Directional-linear data clustering using structural expectation-maximization algorithm

Abstract

The study of plethora of phenomena requires the measurement of their magni- tude and direction as in meteorology (Carta et al. 2009), rhythmometry, medicine or demography (Batschelet 1981, Batschelet et al. 1973). Probabilistic cluster- ing of this data is typically tackled by means of mixtures of Gaussians (Fraley and Raftery 2002, McLachlan and Basford 1988, Melnykov and Maitra 2010), although they tend to underperform due to their inability to handle periodic- ity of directional data. To address this problem several distributions have been proposed to cluster bivariate cylindrical data (Carta et al. 2009, Gatto and Jam- malamadaka 2007, Mardia and Sutton 1978, Qin et al. 2010) and multivariate data having one circular variable (Roy et al. 2014). Recently, an approach (Luengo-Sanchez et al. 2016) based on exploiting the con- ditional independence assumptions encoded by a Bayesian network enables effi- cient clustering of multivariate directional-linear data, distributed as Gaussian and von Mises respectively, even when there is more than one directional variable by means of the structural expectation-maximization algorithm (Friedman 1997). However, strong constraints on the structure of the Bayesian network must be imposed. Here we propose measures of divergence and distance among clusters, as Kullback- Leibler divergence and Bhattacharyya distance, for the previous model to evalu- ate the quality of the clustering outcomes and we extend the model by relaxing the structural constraints to include relations of dependence between directional variables and Gaussians. We present an application for neuroscience to cluster dendritic spines according to a set of morphological features that combine direc- tional and linear variables.