[R] Constrained coefficients in lm (correction)

[Moshe Olshansky]

This problem can be easily solved analytically:

we want to minimize sum(res(i) -a*st(i) -b*mod(i))^2 subject to a+b=1,a,b>=0, so we want to minimize

f(a) = sum((res(i)-mod(i)) - a*(st(i)-mod(i)))^2 for 0<=a<=1

Define Xi = res(i) - mod(i),  Yi = st(i) - mod(i), then

f(a) = sum(Xi - a*Yi)^2

f(0) = sum(Xi^2), f(1) = sum(Xi-Yi)^2, f'(a) = -2*sum(Yi*(Xi-aYi)), so

f'(a) = 0 for a* = sum(Xi*Yi)/sum(Yi^2)

So if 0 <= a* <= 1 then a* is the solution, otherwise take a=0 or a=1 depending on whether f(0) < f(1) or not.






--- On Thu, 24/7/08, Jorge Ivan Velez <jorgeivanvelez at gmail.com> wrote:

> From: Jorge Ivan Velez <jorgeivanvelez at gmail.com>
> Subject: [R] Constrained coefficients in lm (correction)
> To: "R mailing list" <r-help at r-project.org>
> Received: Thursday, 24 July, 2008, 5:39 AM
> Dear list,
> 
> In my previous email, the model I'd like to estimate is
> rea=a*st+b*mod+error, where a+b=1 and a,b>0. My
> apologies for the
> misunderstanding.
> 
> Thanks for all your help,
> 
> Jorge
> 
> 
> On Wed, Jul 23, 2008 at 3:35 PM, Jorge Ivan Velez
> <jorgeivanvelez at gmail.com>
> wrote:
> 
> >
> > Dear list,
> >
> > I have a data set which looks like myDF (see below)
> and I'd like to
> > estimate the linear model rea=a*rea+b*mod+error, where
> a+b=1 and a,b>0. I
> > tried
> >
> > mymodel=lm(rea~ st + mod-1, data=myDF)
> > summary(mymodel)
> >
> > but I couldn't get what I'm looking for. 
> Also, I tried
> > RSiteSearch("constrained coefficients in
> lm") which produces 20 hints, but
> > they're not what I need.
> >
> > Any help would be appreciated.
> >
> > Thanks in advance,
> >
> >
> > Jorge
> >
> >
> > # Data set to estimate the model
> > myDF=read.table(textConnection("rea     st     
> mod
> >  14482734 14305907 14761000
> >  14427969 14502228 14848024
> >  14698723 14408264 14259492
> >  15741064 14797138 14565303
> >  15914249 16162714 14348592
> >  16475594 15977623 15592229
> >  17124613 16688456 14988151
> >  17575281 17383822 15647240
> >  17721548 17757635 15624907
> >  18178273 17779858 15598802
> >  18005810 18364233 16363321
> >  18032618 17938963 16536844
> >  17737470 18043063 17096089
> >  17652014 17624039
> 17056355"),header=TRUE,sep="")
> > closeAllConnections()
> >
> >
> >
> 
> 	[[alternative HTML version deleted]]
> 
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