbitrary (positive definite) covariance matrix - Found word(s) list error in the Text body
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 [EMAIL PROTECTED] wrote:
From: Patrick Burns [EMAIL PROTECTED]
Subject: Re: [R] [SPAM] - constructing arbitrary (positive definite)
covariance matrix - Found word(s) list error in the Text body
To: [EMAIL PROTECTED]
Cc: Mizanur Khondoker [EMAIL PROTECTED], r-help@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
[EMAIL PROTECTED]
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and A Guide for the Unwilling S
User)
[EMAIL PROTECTED] 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: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED]
On Behalf Of Mizanur Khondoker
Sent: Thursday, June 26, 2008 11:11 AM
To: r-help@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@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.
__
R-help@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.
