# How to constrain variables to fit a normal distribution

Given a time series $$s_t$$, I would like to define for each $$t$$ a perturbation $$e_t$$ such that the set of all $$e_t$$ "fits" (within some range) a normal distribution with mean $$\mu=0$$ and a given standard deviation $$\sigma$$. Possibly some other side constraints could be considered.

Is it possible to do this via (mixed integer) linear programming? (a non linear formulation would also be fine).

I was thinking maybe exploiting the fact that if $$e_t$$ follows a normal distribution, the $$Q-Q$$ plot should match the line with equation $$y_t=\frac{e_t-\mu}{\sigma}$$, where $$y_t$$ are the theoretical quantiles of a normal distribution $$\mathcal{N}(0,1)$$.

• You have not stated any criteria for the $e_t.$ Why not just sample the $N(0, \sigma)$ distribution?
– prubin
Commented Apr 10 at 17:59
• I could indeed sample it and add it to $s_t$. This would compute what I want "by construction". But I was wondering if I could do it with (non) linear programming in case additional side constraints must be enforced. Is there a way to express $N(0,\sigma)$ in terms of constraints? Commented Apr 10 at 19:21
• If you are asking whether there is a way to constrain model variables to fit a normal distribution, I'm not sure the question makes sense.
– prubin
Commented Apr 10 at 19:40
• The issue is what you mean by "fit". Suppose your original time series has 10 observations and I give you values for $e_1, \dots, e_{10}.$ How would you decide if they "fit" a $N(0, 3)$ distribution?
– prubin
Commented Apr 10 at 22:46
• The $e_i$ values are not a random sample and, importantly, are not independent (since they are connected through the model constraints).
– prubin
Commented Apr 11 at 15:57