A circular-linear dependence measure under Johnson–Wehrly distributions and its application in Bayesian networks

Abstract

Circular data jointly observed with linear data are common in various disciplines. Since circular data require different techniques than linear data, it is often misleading to use usual dependence measures for joint data of circular and linear observations. Moreover, although a mutual information measure between circular variables exists, the measure has drawbacks in that it is defined only for a bivariate extension of the wrapped Cauchy distribution and has to be approximated using numerical methods. In this paper, we introduce two measures of dependence, namely, (i) circular-linear mutual information as a measure of dependence between circular and linear variables and (ii) circular-circular mutual information as a measure of dependence between two circular variables. It is shown that the expression for the proposed circular-linear mutual information can be greatly simplified for a subfamily of Johnson-Wehrly distributions. We apply these two dependence measures to learn a circular-linear tree-structured Bayesian network that combines circular and linear variables. To illustrate and evaluate our proposal, we perform experiments with simulated data. We also use a real meteorological data set from different European stations to create a circular-linear tree-structured Bayesian network model.

cylcop: An R Package for Circular-Linear Copulae with Angular Symmetry

“…the Kullback-Leibler divergence of the product of the marginal distributions from the joint distribution (see e.g. Leguey et al, 2019).…”

“…We also implemented a correlation measure based on mutual information, i.e., the Kullback-Leibler divergence of the product of the marginal distributions from the joint distribution (see e.g., Leguey, Larrañaga, Bielza, and Kato 2019).…”

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.…”

Clustering based on Bayesian networks with Gaussian and angular predictors : applications in Neuroscience

“…Probabilistic graphical models have been widely applied in research fields whose data present directional variables, as for example biochemistry [Boomsma et al, 2006[Boomsma et al, , 2008Harder et al, 2010;Razavian et al, 2011b,a], neuroscience [Leguey, 2018], meteorology [Leguey et al, 2019] or machine learning [López-Cruz et al, 2013], mainly because they allow to obtain tractable models in a continuous space. It is worth noting that generalisation from the univariate to the multivariate case on directional statistics is not immediate given that high dimensional spaces in these cases encompass several geometric spaces as the sphere, the torus, and the cylinder (see Figure 3.2).…”