Classification of observations into von Mises-Fisher populations with unknown parameters

Abstract

The problem of classification of an observation into von Mises-Fisher populations is considered when the concentration parameters and the mean directions are unknown. For two von Mises-Fisher distributions, we propose the restricted maximum likelihood estimators (MLEs), and Bayes estimators of the concentration parameters. The MLEs and restricted MLEs of the concentration parameters are compared in terms of risks. When the concentration parameters are ordered, we propose classification rules using the restricted MLEs and Bayes estimators of the parameters. For two populations, we also derive predictive Bayes classification rules using informative priors for the concentration parameters. We derive the likelihood ratio-based classification rule. Nonparametric rules such as k-NN rule, support vector machine classifier, and kernel density-based rule are also studied. For more than two populations, we suggest classification rules. Comparisons among the proposed rules have been carried out with respect to the expected probability of misclassification. Applications of the rules are described using directional data sets.

Keywords:

• Bayes classifier
• Discriminant function
• Likelihood approach
• Maximum likelihood classification
• Nonparametric rule
• Probability of misclassification

AMS CLASSIFICATION:

• 62H30
• 62F30

Acknowledgements

We are thankful to the reviewer and the editor-in-chief for their valuable comments and suggestions which led to the substantial improvement of the manuscript. The authors gratefully acknowledge the financial support (Grant No. ECR/2017/000255) received from the Science and Engineering Research Board, Department of Science and Technology, Govt. of India for the research work.

Additional information

Funding

The research work is financially supported by the Science and Engineering Research Board (Department of Science and Technology), Govt. of India (Grant No. ECR/2017/000255).