🤖 AI Summary
Existing methods struggle to effectively model the joint dependence between discrete circular variables—such as finitely many equally spaced directional observations—and continuous linear variables. This work proposes the first analytically tractable joint model, employing a wrapped symmetric geometric distribution for the discrete circular component and a Weibull distribution for the linear component, with their dependence structure induced via a trigonometric linking function. The model yields closed-form marginal and conditional distributions along with conditional moments, offers clear theoretical interpretations of its parameters, and establishes monotonicity of the conditional mean and variance with respect to the dependence parameter. Moreover, it enables direct sampling through the inverse transform method. Simulation studies demonstrate excellent parameter estimation performance, and applications to two real-world environmental datasets underscore the model’s effectiveness and practical utility.
📝 Abstract
Standard statistical methods are often inadequate for modeling the joint dependence between linear and circular variables, and existing methods for modeling this dependence are designed only for continuous variables. However, circular data are frequently observed on a finite set of equally spaced directions, either due to rounding prior to reporting or because of the experimental design employed for data collection. To address this gap, we propose a flexible, analytically tractable model for jointly representing a discrete circular and a continuous linear variable. The construction combines a wrapped symmetric geometric distribution, a Weibull distribution, and a trigonometric linking function. This formulation yields closed-form expressions for the joint, marginal, and conditional distributions. The choice of the Weibull distribution facilitates direct sample generation using the inverse transform technique. Additionally, it provides explicit expressions for conditional moments, enabling a flexible circular-linear regression framework. We detail the theoretical interpretation of the model parameters, mathematically establishing the monotonicity of the conditional mean and variance with respect to the dependence parameters. The performance of the estimators is demonstrated through extensive simulations, and the utility of the model is illustrated by analyzing two empirical environmental datasets.