Erroneous zeros in results using Matrix package

6 messages · Roger Koenker, Simon Urbanek, Brian Ripley +1 more

Original

1

6

Tue, Oct 15, 2013 11:26 AM #

You would be much more likely to get a response to this if you had a _smaller_ reproducible example.

url:    www.econ.uiuc.edu/~roger            Roger Koenker
email    rkoenker at uiuc.edu            Department of Economics
vox:     217-333-4558                University of Illinois
fax:       217-244-6678                Urbana, IL 61801

On Oct 15, 2013, at 11:36 AM, Braun, Michael wrote:

I've come across some interesting behavior using the Matrix package, and I cannot seem to explain it.  I have sent this issue to the Matrix maintainers, and Martin Maechler was unable to replicate it on his Linux system.  Since it might be something Mac-specific, he suggested that I post here.  The complete code is pasted at the bottom of this email, along with my sessionInfo() output.

The function f takes 3 parameters:  p, k and N.  First, it creates A,  a lower triangle of a sparse block diagonal matrix with p blocks, and each block being k x k.  I then let S = AA', and have L be the lower Cholesky decomposition of L. (I use Cholesky instead of chol because, for my "real" application, because I need the permutation matrix also).  I then multiply L by a matrix z, where each column in z is a standard MVN sample.

None of the elements should be zero.  However, if z becomes very large, I get lots of zero entries.

Additionally, you will see that when I try to coerce a large identity matrix to a dtCMatrix, some of the ones are lost. 

A word of warning:  this script does consume a lot of RAM, and it will take several minutes to run.  But it would be helpful if someone with a Mac Pro with 32GB of RAM could try to replicate the problem.

Thanks in advance for your help.

Michael


library(Matrix)

f <- function(p,k,N) {
  A <- tril(kronecker(Diagonal(p),Matrix(1:(k^2),k,k)))
  S <- tcrossprod(A)
  CH <- Cholesky(S)
  L <- expand(CH)$L
  z <- rnorm(N*k*p)
  dim(z) <- c(p*k,N)
  y <- L %*% z
  return(sum(y==0))
}

system.time(small   <- f(p=  100, k=3, N=  200) )
system.time(medium  <- f(p=  200, k=3, N= 5000) )

print(small ) ## 0
print(medium)  ## 0

print( system.time(large.p <- f(p=50000, k=3, N= 5000) ))
print( system.time(large.N <- f(p=10000, k=3, N=20000) ))

print(large.p) ## I get 536870912, but it should be 0
print(large.N) ## I get 536870912, but it should be 0

system.time( W1 <- as(diag(10000),"dtCMatrix") )
print(sum(diag(W1)))  ## should be 10000, and that's what I get

system.time( W3 <- as(diag(30000),"dtCMatrix") )
print(sum(diag(W3)))## should be 30000, I get  12104 

sessionInfo()
R version 3.0.1 (2013-05-16)
Platform: x86_64-apple-darwin12.3.0/x86_64 (64-bit)

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] Matrix_1.0-14   lattice_0.20-15

loaded via a namespace (and not attached):
[1] grid_3.0.1  tools_3.0.1

--------------------------
Michael Braun
Associate Professor of Marketing
Cox School of Business
Southern Methodist University
Dallas, TX 75275
braunm _at_ smu.edu

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

Tue, Oct 15, 2013 12:10 PM #

I suppose that is true.  But the problem does not occur in smaller examples.  It only occurs when working with very large matrices.  That's the whole point.  And to be fair, I did warn the members of the list that this is a memory-intensive example.

On Oct 15, 2013, at 1:26 PM, Roger Koenker <rkoenker at illinois.edu> wrote:

You would be much more likely to get a response to this if you had a _smaller_ reproducible example.

url:    www.econ.uiuc.edu/~roger            Roger Koenker
email    rkoenker at uiuc.edu            Department of Economics
vox:     217-333-4558                University of Illinois
fax:       217-244-6678                Urbana, IL 61801

On Oct 15, 2013, at 11:36 AM, Braun, Michael wrote:

I've come across some interesting behavior using the Matrix package, and I cannot seem to explain it.  I have sent this issue to the Matrix maintainers, and Martin Maechler was unable to replicate it on his Linux system.  Since it might be something Mac-specific, he suggested that I post here.  The complete code is pasted at the bottom of this email, along with my sessionInfo() output.

The function f takes 3 parameters:  p, k and N.  First, it creates A,  a lower triangle of a sparse block diagonal matrix with p blocks, and each block being k x k.  I then let S = AA', and have L be the lower Cholesky decomposition of L. (I use Cholesky instead of chol because, for my "real" application, because I need the permutation matrix also).  I then multiply L by a matrix z, where each column in z is a standard MVN sample.

