# Distribution of a conical combination of n poisson variables?

Does a conical combination of n Poisson distributed variables have a closed-form distribution (linear combination with nonnegative coefficients)? I know that the sum of random Poisson variables would itself still be Poisson distributed, but I also understand that that would not be the case for a conical combination. I also know that scaled Poisson counts are often modelled as being Tweedie distributed. How could one model a conical combination of Poisson variables within a GLM (generalized linear model) framework, i.e. using a distribution from the exponential family (gaussian, Poisson or quasipoisson, Tweedie, negative binomial, Gamma, etc)? Would it still be OK to consider that conical combination as being Tweedie distributed perhaps (even if I know this to not be 100% correct, as it would rather be a mixture distribution of Tweedie distributions)? Or could negative binomial work? Or a quasipoisson framework (not a real distribution, but a good post hoc fix for over or underdispersion)?

PS My question is in the context of a grouped feature selection method I am developing, in this case using a grouped nonnegative identity link Poisson GLM with L0,infinity mixed norm penalty to approximate best subset of group selection (BSGS). In my original dataset I have 500 outcome variables with n=10000 and a covariate matrix with n=p=10000. My fit runs in 500s which is already relatively fast considering the problem size but I would like to get it faster still by working with latent variables where outcome variables would then be either sums of subsets of original variables or conical combinations of them. One approach I tried was to first get a lower dimensional projection using a double nonnegative SVD (sciencedirect.com/science/article/abs/pii/S0031320307004359) which for each extracted component would then produce a conical combination of the original variables. In that case, I wondered what would be the best distribution to model this conical combination. Alternatively, I could also cluster co-correlated variables and then sum those sets together. This would have the advantage that they would stay Poisson distributed, so that would avoid the problem of that conical combination having a rather funky distribution.

• Sorry I meant a conical combination (i.e. a linear combination with nonnegative coefficients)... I suppose what I discribe would have a mixture distribution of Tweedie distributions then. But that can't be modelled within a GLM framework. So what would be the next best thing then? Would it still be OK to regard the conical combination of Tweedie distributions as also being Tweedie distributed (even if that would not be 100% correct as I understand)? Nov 29, 2023 at 6:04
• I was still confused after your last comment, so I tried to search it again and found this Wikipedia article about conical combination. Thank you, I learned something new. Nov 29, 2023 at 6:37
• "How would one model it within a GLM framework" Can you provide more context? Nov 29, 2023 at 7:57
• With a combination of many variables you could try a saddlepoint approximation Nov 29, 2023 at 13:44
• @kjetilbhalvorsen Thanks for that - only issue with using a saddlepoint approximation is that it would give a distribution that still would not fit within a GLM framework. Maybe in the end I'll go for my other idea which is to cluster correlated outcome variables & then sum those clusters together, as in that case they would remain Poisson distributed. And it seems to give good results in practice & is very fast too. Nov 29, 2023 at 14:20

• What sort of linear combinations are you looking for? A sum of a few variables, or a sum of many variables?

With more information about this you could use it to make some approximation. E.g. $$\frac{1}{n}\sum_{i=1}^{n} Y_i$$ may be approximated as a normal distribution.

• If you model it in a GLM framework, what parameters would you have that describe the mean and variance?

Your sum is determined by many parameters and you should have somehow that those parameters are related.

Weird example: if your sum looks like $$Y_1 + a Y_2 \\ Y_1 ,Y_2 \sim Poisson(\lambda)$$ then

$$\begin{array}{rcl} E[X] &=& 1+a \lambda\\ Var[X] &=& 1+a^2 \lambda \end{array}$$

and the relationship between them is

$$Var[X] = E[X]^2-2E[X]$$

in such a case you could possibly model this with a quassi likelihood function where you can specify your own relationship between $$Var[X]$$ and $$E[X]$$.

• Well my question is in the context of a grouped feature selection method I am developing, in this case using a grouped nonnegative identity link Poisson GLM with L0,infinity mixed norm penalty to approximate best subset of group selection (BSGS). In my original dataset I have 500 outcome variables with n=10000 and a covariate matrix with n=p=10000. My fit runs in 500s which is already relatively fast considering the problem size but I would like to get it faster by working with latent variables where outcome variables would then be sums of subsets of original variables or conical combinations. Nov 29, 2023 at 9:10
• One approach I tried was to get a lower dimensional projection using a double nonnegative SVD (sciencedirect.com/science/article/abs/pii/S0031320307004359) which for each extracted component would then produce a conical combination of the original variables, for which I then wondered what would become the best distribution to model this. Nov 29, 2023 at 13:16