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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 $\mathcal{N}(0,1)$.

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    $\begingroup$ You have not stated any criteria for the $e_t.$ Why not just sample the $N(0, \sigma)$ distribution? $\endgroup$
    – prubin
    Commented Apr 10 at 17:59
  • $\begingroup$ 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? $\endgroup$
    – NormalFit
    Commented Apr 10 at 19:21
  • $\begingroup$ 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. $\endgroup$
    – prubin
    Commented Apr 10 at 19:40
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    $\begingroup$ 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? $\endgroup$
    – prubin
    Commented Apr 10 at 22:46
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    $\begingroup$ The $e_i$ values are not a random sample and, importantly, are not independent (since they are connected through the model constraints). $\endgroup$
    – prubin
    Commented Apr 11 at 15:57

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