None of the elements should be zero.  However, if z becomes very large, I get lots of zero entries.

Additionally, you will see that when I try to coerce a large identity matrix to a dtCMatrix, some of the ones are lost. 

A word of warning:  this script does consume a lot of RAM, and it will take several minutes to run.  But it would be helpful if someone with a Mac Pro with 32GB of RAM could try to replicate the problem.

Thanks in advance for your help.

Michael


library(Matrix)

f <- function(p,k,N) {
 A <- tril(kronecker(Diagonal(p),Matrix(1:(k^2),k,k)))
 S <- tcrossprod(A)
 CH <- Cholesky(S)
 L <- expand(CH)$L
 z <- rnorm(N*k*p)
 dim(z) <- c(p*k,N)
 y <- L %*% z
 return(sum(y==0))
}

system.time(small   <- f(p=  100, k=3, N=  200) )
system.time(medium  <- f(p=  200, k=3, N= 5000) )

print(small ) ## 0
print(medium)  ## 0

print( system.time(large.p <- f(p=50000, k=3, N= 5000) ))
print( system.time(large.N <- f(p=10000, k=3, N=20000) ))

print(large.p) ## I get 536870912, but it should be 0
print(large.N) ## I get 536870912, but it should be 0

system.time( W1 <- as(diag(10000),"dtCMatrix") )
print(sum(diag(W1)))  ## should be 10000, and that's what I get

system.time( W3 <- as(diag(30000),"dtCMatrix") )
print(sum(diag(W3)))## should be 30000, I get  12104 

sessionInfo()
R version 3.0.1 (2013-05-16)
Platform: x86_64-apple-darwin12.3.0/x86_64 (64-bit)

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] Matrix_1.0-14   lattice_0.20-15

loaded via a namespace (and not attached):
[1] grid_3.0.1  tools_3.0.1

--------------------------
Michael Braun
Associate Professor of Marketing
Cox School of Business
Southern Methodist University
Dallas, TX 75275
braunm _at_ smu.edu

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

Tue, Oct 15, 2013 12:37 PM #

On Oct 15, 2013, at 3:10 PM, Braun, Michael wrote:

Isn't this related to the mysterious issue with 32-bit overflow in matrices on some OS X systems? Unfortunately I can't seem to find that thread - I recall Brian committing a work-around, so can you test more recent R? In addition, which BLAS implementation are you using?

Thanks,
Simon

On Oct 15, 2013, at 1:26 PM, Roger Koenker <rkoenker at illinois.edu> wrote:

You would be much more likely to get a response to this if you had a _smaller_ reproducible example.

url:    www.econ.uiuc.edu/~roger            Roger Koenker
email    rkoenker at uiuc.edu            Department of Economics
vox:     217-333-4558                University of Illinois
fax:       217-244-6678                Urbana, IL 61801

On Oct 15, 2013, at 11:36 AM, Braun, Michael wrote:

I've come across some interesting behavior using the Matrix package, and I cannot seem to explain it.  I have sent this issue to the Matrix maintainers, and Martin Maechler was unable to replicate it on his Linux system.  Since it might be something Mac-specific, he suggested that I post here.  The complete code is pasted at the bottom of this email, along with my sessionInfo() output.

The function f takes 3 parameters:  p, k and N.  First, it creates A,  a lower triangle of a sparse block diagonal matrix with p blocks, and each block being k x k.  I then let S = AA', and have L be the lower Cholesky decomposition of L. (I use Cholesky instead of chol because, for my "real" application, because I need the permutation matrix also).  I then multiply L by a matrix z, where each column in z is a standard MVN sample.

None of the elements should be zero.  However, if z becomes very large, I get lots of zero entries.

Additionally, you will see that when I try to coerce a large identity matrix to a dtCMatrix, some of the ones are lost. 

A word of warning:  this script does consume a lot of RAM, and it will take several minutes to run.  But it would be helpful if someone with a Mac Pro with 32GB of RAM could try to replicate the problem.

Thanks in advance for your help.

