[R] Constrained Regression

[David Winsemius]

On Oct 31, 2010, at 2:44 AM, Jim Silverton wrote:

> Hello everyone,
> I have 3 variables Y, X1 and X2. Each variables lies between 0 and  
> 1. I want
> to do a constrained regression such that a>0 and (1-a) >0
>
> for the model:
>
> Y = a*X1 + (1-a)*X2


It would not accomplish the constraint that a > 0 but you could  
accomplish the other constraint within an lm fit:

X3 <- X1-X2
lm(Y ~ X3 + offset(X2) )

Since beta1 is for the model Y ~ 1 + beta1(X1- X2) + 1*X2)
                              Y ~ intercept + beta1*X1 + (1 -beta1)*X2

... so beta1 is a.

In the case beta < 0 then I suppose a would be assigned 0. This might  
be accomplished within an iterative calculation framework by a large  
penalization for negative values. In a reply (1) to a question by  
Carlos Alzola in 2008 on rhalp, Berwin Turlach offered a solution to a  
similar problem ( sum(coef) == 1 AND coef non-negative). Modifying his  
code to incorporate the above strategy (and choosing two variables for  
which parameter values might be inside the constraint boundaries) we  
get:

library(MASS)   ## to access the Boston data
   designmat <- model.matrix(medv~I(age-lstat) +offset(lstat),  
data=Boston)
   Dmat <-crossprod(designmat, designmat); dvec <- crossprod(designmat,
   Boston$medv)
   Amat <- cbind(1, diag(NROW(Dmat)))
   bvec <- c(1,rep(0,NROW(Dmat)))
   meq <- 1
library(quadprog)
   res <- solve.QP(Dmat, dvec, Amat, bvec, meq)

 > zapsmall(res$solution)
[1] 0.686547 0.313453

Turlach specifically advised against any interpretation of this  
particular result which was only contructed to demonstrate the  
mathematical mechanics.

>
> I tried the help on the constrained regression in R but I concede  
> that it
> was not helpful.

I must not have that package installed because I got nothing that  
appeared to be useful with ??"constrained regression" .


David Winsemius, MD
West Hartford, CT

1) http://finzi.psych.upenn.edu/Rhelp10/2008-March/155990.html