Using von Mises-Fisher distributions for geo-spatial machine learning There's an interesting paper about predicting the geographical co-ordinates of Twitter users based on the kinds of words that they use in their posts. I'd like to do something similar that involves taking text and using it to predict a subject's latitude and longitude.
However, the approach mentioned in the above paper isn't ideal, because it ignores the spherical-ness of the Earth. Here's a quote from another paper:

The earth’s surface is continuous, so a natural approach is to predict
  locations using a continuous distribution. For example, Eisenstein et
  al. (2010) use Gaussian distributions to model the locations of
  Twitter users in the United States of America. This appears to work
  reasonably well for that restricted region, but is likely to run into
  problems when predicting locations for anywhere on earth—-instead,
  spherical distributions like the von Mises-Fisher distribution would
  need to be employed.

I'd like to try using the von Mises-Fisher distribution to perform ML tasks with spatial data, but I just don't know how. Does anyone know of a paper or text book that describes in more detail how to do something like this? I've read about some approaches that involve discretizing the earth into little squares, but I'd prefer to avoid that sort of solution, if I can. Also, if using von Mises-Fisher is the wrong way to go, feel free to suggest going in a completely different direction.
Thanks for your help.
Edit: This question is not an easy one to answer. At this point, I think that I've done a pretty exhaustive search through the literature, and I haven't found a paper that both (a) predicts lat/lon co-ordinates with text or whatever other features and (b) doesn't treat the earth like a flat plane. If anyone else found a paper that met those two requirements, I would be quite surprised. Anyway, as a last-ditch effort, I'll add a bounty to this question. I'll also list all of the lat/lon prediction papers that I know of below, for others that may research this topic later on:


*

*A latent variable model for geographic lexical variation <-- The one mentioned above in the question

*Estimating User Location in Social Media with Stacked Denoising Auto-encoders <-- Uses deep learning techniques, but I don't see any directional statistics

*Inferring the Origin Locations of Tweets with Quantitative Confidence <-- Returns probability densities, the paper contains a note that mentions "plate carrée"

*Sparse Additive Generative Models of Text <-- See "Application 4" for lat/lon prediction


Edit 2: One of the below comments asked me to briefly describe the model used in the paper by Eisenstein et al. Unfortunately, that paper is quite dense and so I think that it may make more sense to summarize a simpler method used by one of the other papers instead.
The simpler paper is based on Gaussian Mixture Models (GMMs). Training consists of tokenizing all of the tweets to create a list of all the words used; if a word happens to be used 1000 times (for example), then that means that there will be 1000 locations that map to that word in the data (taken from the geotagged tweets). The locations are used to fit a 2D GMM for each of the words. This collection of many fitted GMMs is the trained model.
When the user wishes to estimate the location for a tweet, it is tokenized in the same way, and the GMMs for each of the words are combined. The sum of all the GMMs can then be examined to find the location with the highest probability of producing the tweet under consideration.
Unfortunately, the use of 2D Gaussian distributions could be problematic, since they have no way of taking into account the 'wrapping around' at the international date line. As mentioned above, the use of von Mises-Fisher distributions have been suggested as a possible method for resolving this problem, but I'm not knowledgeable enough to say for sure if this is the best way forward. Any suggestions or ideas would be welcome.
 A: During my research in search systems design I met that article once upon a time: http://www.jmlr.org/papers/volume6/banerjee05a/banerjee05a.pdf
This paper is pretty old, but is well written and proposes a generative mixture-model approach to clustering directional data  based on the von Mises-Fisher (vMF) distribution, which arises naturally for data distributed on the unit hypersphere.  In particular, they
derive and analyze two variants of the Expectation Maximization (EM) framework for estimating the mean and concentration parameters of this mixture. 
But my extensive experience in solving exactly your task suggests, that when you rely totally on association of words and locations you are not in a right track. Even short twitter texts are very free form, contain slang and homonyms and if some tweet contain no toponyms at all you have no chance to reliably estimate location of tweet.
I don't know what task you really solve, but if I had to estimate location from tweets, I would consider all possible sources of information and form an ensemble.
Those sources could be, for example:


*

*Social connections ( if I know for sure, that for some user X his friends just checked in at Paris and user X mentioned some French toponyms in some tweets, then my prior ( in Bayesian sense ) that X is in Paris is more, that if I had no idea about X's friends location ). Also bursts of mutual amount of messeges via connections could be important.

*Tweets sequence for each user. No one mention Paris and Seine in each tweet just to please data scientist :-) So, in Bayesian sense you are better to increase scores for Paris for each tweet, which coincides in time with tweets with toponyms.

*User photos. This resource is very valuable and using recent advances in deep learning and convolutional networks it's typically very simple to uncover full potential of photos

*Time difference between 'typical' user tweets and tweets for last day/3 days/week. If a user is in trip over a Paris ( but typically he lives in US ) then it's likely, that his tweets will suddenly have different time zone ( during this trip )
Actually, these sources are fractal-like, and can be combined in multiple ways, so I know no better tool than ensembling these evidences.