Michael


library(Matrix)

f <- function(p,k,N) {
A <- tril(kronecker(Diagonal(p),Matrix(1:(k^2),k,k)))
S <- tcrossprod(A)
CH <- Cholesky(S)
L <- expand(CH)$L
z <- rnorm(N*k*p)
dim(z) <- c(p*k,N)
y <- L %*% z
return(sum(y==0))
}

system.time(small   <- f(p=  100, k=3, N=  200) )
system.time(medium  <- f(p=  200, k=3, N= 5000) )

print(small ) ## 0
print(medium)  ## 0

print( system.time(large.p <- f(p=50000, k=3, N= 5000) ))
print( system.time(large.N <- f(p=10000, k=3, N=20000) ))

print(large.p) ## I get 536870912, but it should be 0
print(large.N) ## I get 536870912, but it should be 0

system.time( W1 <- as(diag(10000),"dtCMatrix") )
print(sum(diag(W1)))  ## should be 10000, and that's what I get

system.time( W3 <- as(diag(30000),"dtCMatrix") )
print(sum(diag(W3)))## should be 30000, I get  12104 

sessionInfo()
R version 3.0.1 (2013-05-16)
Platform: x86_64-apple-darwin12.3.0/x86_64 (64-bit)

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] Matrix_1.0-14   lattice_0.20-15

loaded via a namespace (and not attached):
[1] grid_3.0.1  tools_3.0.1

--------------------------
Michael Braun
Associate Professor of Marketing
Cox School of Business
Southern Methodist University
Dallas, TX 75275
braunm _at_ smu.edu

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

Brian Ripley

Tue, Oct 15, 2013 1:03 PM #

On 15/10/2013 20:37, Simon Urbanek wrote:

That was my guess, and the workaround is in 3.0.2 so please try current 
R (as the posting guide asked to be done before posting).

Thanks,
Simon

On Oct 15, 2013, at 1:26 PM, Roger Koenker <rkoenker at illinois.edu> wrote:

You would be much more likely to get a response to this if you had a _smaller_ reproducible example.

url:    www.econ.uiuc.edu/~roger            Roger Koenker
email    rkoenker at uiuc.edu            Department of Economics
vox:     217-333-4558                University of Illinois
fax:       217-244-6678                Urbana, IL 61801

On Oct 15, 2013, at 11:36 AM, Braun, Michael wrote:

I've come across some interesting behavior using the Matrix package, and I cannot seem to explain it.  I have sent this issue to the Matrix maintainers, and Martin Maechler was unable to replicate it on his Linux system.  Since it might be something Mac-specific, he suggested that I post here.  The complete code is pasted at the bottom of this email, along with my sessionInfo() output.

The function f takes 3 parameters:  p, k and N.  First, it creates A,  a lower triangle of a sparse block diagonal matrix with p blocks, and each block being k x k.  I then let S = AA', and have L be the lower Cholesky decomposition of L. (I use Cholesky instead of chol because, for my "real" application, because I need the permutation matrix also).  I then multiply L by a matrix z, where each column in z is a standard MVN sample.

None of the elements should be zero.  However, if z becomes very large, I get lots of zero entries.

Additionally, you will see that when I try to coerce a large identity matrix to a dtCMatrix, some of the ones are lost.

A word of warning:  this script does consume a lot of RAM, and it will take several minutes to run.  But it would be helpful if someone with a Mac Pro with 32GB of RAM could try to replicate the problem.

Thanks in advance for your help.

Michael


library(Matrix)

f <- function(p,k,N) {
A <- tril(kronecker(Diagonal(p),Matrix(1:(k^2),k,k)))
S <- tcrossprod(A)
CH <- Cholesky(S)
L <- expand(CH)$L
z <- rnorm(N*k*p)
dim(z) <- c(p*k,N)
y <- L %*% z
return(sum(y==0))
}

system.time(small   <- f(p=  100, k=3, N=  200) )
system.time(medium  <- f(p=  200, k=3, N= 5000) )

print(small ) ## 0
print(medium)  ## 0

print( system.time(large.p <- f(p=50000, k=3, N= 5000) ))
print( system.time(large.N <- f(p=10000, k=3, N=20000) ))

print(large.p) ## I get 536870912, but it should be 0
print(large.N) ## I get 536870912, but it should be 0

system.time( W1 <- as(diag(10000),"dtCMatrix") )
print(sum(diag(W1)))  ## should be 10000, and that's what I get

system.time( W3 <- as(diag(30000),"dtCMatrix") )
print(sum(diag(W3)))## should be 30000, I get  12104

sessionInfo()
R version 3.0.1 (2013-05-16)
Platform: x86_64-apple-darwin12.3.0/x86_64 (64-bit)

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base

other attached packages:
[1] Matrix_1.0-14   lattice_0.20-15

loaded via a namespace (and not attached):
[1] grid_3.0.1  tools_3.0.1

--------------------------
Michael Braun
Associate Professor of Marketing
Cox School of Business
Southern Methodist University
Dallas, TX 75275
braunm _at_ smu.edu

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

Tue, Oct 15, 2013 1:08 PM #

I was unaware of the 32-bit overflow issue, and I must have missed the announcement of R-3.0.2.  I cannot update from 3.0.1 to 3.0.2 right away, for some unrelated reasons, but I will do that ASAP.  I did peruse the NEWS file for R-3.0.2, R-3.0.2-patched, and R-devel, but I did not see any items that immediately jumped out as being connected.

