Talk:Wrapped normal distribution

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

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

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

My Proposal

$f_{WN}(\theta ;\mu ,\sigma )={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{-\sigma ^{2}n^{2}/2+in(\theta -\mu )}={\frac {1}{2\pi }}\vartheta _{3}\left({\frac {\theta -\mu }{2}},e^{-\sigma ^{2}/2}\right)$

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)

Comments on Points

• 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 thi
• "Jacobi elliptic" was an abuse of the language on my part. Jacobi theta functions are viewed by some as elliptic versions of the exponential functions.
• The summation is correct as stated. I think the misleading part is the Jacobi theta article, and the different notations used.
• Hmm, so what would be a good definition? I'm off to make sense of the Jacobi theta article.

Shawn@garbett.org (talk) 20:46, 27 January 2010 (UTC)

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)

It is always much, much better ...

... if an article like this does not immediately define its subject in the greatest possible generality.

Rather, it is far better to define the most common simple case (zero mean, unit variance) before describing the most general formulation.

A lot of contributors seem to forget or not care that readers who don't know much about the subject of a Wikipedia article absorb the information best by seeing the simple version before seeing the situation in full generality.

But this article, heedlessly, barrels into the full generality version instead.