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

[Spencer Graves]

      How about generating matrices (X1|X2), dim(X1) = c(n, k1), dim(X2) 
= c(n, k2), with mean 0 and covariance matrix as follows: 

(S11 | S12)
(S21 | S22),

with S12 = W, S22 = Z and S11 = whatever you want?  With X = t(X1), and 
Q = X2, we have E(XQ) = W and E(Q'Q) = Z. 

      This can be done using rmvnorm in package mvtnorm. 

      hope this helps.  spencer graves    

Stephane DRAY wrote:

> 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éphane DRAY
>>> -------------------------------------------------------------------------------------------------- 
>>>
>>> Département des Sciences Biologiques
>>> Université de Montréal, C.P. 6128, succursale centre-ville
>>> Montréal, Québec 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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>>> PLEASE do read the posting guide! 
>>> http://www.R-project.org/posting-guide.html
>>
>>
>
> Stéphane DRAY
> -------------------------------------------------------------------------------------------------- 
>
> Département des Sciences Biologiques
> Université de Montréal, C.P. 6128, succursale centre-ville
> Montréal, Québec H3C 3J7, Canada
>
> Tel : 514 343 6111 poste 1233
> E-mail : stephane.dray at umontreal.ca
> -------------------------------------------------------------------------------------------------- 
>
> Web                                          
> http://www.steph280.freesurf.fr/
> -------------------------------------------------------------------------------------------------- 
>
>