My installation of R links to the Intel MKL BLAS, version 11.0, Update 3.  I compiled R using clang, with flags -msse4.2 -mtune=corei7 -O2 (among some others).  I would doubt that the BLAS choice is the problem, since a simple coercion of a large identity matrix to its sparse counterpart should not have to call a BLAS routine.  I can send my Makeconf file if you think that would be useful.

On Oct 15, 2013, at 2:37 PM, Simon Urbanek <simon.urbanek at r-project.org> wrote:

On Oct 15, 2013, at 3:10 PM, Braun, Michael wrote:

I suppose that is true.  But the problem does not occur in smaller examples.  It only occurs when working with very large matrices.  That's the whole point.  And to be fair, I did warn the members of the list that this is a memory-intensive example.

Isn't this related to the mysterious issue with 32-bit overflow in matrices on some OS X systems? Unfortunately I can't seem to find that thread - I recall Brian committing a work-around, so can you test more recent R? In addition, which BLAS implementation are you using?

Thanks,
Simon

On Oct 15, 2013, at 1:26 PM, Roger Koenker <rkoenker at illinois.edu> wrote:

You would be much more likely to get a response to this if you had a _smaller_ reproducible example.

url:    www.econ.uiuc.edu/~roger            Roger Koenker
email    rkoenker at uiuc.edu            Department of Economics
vox:     217-333-4558                University of Illinois
fax:       217-244-6678                Urbana, IL 61801

On Oct 15, 2013, at 11:36 AM, Braun, Michael wrote:

I've come across some interesting behavior using the Matrix package, and I cannot seem to explain it.  I have sent this issue to the Matrix maintainers, and Martin Maechler was unable to replicate it on his Linux system.  Since it might be something Mac-specific, he suggested that I post here.  The complete code is pasted at the bottom of this email, along with my sessionInfo() output.

The function f takes 3 parameters:  p, k and N.  First, it creates A,  a lower triangle of a sparse block diagonal matrix with p blocks, and each block being k x k.  I then let S = AA', and have L be the lower Cholesky decomposition of L. (I use Cholesky instead of chol because, for my "real" application, because I need the permutation matrix also).  I then multiply L by a matrix z, where each column in z is a standard MVN sample.

None of the elements should be zero.  However, if z becomes very large, I get lots of zero entries.

Additionally, you will see that when I try to coerce a large identity matrix to a dtCMatrix, some of the ones are lost. 

A word of warning:  this script does consume a lot of RAM, and it will take several minutes to run.  But it would be helpful if someone with a Mac Pro with 32GB of RAM could try to replicate the problem.

Thanks in advance for your help.

Michael


library(Matrix)

f <- function(p,k,N) {
A <- tril(kronecker(Diagonal(p),Matrix(1:(k^2),k,k)))
S <- tcrossprod(A)
CH <- Cholesky(S)
L <- expand(CH)$L
z <- rnorm(N*k*p)
dim(z) <- c(p*k,N)
y <- L %*% z
return(sum(y==0))
}

system.time(small   <- f(p=  100, k=3, N=  200) )
system.time(medium  <- f(p=  200, k=3, N= 5000) )

print(small ) ## 0
print(medium)  ## 0

print( system.time(large.p <- f(p=50000, k=3, N= 5000) ))
print( system.time(large.N <- f(p=10000, k=3, N=20000) ))

print(large.p) ## I get 536870912, but it should be 0
print(large.N) ## I get 536870912, but it should be 0

system.time( W1 <- as(diag(10000),"dtCMatrix") )
print(sum(diag(W1)))  ## should be 10000, and that's what I get

system.time( W3 <- as(diag(30000),"dtCMatrix") )
print(sum(diag(W3)))## should be 30000, I get  12104 

sessionInfo()
R version 3.0.1 (2013-05-16)
Platform: x86_64-apple-darwin12.3.0/x86_64 (64-bit)

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] Matrix_1.0-14   lattice_0.20-15

loaded via a namespace (and not attached):
[1] grid_3.0.1  tools_3.0.1

--------------------------
Michael Braun
Associate Professor of Marketing
Cox School of Business
Southern Methodist University
Dallas, TX 75275
braunm _at_ smu.edu

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

--------------------------
Michael Braun
Associate Professor of Marketing
Cox School of Business
Southern Methodist University
Dallas, TX 75275
braunm at smu.edu

Tue, Oct 15, 2013 2:59 PM #

I just installed R-3.0.2, and the problem remains, with the exact same results.  And I did hear back from another user off-list, who also gets the same erroneous output.

