# Talk:Wrapped normal distribution

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## Wrapped Normal Distribution

(This contrib copied from Stats project talk page by me: Melcombe (talk) 09:41, 21 January 2010 (UTC) )

The pdf for the wrapped normal doesn't appear correct to me. If I type it in Mathematica, I get imaginary values out. The Jacobi description that follows is a mix of variables that have the same name in different formulas, and is confusing at best. As stated it appears as such.

### Current

$f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-\sigma^2n^2/2+in(\theta-\mu)} =\frac{1}{2\pi}\vartheta\left(\frac{\theta-\mu}{2\pi},\frac{i\sigma^2}{2\pi}\right)$

where $\vartheta(\theta,\tau)$ is the Jacobi theta function:

### My Proposal

$f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-\sigma^2n^2/2+in(\theta-\mu)} =\frac{1}{2\pi}\vartheta_3\left(\frac{\theta-\mu}{2},e^{-\sigma^2/2}\right)$

where $\vartheta_3(\theta,\tau)$ is the 3rd Jacobi theta function:

If I type this in to Mathematica, it works, and matches know results that I have to compare with. Also, I propose deleting the Jacobi elliptic explanation. The summation form is also suspect, but I'll look this up later.

I would prefer someone to validate this, otherwise in 2 weeks i will change it.

Shawn@garbett.org (talk) 20:08, 20 January 2010 (UTC)

A few points
• The current definition is consistent if you accept the definitions given in this article, so there's not an error in that sense. Its easy to do the substitution.
• The current definition is in terms of the Jacobi theta function as defined in the Wikipedia Jacobi theta function article under the "nome definition", except its written $\vartheta_{00}(w,q)$. Further up, it is stated that $\vartheta_{00}(z,\tau)=\vartheta(z,\tau)$. This is kind of confusing but I assumed that $\vartheta_{00}(w,q)$ is a different function with the same name. I just avoided the problem by assuming that $\vartheta(z,\tau)=\vartheta_{00}(w,q)$ was what was meant. This is a notational problem, not a mathematical problem. Maybe this is something that should be cleared up in the Jacobi theta function article.
• Mathematica gives a correct answer in terms of the $\vartheta_3$ function, but this function is not defined in the Wikipedia article. I wanted to stick with functions that could easily be accessed within Wikipedia. Maybe the $\vartheta_3$ function should be included in the Jacobi theta article.
• I don't understand what you mean by "the Jacobi elliptic explanation". There is no reference to "Jacobi elliptic" in the article.
• By all means, check the summation form.
• I'm beginning not to like the present definition, because the $z=e^{i\theta}$ is really the preferred variable for circular statistics and I think we should stick to using that variable as much as possible. That means using the nome variables themselves, not the present definition nor the $\vartheta_3$ function. PAR (talk) 07:34, 22 January 2010 (UTC)

• Yes it is consistent. I stumbled on the $\theta$ having two different meanings.
• Okay, I read that article, and using $\vartheta_{00}$ would be better. I read through the Jacobi theta function article, and it's different than the reference I have on the subject, but as you stated it appears notational.
I'm in favor of a Jacobi function that uses the nome variables w and q because they are the more natural variables for circular statistics, along with $z=e^{i\theta}$. I haven't worked it all out yet, but I believe it would be an improvement. I don't know what you mean by $\theta$ having two different meanings. There is $\phi$ which is the "true" or "unwrapped" angle, that lies in $[-\infty,\infty]$ and the "measured" or "wrapped" angle that lies in some interval of length $2\pi$. PAR (talk) 23:10, 27 January 2010 (UTC)