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.