On Oct 15, 2013, at 2:37 PM, Simon Urbanek <simon.urbanek at r-project.org> wrote:

On Oct 15, 2013, at 3:10 PM, Braun, Michael wrote:

I suppose that is true.  But the problem does not occur in smaller examples.  It only occurs when working with very large matrices.  That's the whole point.  And to be fair, I did warn the members of the list that this is a memory-intensive example.

Isn't this related to the mysterious issue with 32-bit overflow in matrices on some OS X systems? Unfortunately I can't seem to find that thread - I recall Brian committing a work-around, so can you test more recent R? In addition, which BLAS implementation are you using?

Thanks,
Simon

On Oct 15, 2013, at 1:26 PM, Roger Koenker <rkoenker at illinois.edu> wrote:

You would be much more likely to get a response to this if you had a _smaller_ reproducible example.

url:    www.econ.uiuc.edu/~roger            Roger Koenker
email    rkoenker at uiuc.edu            Department of Economics
vox:     217-333-4558                University of Illinois
fax:       217-244-6678                Urbana, IL 61801

On Oct 15, 2013, at 11:36 AM, Braun, Michael wrote:

I've come across some interesting behavior using the Matrix package, and I cannot seem to explain it.  I have sent this issue to the Matrix maintainers, and Martin Maechler was unable to replicate it on his Linux system.  Since it might be something Mac-specific, he suggested that I post here.  The complete code is pasted at the bottom of this email, along with my sessionInfo() output.

The function f takes 3 parameters:  p, k and N.  First, it creates A,  a lower triangle of a sparse block diagonal matrix with p blocks, and each block being k x k.  I then let S = AA', and have L be the lower Cholesky decomposition of L. (I use Cholesky instead of chol because, for my "real" application, because I need the permutation matrix also).  I then multiply L by a matrix z, where each column in z is a standard MVN sample.

None of the elements should be zero.  However, if z becomes very large, I get lots of zero entries.

Additionally, you will see that when I try to coerce a large identity matrix to a dtCMatrix, some of the ones are lost. 

A word of warning:  this script does consume a lot of RAM, and it will take several minutes to run.  But it would be helpful if someone with a Mac Pro with 32GB of RAM could try to replicate the problem.

Thanks in advance for your help.

Michael


library(Matrix)

f <- function(p,k,N) {
A <- tril(kronecker(Diagonal(p),Matrix(1:(k^2),k,k)))
S <- tcrossprod(A)
CH <- Cholesky(S)
L <- expand(CH)$L
z <- rnorm(N*k*p)
dim(z) <- c(p*k,N)
y <- L %*% z
return(sum(y==0))
}

system.time(small   <- f(p=  100, k=3, N=  200) )
system.time(medium  <- f(p=  200, k=3, N= 5000) )

print(small ) ## 0
print(medium)  ## 0

print( system.time(large.p <- f(p=50000, k=3, N= 5000) ))
print( system.time(large.N <- f(p=10000, k=3, N=20000) ))

print(large.p) ## I get 536870912, but it should be 0
print(large.N) ## I get 536870912, but it should be 0

system.time( W1 <- as(diag(10000),"dtCMatrix") )
print(sum(diag(W1)))  ## should be 10000, and that's what I get

system.time( W3 <- as(diag(30000),"dtCMatrix") )
print(sum(diag(W3)))## should be 30000, I get  12104 

sessionInfo()
R version 3.0.1 (2013-05-16)
Platform: x86_64-apple-darwin12.3.0/x86_64 (64-bit)

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] Matrix_1.0-14   lattice_0.20-15

loaded via a namespace (and not attached):
[1] grid_3.0.1  tools_3.0.1

--------------------------
Michael Braun
Associate Professor of Marketing
Cox School of Business
Southern Methodist University
Dallas, TX 75275
braunm _at_ smu.edu

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

_______________________________________________
R-SIG-Mac mailing list
R-SIG-Mac at r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-mac

--------------------------
Michael Braun
Associate Professor of Marketing
Cox School of Business
Southern Methodist University
Dallas, TX 75275
braunm at smu.edu