R help - Mar 2005 - Optimization of constrained linear least-squares problem

Dear R-ians,

I want to perform an linear unmixing of image pixels in fractions of 
pure endmembers. Therefore I need to perform a constrained linear 
least-squares problem that looks like :

min || Cx - d || ? where sum(x) = 1.

I have a 3x3 matrix C, containing the values for endmembers and I have a 
3x1 column vector d (for every pixel in the image). In theory my x 
values should all be in the (0,1) interval but I don't want to force 
them so I can check the validity of my solution. I just want to 
calculate the x values. Can anyone help me with this problem? I've been 
checking the optim, optimize, constrOptim and nlm help files, burt  I 
don't understand it very well. Wich function should I use for my 
problem? I did a first test using optim:

# Make my C matrix
EM<- c(4.5000,6.0000,10.5000,5.0000,27.0000,20.7500,16.7500,23.6666,38.7500)
C <- array(EM, c(3,3))

# Take an arbitrary d
d<-c(10, 20, 20)

# Define the function
 fr <- function(x) {
C[1,]*x=d
C[2,]*x=d
C[3,]*x=d
sum(x)=1}

# Perform the optimization
optim(c(0.25,0.25,0.25),fr)

But it did not work. I got the eror couldn't find function. Can anyone 
tell me what functyion I should use for my problem and how should I 
program it?

Thanx in advance,
Stef

you could re-parameterize, e.g.,

EM <- 
c(4.5000,6.0000,10.5000,5.0000,27.0000,20.7500,16.7500,23.6666,38.7500)
W <- array(EM, c(3,3))
d <- c(10, 20, 20)
##############33
fn <- function(x){
    x <- exp(x) / sum(exp(x))
    r <- W%*%x - d
    crossprod(r, r)[1,1]
}
opt <- optim(rnorm(3), fn)
res <- exp(opt$par) / sum(exp(opt$par))
res


I hope it helps.

Best,
Dimitris

----
Dimitris Rizopoulos
Ph.D. Student
Biostatistical Centre
School of Public Health
Catholic University of Leuven

Address: Kapucijnenvoer 35, Leuven, Belgium
Tel: +32/16/336899
Fax: +32/16/337015
Web: http://www.med.kuleuven.ac.be/biostat/
     http://www.student.kuleuven.ac.be/~m0390867/dimitris.htm


----- Original Message ----- 
From: "Stefaan Lhermitte" <stefaan.lhermitte at biw.kuleuven.be>
To: <r-help at stat.math.ethz.ch>
Sent: Thursday, March 17, 2005 3:13 PM
Subject: [R] Optimization of constrained linear least-squares problem

> Dear R-ians,
>
> I want to perform an linear unmixing of image pixels in fractions of 
> pure endmembers. Therefore I need to perform a constrained linear 
> least-squares problem that looks like :
>
> min || Cx - d || ? where sum(x) = 1.
>
> I have a 3x3 matrix C, containing the values for endmembers and I 
> have a 3x1 column vector d (for every pixel in the image). In theory 
> my x values should all be in the (0,1) interval but I don't want to 
> force them so I can check the validity of my solution. I just want 
> to calculate the x values. Can anyone help me with this problem? 
> I've been checking the optim, optimize, constrOptim and nlm help 
> files, burt  I don't understand it very well. Wich function should I 
> use for my problem? I did a first test using optim:
>
> # Make my C matrix
> EM<- 
> c(4.5000,6.0000,10.5000,5.0000,27.0000,20.7500,16.7500,23.6666,38.7500)
> C <- array(EM, c(3,3))
>
> # Take an arbitrary d
> d<-c(10, 20, 20)
>
> # Define the function
> fr <- function(x) {
> C[1,]*x=d
> C[2,]*x=d
> C[3,]*x=d
> sum(x)=1}
>
> # Perform the optimization
> optim(c(0.25,0.25,0.25),fr)
>
> But it did not work. I got the eror couldn't find function. Can 
> anyone tell me what functyion I should use for my problem and how 
> should I program it?
>
> Thanx in advance,
> Stef
>
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