Talk:Wrapped normal distribution

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Wrapped Normal Distribution[edit]

(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.


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[edit]

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. (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[edit]

  • 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. (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)