R help - Sep 2003 - truncated multivariate normal

Please,    

I would like to know how to generate a truncated multivariate normal
distribution k - dimensional,    X ~ NT(mu, Sigma),  where the
elements of X to be non-negative (except the first), and the first
dimension is strictly larger than zero.    

Example:  

X ~ NT_2(mu, Sigma),   

where mu=c(0.5, 0.5) and Sigma=c([120, 191], [191,154]), with X_1>0
and X_2>=0  
Could anybody help me?     
  
Thanks in advanced.

Juliana G. Cespedes
Mestranda em Estat?stica e Experimenta??o Agron?mica
Departamento de Ci?ncias Exatas - ESALQ/USP
S?o Paulo - Brasil

Do you mean you want to generate deviates from this distribution or that 
you want the density function (or distribution function)?

mvrnorm in package MASS (library(MASS)) will generate multivariate normal
deviates.  A brute-force approach would just keep picking values until you
get ones in the right quadrant, although with your example (large
variance, small mean) that would mean you would be wasting 3/4 of your 
values -- so there are probably more efficient approaches.  Depends how 
much time you want to spend programming, how many values you want to draw, 
and how fast your computer is ...

   Ben

On Tue, 30 Sep 2003, Juliana Garcia Cespedes wrote:
> Please,    
> 
> I would like to know how to generate a truncated multivariate normal
> distribution k - dimensional,    X ~ NT(mu, Sigma),  where the
> elements of X to be non-negative (except the first), and the first
> dimension is strictly larger than zero.    
> 
> Example:  
> 
> X ~ NT_2(mu, Sigma),   
> 
> where mu=c(0.5, 0.5) and Sigma=c([120, 191], [191,154]), with X_1>0
> and X_2>=0  
> Could anybody help me?     
>   
> Thanks in advanced.
> 
> Juliana G. Cespedes
> Mestranda em Estat?stica e Experimenta??o Agron?mica
> Departamento de Ci?ncias Exatas - ESALQ/USP
> S?o Paulo - Brasil
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> 
-- 
620B Bartram Hall                            bolker at zoo.ufl.edu
Zoology Department, University of Florida    http://www.zoo.ufl.edu/bolker
Box 118525                                   (ph)  352-392-5697
Gainesville, FL 32611-8525                   (fax) 352-392-3704

< Depends how much time you want to spend programming, how many values you
want to draw, and how fast your computer is ...>
There are examples of 'efficient' Gibbs samplers for truncated
multivariate
normal variables on the web (try a search on 'truncated multivariate
normal').

Use the search engine at CRAN (http://cran.r-project.org/) to search for
Gibbs - in the MCMC package there is an example of generating a truncated
(univariate) Normal which may help you get started.

However...with a mean of (.5, .5) and variances of 120 and 154 you might be
able to live with an approximation.  Generate MVN from mu = c(0,0)
Sigma=c([120, 191], [191,154]) using the mvnorm that Ben suggested.  Then
take absolute values (throwing away any generated pair with a 0 in the first
coordinate).  This is not precisely the distribution you want and maybe a
bad idea for your purposes but the absolute difference between the joint CDF
of what you exactly want and the joint CDF of this approximation is at most
about 0.01.

And bytheway Ben: go gators :)

Good luck with it
Bob
Usual disclaimers...


-----Original Message-----
From: Ben Bolker [mailto:bolker@zoo.ufl.edu] 
Sent: Tuesday, September 30, 2003 5:30 PM
To: Juliana Garcia Cespedes
Cc: r-help@stat.math.ethz.ch
Subject: Re: [R] truncated multivariate normal 


  Do you mean you want to generate deviates from this distribution or that 
you want the density function (or distribution function)?

mvrnorm in package MASS (library(MASS)) will generate multivariate normal
deviates.  A brute-force approach would just keep picking values until you
get ones in the right quadrant, although with your example (large
variance, small mean) that would mean you would be wasting 3/4 of your 
values -- so there are probably more efficient approaches.  Depends how 
much time you want to spend programming, how many values you want to draw, 
and how fast your computer is ...

   Ben

On Tue, 30 Sep 2003, Juliana Garcia Cespedes wrote:
> Please,    
> 
> I would like to know how to generate a truncated multivariate normal
> distribution k - dimensional,    X ~ NT(mu, Sigma),  where the
> elements of X to be non-negative (except the first), and the first
> dimension is strictly larger than zero.    
> 
> Example:  
> 
> X ~ NT_2(mu, Sigma),   
> 
> where mu=c(0.5, 0.5) and Sigma=c([120, 191], [191,154]), with X_1>0
> and X_2>=0  
> Could anybody help me?     
>   
> Thanks in advanced.
> 
> Juliana G. Cespedes
> Mestranda em Estatística e Experimentação Agronômica
> Departamento de Ciências Exatas - ESALQ/USP
> São Paulo - Brasil
> 
> ______________________________________________
> R-help@stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> 
-- 
620B Bartram Hall                            bolker@zoo.ufl.edu
Zoology Department, University of Florida    http://www.zoo.ufl.edu/bolker
Box 118525                                   (ph)  352-392-5697
Gainesville, FL 32611-8525                   (fax) 352-392-3704

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