R help - Sep 2007 - generate fourth vector based on known correlations

I am trying to generate a fourth vector,z, given three known and fixed 
vectors, x1,x2,x3 with corresponding known and fixed correlations with 
themeselves and with z. That is, all correlations are known and 
prespecified. How can I do this?
Thank you,
ben

I am trying to generate a fourth vector,z, given three known and fixed 
vectors, x1,x2,x3with corresponding known and fixed correlations with 
themeselves and with z. That is, all correlations are known and 
prespecified. How can I do this?
Thank you,
ben

Here is one approach (someone with better linear algebra skills may be
able to shorten this, some steps could be combined, but I wanted to show
each step):


# create x1-x3

x1 <- rnorm(100, 50, 3)
x2 <- rnorm(100) + x1/5
x3 <- rnorm(100) + x2/5

# find current correlations

cor1 <- cor( cbind(x1,x2,x3) )
cor1

# create 1st version of z

z <- rnorm(100)

# combine in a matrix

m1 <- cbind( x1, x2, x3, z )

# center and scale

m2 <- scale(m1)

# find cholesky decomp

c1 <- chol(var(m2))

# force to be independent

m3 <- m2 %*% solve(c1)

zapsmall(cor(m3)) # check

# create new correlation matrix:

cor2 <- cbind( rbind( cor1, z=c(.5,.3,.1) ), z=c(.5,.3,.1,1) )

# create new matrix

m4 <- m3 %*% chol(cor2)

# uncenter and unscale

m5 <- sweep( m4, 2, attr(m2, 'scaled:scale'), '*')
m5 <- sweep( m5, 2, attr(m2, 'scaled:center'), '+')

# check to see if we succeeded

all.equal( cbind(x1,x2,x3), m5[, 1:3] )
all.equal( cor(m5)[1:3,1:3], cor1 )
all.equal( cor(m5), cor2 )

cor(m5)

 
Hope this helps,


-- 
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.snow at intermountainmail.org
(801) 408-8111
 
 
> -----Original Message-----
> From: r-help-bounces at r-project.org 
> [mailto:r-help-bounces at r-project.org] On Behalf Of bkelcey at umich.edu
> Sent: Wednesday, September 26, 2007 2:25 PM
> To: r-help at stat.math.ethz.ch
> Subject: [R] generate fourth vector based on known correlations
> 
> 
> 
> I am trying to generate a fourth vector,z, given three known 
> and fixed vectors, x1,x2,x3 with corresponding known and 
> fixed correlations with themeselves and with z. That is, all 
> correlations are known and prespecified. How can I do this?
> Thank you,
> ben
> 
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