Model-Based Clustering for Cylindrical Data

Abstract

The objective of this paper is to perform clustering based on data consisting of both linear and circular variables, that is the data that lie on the surface of a cylinder. There are many circular–linear distributions available in the literature. We use the pragmatic approach of specifying the conditional rather than the marginal, which is often easier. Adopting Arnold et al. (Lecture Notes in Statistics: Conditionally Specified Distributions, Springer Verlag Publisher, Berlin Heidelberg, 1992), we provide the conditional distribution of θ given x and that of x given θ. Here, a mixture model approach based on the joint distribution of the linear and the circular variable is proposed. In particular, two types of such mixture models are used. One is based on the joint distribution of the marginal distribution of the linear variable and the conditional distribution of the circular variable given the linear variable and the other vice versa. Convergence property of Expectation Maximization (EM) algorithm for the members of the curved exponential family used for our models is studied. A real-life application on meteorological data is made of the proposed approaches. Comparison of the two models is done based on this example. The distinctive and important feature of preserving the geometry of the cylindrical manifold by our clustering method and its marked deviation from that for data on $\Re ^p$ is also revealed by this example.