Sequentially spherical data modeling with hidden Markov models and its application to fMRI data analysis

Wentao Fan, Lin Yang, Nizar Bouguila, Yewang Chen

Abstract: Due to the reason that spherical data (i.e. L2<math><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math> normalized vectors) are often involved with various real-life applications (such as anomaly detection, gesture recognition, intrusion detection in networks, gene expression data analysis, etc.), spherical data modeling has recently become an important research topic. In this work, we address the problem of modeling sequentially spherical data through continuous hidden Markov models (HMMs). Instead of adopting Gaussian mixture models (GMMs) as the emission distributions as in common continuous HMMs, we propose a continuous HMM by considering the mixture of von Mises–Fisher (VMF) distributions as its emission densities. Then, we systematically propose an effective method based on variational Bayes (VB) to learn the VMF-based HMM. The developed learning method has the following merits: (1) It is convergence-guaranteed; (2) It can be optimized with closed-form solutions. The proposed VMF–HMM with VB learning is validated by conducting experiments on both simulated sequential spherical data and a real application about fMRI data analysis.