An Expectation–Maximization algorithm for the Wishart mixture model: Application to movement clustering

This article presents an Expectation–Maximization algorithm for the Wishart mixture model in which realizations are matrices. Given a set of matrices, an iterative algorithm for estimating the parameters of such a mixture model is proposed. The obtained estimates can be interpreted in terms of mean matrices and scale factors. By applying the maximum a posteriori rule, we get an algorithm for the clustering of a set of matrices. This mixture model is then modified in order to deal with a set of samples. Unfortunately, the samples may be of different sizes. We propose to tackle this problem by considering the cross-product matrix as a signature for each sample. This set of cross-product matrices may be fitted with the proposed Wishart pseudo-mixture model in which the scale parameters of the distribution are not estimated but fixed. Again, we easily get a clustering algorithm from final parameter estimates. The different estimators are studied empirically through an analysis of their bias and variance and are validated onto an artificial dataset. Finally, we apply the Wishart pseudo-mixture model for analyzing motion-captured movements. Given the successive 3D positions of markers over the time, a cross-product matrix is constructed for each movement and put into the proposed classifier. We observe that the recognition rates are higher with our proposed approach than those with other geometric methods. Limits and constraints of the provided models are finally discussed.