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I've been doing a bunch of work as of late with data that is well-described by (mixtures of) von Mises-Fisher (vmf) distributions:

$$ \mathscr{F} \left(x \, \lvert \, \mu, \lambda \right) = C_D \left(\lambda \right) e^{\lambda \left<\mu, x\right>} \,\,\, x,\mu \in \mathbb{S}^{D-1}, \, \lambda \in \mathbb{R}_+ $$

which have the drawback for my situation that the density of this distribution will always be axially symmetric about the centroid $\mu$. What I'm wondering is if there exists a modification / variant of this distribution that can provide a density that is more 'elliptical' relative to $\mu$? Intuitively, it seems like this should be possible via modification of the parameter $\lambda$, but the ideas I'm coming up with are... difficult, and it's generally been my experience that most things that seem this tricky have already been worked out by somebody else smarter than myself.

So basically, the question is does the distribution I'm looking for exist? And if so what is it referred to as? Then I can go about locating some references on the subject.

Thanks!

Matt
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1 Answers1

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It does indeed exist and is called the Kent distribution https://en.wikipedia.org/wiki/Kent_distribution. Also note that the axial symmetry you are referring to is caused by the inner product $\langle\mu,x\rangle$. Thus, modification of $\lambda$ alone is insufficient to make the distribution more 'elliptical'.

In general, when dealing with directional statistics, Mardia's book on the topic is certainly worth a look.

LoS
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  • Thanks! I didn't notice your reply until just now, and it's exactly what I was looking for! Also, I realize that the $\left<\mu,x\right>$ term is what's causing the symmetry I'm referring to, so I was in fact looking for a modification along the lines of $\left<x,Mx\right> + \left<\mu,x\right>$ and there it is! Thanks! – Matt May 04 '16 at 02:03