# [R] Constrained Poisson model / Bayesian Poisson model

mara.pfleiderer at uni-ulm.de mara.pfleiderer at uni-ulm.de
Sat Jan 23 14:24:00 CET 2016

```Hi David,

I'm sorry. I'm not familiar with posting problems on helppages.
As the data I deal with is confidential I can't provide all details,
but I will try to be as precise as possible about my problem:

As I said I'm working on a Poisson regression model with a linear
predictor and the identity as link function.
In particular: I have data which consists of observations of two
random variables X and Y over the period of about 3 months. I have
clustered this data into hours, so I received 2088 observations of
those variables and each observation represents the values of the
variables in one hour. Now I assume Y to be Poisson distributed
(Y_i~Poi(lambda_i)) and that the values of Y come from two different
impacts, so I split Y_i into two Poisson-distributed variables Y_i1
and Y_i2. I assume that the values of X have a long-term effect (of at
least one day) on the part Y_i1 and I have estimators for the
parameters lambda_i2, but I have no exact values of the variables of
Y_i1 and Y_i2 but only of Y_i as a whole.
So my model looks as follows:
fml=Y_i1 ~ b_1*X_i+....+b_n*X_(i-n+1) - lambda_i2 -1 with n>=24
That means that the last 24 (or more) values of X influence the value
of Y_i1 in an additive way and I have no intercept(b_0=0).
Now I made a matrix whose rows represent Y_i, all of its 24 (or more)
regressors for each observation of Y_i and the corresponding estimator
lambda_i2.
Then I used glm(fml, family=poisson(link="identity"), data=matrix) and
tried it for different values of n (=24,48,36,...).
But always some of the coefficients received negative values which
doesnt't make sense in interpretation. (The values of X represent
certain events which can only have a positive or none effect on the
value of Y.)

Now I want to use the constraint b_i >=0 in my model, but I don't know
how I can do this.

I also thought of analyzing this model in a Bayesian way, but yet I
haven't found a Bayesian version of glm() for the Poisson distribution
such that I can specify the prior for the b_i on my own. (Then I could
include the positivity in the prior.) Do you have a hint for me?

I'm sorry if I went too much into detail now...
I hope you understand my point now and have some answers for me!

Best,
Mara

Zitat von David Winsemius <dwinsemius at comcast.net>:

>
>> On Jan 22, 2016, at 7:01 AM, mara.pfleiderer at uni-ulm.de wrote:
>>
>> Hi all,
>>
>> I am dealing with a problem about my linear Poisson regression
>> model (link function=identity).
>>
>> I am using the glm()-function which results in negative
>> coefficients, but a negative influence of the regressors wouldn't
>> make sense.
>
> Negative coefficients merely indicate a lower relative rate. You
> need to be more specific about the exactly data and model output
> before you can raise our concern to a level where further comment
> can be made.
>
>
>>
>> (i) Is there a possibility to set constraints on the regression
>> parameters in glm() such that all coefficients are positive? Or is
>>  there another function in R for which this is possible?
>>
>> (ii) Is there a Bayesian version of the glm()-function where I can
>>  specify the prior distribution for my regression parameters? (e.g.
>>   a Dirichlet prior s.t. the parameters are positive)
>>
>> All this with respect to the linear Poisson model...
>
> As I implied above, the word "linear" means something different than
>   "additive" when the link is log().
>
> --
>
> David Winsemius
> Alameda, CA, USA
>
>

```