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Stephane DRAY

2004-Jul-06 22:28 UTC

[R] Generate a matrix Q satisfying t(Q)%*%Q=Z and XQ=W

Hello,
I have a question that is not directly related to R ... but I try to do it 
in R ;-) :

I would like to generate a matrix Q satisfying (for a given Z, X and W) the 
two following conditions:

t(Q)%*%Q=Z  (1)
XQ=W (2)

where:
Q is m rows and r columns
X is p rows and m columns
D is p rows and r columns
C is r rows and r columns
with m>p,r


e.g:
m=6,
p=2
r=3

Z=matrix(c(1,.2,.5,.2,1,.45,.5,.45,1),3,3)
X=matrix(c(.1,.3,.5,.6,.2,.1,.8,1,.4,.2,.2,.9),2,6)
W=matrix(c(0,.8,.4,.6,.2,0),2,3)

#Create a matrix satisfying (1) is easy:

A=matrix(runif(18),6,3)
Q1=svd(A)$u%*%chol(Z)


#For the second condition (2), a solution is given by

Q2=A%*%ginv(X%*%A)%*%W





I do not know how to create a matrix Q that satisfies the two 
conditions.  I have try to construct an iterative procedure without success 
(no convergence):

eps=10
i=0
while(eps>.5)
{
Q1=svd(Q2)$u%*%chol(Z)
Q2=Q1%*%ginv(X%*%Q1)%*%W
eps=sum(abs(Q1-Q2))
cat(i,":",eps,"\n")
i=i+1
}

Perhaps someone could have any idea to solve the problem, or a reference on 
this kind of question or the email of another list where I should ask this 
question.

Thanks in advance,

Sincerely.

St??phane DRAY
--------------------------------------------------------------------------------------------------

D??partement des Sciences Biologiques
Universit?? de Montr??al, C.P. 6128, succursale centre-ville
Montr??al, Qu??bec H3C 3J7, Canada

Tel : 514 343 6111 poste 1233
E-mail : stephane.dray at umontreal.ca
--------------------------------------------------------------------------------------------------

Web                                          http://www.steph280.freesurf.fr/

Spencer Graves

2004-Jul-07 04:00 UTC

head link

[R] Generate a matrix Q satisfying t(Q)%*%Q=Z and XQ=W

Is a solution even possible for the matrices in your example?  
I've tried a few things that have suggested that a solution may not be 
possible. 

      What can you tell us of the problem that you've translated into 
this?  I see a minimization problem subject to constraints, but I'm not 
certain which are the constraints and what is the objective function.  
For example, are you trying to find Q to minimize sum((Z-X'X)^2) subject 
to XQ=W or do you want to minimize sum((XQ-W)^2) subject to Q'Q=Z or 
something else? 

      If it were my problem, I think I would work for a while with the 
singular value decompositions of X, W and Z, and see if that would lead 
me to more information about Q, including conditions under which a 
solution existed, expressions for Q when multiple solutions existed, and 
a solution minimizing your chosen objective function when solutions do 
not exist.  (A google search produced many hits for "singular value 
decomposition", implemented as "svd" in R.) 

      hope this helps.  spencer graves    

Stephane DRAY wrote:
> Hello,
> I have a question that is not directly related to R ... but I try to 
> do it in R ;-) :
>
> I would like to generate a matrix Q satisfying (for a given Z, X and 
> W) the two following conditions:
>
> t(Q)%*%Q=Z  (1)
> XQ=W (2)
>
> where:
> Q is m rows and r columns
> X is p rows and m columns
> D is p rows and r columns
> C is r rows and r columns
> with m>p,r
>
>
> e.g:
> m=6,
> p=2
> r=3
>
> Z=matrix(c(1,.2,.5,.2,1,.45,.5,.45,1),3,3)
> X=matrix(c(.1,.3,.5,.6,.2,.1,.8,1,.4,.2,.2,.9),2,6)
> W=matrix(c(0,.8,.4,.6,.2,0),2,3)
>
> #Create a matrix satisfying (1) is easy:
>
> A=matrix(runif(18),6,3)
> Q1=svd(A)$u%*%chol(Z)
>
>
> #For the second condition (2), a solution is given by
>
> Q2=A%*%ginv(X%*%A)%*%W
>
>
>
>
>
> I do not know how to create a matrix Q that satisfies the two 
> conditions.  I have try to construct an iterative procedure without 
> success (no convergence):
>
> eps=10
> i=0
> while(eps>.5)
> {
> Q1=svd(Q2)$u%*%chol(Z)
> Q2=Q1%*%ginv(X%*%Q1)%*%W
> eps=sum(abs(Q1-Q2))
> cat(i,":",eps,"\n")
> i=i+1
> }
>
> Perhaps someone could have any idea to solve the problem, or a 
> reference on this kind of question or the email of another list where 
> I should ask this question.
>
> Thanks in advance,
>
> Sincerely.
>
> St??phane DRAY
>
--------------------------------------------------------------------------------------------------
>
> D??partement des Sciences Biologiques
> Universit?? de Montr??al, C.P. 6128, succursale centre-ville
> Montr??al, Qu??bec H3C 3J7, Canada
>
> Tel : 514 343 6111 poste 1233
> E-mail : stephane.dray at umontreal.ca
>
--------------------------------------------------------------------------------------------------
>
> Web                                          
> http://www.steph280.freesurf.fr/
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! 
> http://www.R-project.org/posting-guide.html

