Parameter estimation for von mises-fisher mixture model via Gaussian distribution

Directional statistics deal with direction, such as angles and phases. A well-known distribution in directional statistics is von Mises-Fisher (vMF) distribution, which is Gaussian distribution on a unit hypersphere. For a vMF mixture model, a maximum likelihood estimator and a variational Bayes estimator have already been derived. However, an iterative algorithm for finding the maximum likelihood estimator may accumulate approximation error. Besides, the variational Bayes estimator cannot estimate some parameters. This paper derives an estimator of the parameters in the vMF mixture model via the Gaussian distribution to solve these problems. We focus on the fact that the vMF distribution is derived from the Gaussian distribution. At first, we apply the estimation for the Gaussian mixture model to observed samples. Then, we convert the estimated Gaussian mixture distribution to a vMF mixture distribution. Experimental results support the analysis.