Circular Bayesian classifiers using wrapped Cauchy distributions

Abstract

A B S T R A C TCapturing the dependences among circular variables within supervised classification models is a challenging task. In this paper, we propose four different supervised Bayesian classification algorithms where the predictor variables follow all circular wrapped Cauchy distributions. For this purpose, we introduce four wrapped Cauchy classifiers. The bivariate wrapped Cauchy distribution is the only bivariate circular distribution whose marginals and conditionals are also wrapped Cauchy distributions, a property that makes it possible to define these models easily. Furthermore, the wrapped Cauchy tree-augmented naive Bayes classifier requires the definition of a conditional circular mutual information measure between variables that follow wrapped Cauchy distributions. Synthetic data is used to illustrate, compare and evaluate the classification algorithms (including a comparison with the Gaussian TAN classifier, decision tree, random forest, multinomial logistic regression, support vector machine and simple neural network), leading to satisfactory predictive results. We also use a real neuromorphological dataset obtained from juvenile rat somatosensory cortex cells, where we measure the bifurcation angles of the dendritic basal arbors.characteristic that each variable is conditionally independent of those that are non-descendants in the graph given the value of their parents. Therefore the joint probability distribution is expressed as the product of the local distributions conditioned to their parents. For these reasons, Bayesian networks can deal efficiently with supervised classification i.e., the Bayesian network classifiers [13] and offer an explicit, graphical and interpretable representation of uncertain knowledge, which has made it possible to successfully apply them to real-world problems.Supervised classification [19] deals with the problem of assigning a label to an instance, based on a set of variables that characterize it. Yet circular data has been commonly treated as linear data in supervised classification tasks. Only a few circular classifiers exist, and almost none of them are based on the principles of Bayesian networks, capable of capturing multivariate relationships among variables. Most of them focus on discriminant analysis and assume several circular distributions such as the von Mises distribution [20], later extended to the von Mises-Fisher distribution [21]. There are also circular discriminant analysis studies for the Watson, Selby and Arnold distributions on the sphere [22,23]. SenGupta and Roy [24] used a classification discriminant rule based on the mean chord-length to classify a new observation into one of two different circular populations that are von Mises, when training samples are available for each of them. Also a likelihood ratio test based on a bootstrapping approach for classifying into two populations was proposed for linear and circular data [25]. Kirby and Miranda [26] proposed a variation of a neural network, including a circular node, which was able to ke...

Recent advances in directional statistics

“…Recently, Zhan et al (2019) reviewed the correlation coefficients available for toroidal data and proposed two new ones. In the context of Bayesian network modelling, Leguey et al (2019b) and Leguey et al (2019a) introduced mutual information measures of the dependence between circular and linear variables, and between two circular variables, respectively.…”

“…More recently, DiMarzio et al (2018b) considered nonparametric circular classification based on KDE and local logistic regression Pandolfo et al (2018a). studied the depth-versus-depth classifier for circular data Leguey et al (2019a). proposed Bayesian classification algorithms for WC-distributed circular predictors.…”

Recent advances in directional statistics

“…Pandolfo et al (2018a) studied the depth-versus-depth classifier for circular data. Leguey et al (2019a) proposed Bayesian classification algorithms for WC-distributed circular predictors.…”