# [R] Linear Regressions with constraint coefficients

Gabor Grothendieck ggrothendieck at gmail.com
Tue Apr 26 17:59:14 CEST 2016

```This is a quadratic programming problem that you can solve using
either a quadratic programming solver with constraints or a general
nonlinear solver with constraints.  See
https://cran.r-project.org/web/views/Optimization.html

Here is an example using a nonlinear least squares solver and
non-negative bound constraints. The constraint that the coefficients
sum to 1 is implied by dividing them by their sum and then dividing
the coefficients found by their sum at the end:

# test data
set.seed(123)
n <- 1000
X1 <- rnorm(n)
X2 <- rnorm(n)
X3 <- rnorm(n)
Y <- .2 * X1 + .3 * X2 + .5 * X3 + rnorm(n)

# fit
library(nlmrt)
fm <- nlxb(Y ~ (b1 * X1 + b2 * X2 + b3 * X3)/(b1 + b2 + b3),
data = list(Y = Y, X1 = X1, X2 = X2, X3 = X3),
lower = numeric(3),
start = list(b1 = 1, b2 = 2, b3 = 3))

giving the following non-negative coefficients which sum to 1 that are
reasonably close to the true values of 0.2, 0.3 and 0.5:

> fm\$coefficients / sum(fm\$coefficients)
b1      b2      b3
0.18463 0.27887 0.53650

On Tue, Apr 26, 2016 at 8:39 AM, Aleksandrovic, Aljosa (Pfaeffikon)
<Aljosa.Aleksandrovic at man.com> wrote:
> Hi all,
>
> I hope you are doing well?
>
> I’m currently using the lm() function from the package stats to fit linear multifactor regressions.
>
> Unfortunately, I didn’t yet find a way to fit linear multifactor regressions with constraint coefficients? I would like the slope coefficients to be all inside an interval, let’s say, between 0 and 1. Further, if possible, the slope coefficients should add up to 1.
>
> Is there an elegant and not too complicated way to do such a constraint regression estimation in R?
>
> I would very much appreciate if you could help me with my issue?
>
> Thanks a lot in advance and kind regards,
> Aljosa Aleksandrovic
>
>
>
> Aljosa Aleksandrovic, FRM, CAIA
> Quantitative Analyst - Convertibles
> aljosa.aleksandrovic at man.com
> Tel +41 55 417 7603
>
> Man Investments (CH) AG
> Huobstrasse 3 | 8808 Pfäffikon SZ | Switzerland
>
>
> -----Original Message-----
> From: Kevin E. Thorpe [mailto:kevin.thorpe at utoronto.ca]
> Sent: Dienstag, 26. April 2016 14:35
> To: Aleksandrovic, Aljosa (Pfaeffikon)
> Subject: Re: Linear Regressions with constraint coefficients
>
> You need to send it to r-help at r-project.org however.
>
> Kevin
>
> On 04/26/2016 08:32 AM, Aleksandrovic, Aljosa (Pfaeffikon) wrote:
>> Ok, will do! Thx a lot!
>>
>> Please find below my request:
>>
>> Hi all,
>>
>> I hope you are doing well?
>>
>> I’m currently using the lm() function from the package stats to fit linear multifactor regressions.
>>
>> Unfortunately, I didn’t yet find a way to fit linear multifactor regressions with constraint coefficients? I would like the slope coefficients to be all inside an interval, let’s say, between 0 and 1. Further, if possible, the slope coefficients should add up to 1.
>>
>> Is there an elegant and not too complicated way to do such a constraint regression estimation in R?
>>
>> I would very much appreciate if you could help me with my issue?
>>
>> Thanks a lot in advance and kind regards, Aljosa Aleksandrovic
>>
>>
>>
>> Aljosa Aleksandrovic, FRM, CAIA
>> Quantitative Analyst - Convertibles
>> aljosa.aleksandrovic at man.com
>> Tel +41 55 417 7603
>>
>> Man Investments (CH) AG
>> Huobstrasse 3 | 8808 Pfäffikon SZ | Switzerland
>>
>>
>> -----Original Message-----
>> From: Kevin E. Thorpe [mailto:kevin.thorpe at utoronto.ca]
>> Sent: Dienstag, 26. April 2016 14:28
>> To: Aleksandrovic, Aljosa (Pfaeffikon); r-help-owner at r-project.org
>> Subject: Re: Linear Regressions with constraint coefficients
>>
>> I believe I approved a message with such a subject. Perhaps there was another layer that subsequently rejected it after that. I didn't notice any unusual content. Try again, making sure you send the message in plain text only.
>>
>> Kevin
>>
>> On 04/26/2016 08:16 AM, Aleksandrovic, Aljosa (Pfaeffikon) wrote:
>>> Do you know where I get help for my issue?
>>>
>>> Thanks in advance and kind regards,
>>> Aljosa
>>>
>>>
>>> Aljosa Aleksandrovic, FRM, CAIA
>>> Quantitative Analyst - Convertibles
>>> aljosa.aleksandrovic at man.com
>>> Tel +41 55 417 7603
>>>
>>> Man Investments (CH) AG
>>> Huobstrasse 3 | 8808 Pfäffikon SZ | Switzerland
>>>
>>> -----Original Message-----
>>> From: R-help [mailto:r-help-bounces at r-project.org] On Behalf Of
>>> r-help-owner at r-project.org
>>> Sent: Dienstag, 26. April 2016 14:10
>>> To: Aleksandrovic, Aljosa (Pfaeffikon)
>>> Subject: Linear Regressions with constraint coefficients
>>>
>>> The message's content type was not explicitly allowed
>>>
>
>
> --
> Kevin E. Thorpe
> Head of Biostatistics,  Applied Health Research Centre (AHRC)
> Li Ka Shing Knowledge Institute of St. Michael's Hospital
> Assistant Professor, Dalla Lana School of Public Health
> University of Toronto
> email: kevin.thorpe at utoronto.ca  Tel: 416.864.5776  Fax: 416.864.3016
>
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--
Statistics & Software Consulting
GKX Group, GKX Associates Inc.
tel: 1-877-GKX-GROUP
email: ggrothendieck at gmail.com

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