Constrained regression

["Carlos Alzola" <calzola@cox.net>]

I would like to acknowledge the answers I received from Tom Filloon, Mike Cheung and Berwyn Turlach.
Berwyn's response was exactly what I needed. Use solve.QP from the quadprog package in R. S-Plus has the equivalent function solveQP in the NuOpt module.

Berwyn's response is below

G'day Carlos,

On Mon, Mar 3, 2008 at 11:52 AM
Carlos Alzola <calzola@cox.net> wrote:

>  I am trying to get information on how to fit a linear regression with
> constrained parameters. Specifically, I have 8 predictors , their
> coeffiecients should all be non-negative and add up to 1. I understand
> it is a quadratic programming problem but I have no experience in the
> subject. I searched the archives but the results were inconclusive.
>
>  Could someone provide suggestions and references to the literature,
> please?

A suggestion:

> library(MASS)   ## to access the Boston data
> designmat <- model.matrix(medv~., data="" 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)

The solution seems to contain values that are, for all practical purposes, actually zero:

> res$solution
 [1]  4.535581e-16  2.661931e-18  1.016929e-01 -1.850699e-17  [5]  1.458219e-16 -3.892418e-15  8.544939e-01  0.000000e+00  [9]  2.410742e-16  2.905722e-17 -5.700600e-20 -4.227261e-17 [13]  4.381328e-02 -3.723065e-18

So perhaps better:

> zapsmall(res$solution)
 [1] 0.0000000 0.0000000 0.1016929 0.0000000 0.0000000 0.0000000  [7] 0.8544939 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 [13] 0.0438133 0.0000000

So the estimates seem to follow the constraints.

And the unconstrained solution is:

> res$unconstrainted.solution
 [1]  3.645949e+01 -1.080114e-01  4.642046e-02  2.055863e-02  [5]  2.686734e+00 -1.776661e+01  3.809865e+00  6.922246e-04  [9] -1.475567e+00  3.060495e-01 -1.233459e-02 -9.527472e-01 [13]  9.311683e-03 -5.247584e-01

which seems to coincide with what lm() thinks it should be:

> coef(lm(medv~., Boston))
  (Intercept)          crim            zn         indus          chas
 3.645949e+01 -1.080114e-01  4.642046e-02  2.055863e-02  2.686734e+00
          nox            rm           age           dis           rad
-1.776661e+01  3.809865e+00  6.922246e-04 -1.475567e+00  3.060495e-01
          tax       ptratio         black         lstat
-1.233459e-02 -9.527472e-01  9.311683e-03 -5.247584e-01

So there seem to be no numeric problems.  Otherwise we could have done something else (e.g calculate the QR factorization of the design matrix, say X, and give the R factor to solve.QP, instead of calculating X'X and giving that one to solve.QP).

If the intercept is not supposed to be included in the set of constrained estimates, then something like the following can be done:

> Amat[1,] <- 0
> res <- solve.QP(Dmat, dvec, Amat, bvec, meq)
> zapsmall(res$solution)
 [1] 6.073972 0.000000 0.109124 0.000000 0.000000 0.000000 0.863421  [8] 0.000000 0.000000 0.000000 0.000000 0.000000 0.027455 0.000000

Of course, since after the first command in that last block the second column of Amat contains only zeros
> Amat[,2]
 [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0
we might as well have removed it (and the corresponding entry in bvec)
> Amat <- Amat[, -2]
> bvec <- bvec[-2]
before calling solve.QP().

Note, the Boston data set was only used to illustrate how to fit such models, I do not want to imply that these models are sensible for these data. :-)

Hope this helps.

Cheers,

        Berwin

Carlos Alzola
calzola@cox.net
(703) 242-6747