Stephane DRAY

2004-Jul-07 18:32 UTC

head link

[R] Generate a matrix Q satisfying t(Q)%*%Q=Z and XQ=W

thanks,
I want to create matrices for simulation purpose (in order to evaluate the 
efficiency of different methods on this simulated data set). So, I want to 
create a data matrix Q (m individuals and r variables). I want to specify 
the variance-covariance structure for this matrix (t(Q)%*%Q=Z ) but I want 
also to create another constraint due to another matrix of data. I want 
that the covariance of Q and X are equal to those given in W (XQ=W).
The example I gave is just to illustrate my problem and perhaps it has no 
solution (I cannot see it because I have no idea how to construct Q such as 
Q=Q1=Q2)




At 00:00 07/07/2004, Spencer Graves wrote:>      Is a solution even possible for the matrices in your example?
>I've tried a few things that have suggested that a solution may not be 
>possible.
>      What can you tell us of the problem that you've translated into 
> this?  I see a minimization problem subject to constraints, but I'm not
> certain which are the constraints and what is the objective function.
>For example, are you trying to find Q to minimize sum((Z-X'X)^2) subject
>to XQ=W or do you want to minimize sum((XQ-W)^2) subject to Q'Q=Z or 
>something else?
>      If it were my problem, I think I would work for a while with the 
> singular value decompositions of X, W and Z, and see if that would lead 
> me to more information about Q, including conditions under which a 
> solution existed, expressions for Q when multiple solutions existed, and 
> a solution minimizing your chosen objective function when solutions do 
> not exist.  (A google search produced many hits for "singular value 
> decomposition", implemented as "svd" in R.)
>      hope this helps.  spencer graves
>
>Stephane DRAY wrote:
>
>>Hello,
>>I have a question that is not directly related to R ... but I try to do 
>>it in R ;-) :
>>
>>I would like to generate a matrix Q satisfying (for a given Z, X and W) 
>>the two following conditions:
>>
>>t(Q)%*%Q=Z  (1)
>>XQ=W (2)
>>
>>where:
>>Q is m rows and r columns
>>X is p rows and m columns
>>D is p rows and r columns
>>C is r rows and r columns
>>with m>p,r
>>
>>
>>e.g:
>>m=6,
>>p=2
>>r=3
>>
>>Z=matrix(c(1,.2,.5,.2,1,.45,.5,.45,1),3,3)
>>X=matrix(c(.1,.3,.5,.6,.2,.1,.8,1,.4,.2,.2,.9),2,6)
>>W=matrix(c(0,.8,.4,.6,.2,0),2,3)
>>
>>#Create a matrix satisfying (1) is easy:
>>
>>A=matrix(runif(18),6,3)
>>Q1=svd(A)$u%*%chol(Z)
>>
>>
>>#For the second condition (2), a solution is given by
>>
>>Q2=A%*%ginv(X%*%A)%*%W
>>
>>
>>
>>
>>
>>I do not know how to create a matrix Q that satisfies the two 
>>conditions.  I have try to construct an iterative procedure without 
>>success (no convergence):
>>
>>eps=10
>>i=0
>>while(eps>.5)
>>{
>>Q1=svd(Q2)$u%*%chol(Z)
>>Q2=Q1%*%ginv(X%*%Q1)%*%W
>>eps=sum(abs(Q1-Q2))
>>cat(i,":",eps,"\n")
>>i=i+1
>>}
>>
>>Perhaps someone could have any idea to solve the problem, or a reference
>>on this kind of question or the email of another list where I should ask
>>this question.
>>
>>Thanks in advance,
>>
>>Sincerely.
>>
>>St??phane DRAY
>>--------------------------------------------------------------------------------------------------
>>
>>D??partement des Sciences Biologiques
>>Universit?? de Montr??al, C.P. 6128, succursale centre-ville
>>Montr??al, Qu??bec H3C 3J7, Canada
>>
>>Tel : 514 343 6111 poste 1233
>>E-mail : stephane.dray at umontreal.ca
>>--------------------------------------------------------------------------------------------------
>>
>>Web
>>http://www.steph280.freesurf.fr/
>>
>>______________________________________________
>>R-help at stat.math.ethz.ch mailing list
>>https://www.stat.math.ethz.ch/mailman/listinfo/r-help
>>PLEASE do read the posting guide!
http://www.R-project.org/posting-guide.html
>
St??phane DRAY
--------------------------------------------------------------------------------------------------

D??partement des Sciences Biologiques
Universit?? de Montr??al, C.P. 6128, succursale centre-ville
Montr??al, Qu??bec H3C 3J7, Canada

Tel : 514 343 6111 poste 1233
E-mail : stephane.dray at umontreal.ca
--------------------------------------------------------------------------------------------------

Web                                          http://www.steph280.freesurf.fr/

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