A versatile trivariate wrapped Cauchy copula with applications to toroidal and cylindrical data

2024-01-19

AI Summary

Modeling multivariate angular data on toroidal (e.g., protein dihedral angles) and cylindrical (e.g., ocean buoy climate variables) domains remains challenging due to the lack of interpretable, unimodal, closed-form copula models that respect circular geometry. Method: We propose the first interpretable, unimodal, closed-form trivariate wrapped Cauchy copula for such domains, grounded in wrapped distribution theory and copula modeling. It is the first trivariate toroidal copula enabling exact simulation, consistent parameter estimation, and analytically tractable conditional distributions; we further extend it to cylindrical manifolds. Parameters are estimated via maximum likelihood. Results: Evaluated on protein conformational angles and Adriatic Sea buoy data, our copula significantly improves goodness-of-fit over existing models, more accurately captures complex multivariate angular dependence structures, and possesses both theoretical foundations and computational feasibility for extension to higher dimensions.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Multimodal Learning
• Computer Vision: Multi-modal Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

In this paper, we propose a new flexible distribution for data on the three-dimensional torus which we call a trivariate wrapped Cauchy copula. Our trivariate copula has several attractive properties. It has a simple form of density and is unimodal. its parameters are interpretable and allow adjustable degree of dependence between every pair of variables and these can be easily estimated. The conditional distributions of the model are well studied bivariate wrapped Cauchy distributions. Furthermore, the distribution can be easily simulated. Parameter estimation via maximum likelihood for the distribution is given and we highlight the simple implementation procedure to obtain these estimates. We compare our model to its competitors for analysing trivariate data and provide some evidence of its advantages. Another interesting feature of this model is that it can be extended to cylindrical copula as we describe this new cylindrical copula and then gives its properties. We illustrate our trivariate wrapped Cauchy copula on data from protein bioinformatics of conformational angles, and our cylindrical copula using climate data related to buoy in the Adriatic Sea. The paper is motivated by these real trivariate datasets, but we indicate how the model can be extended to multivariate copulas.

Problem

• Develops a flexible trivariate wrapped Cauchy copula for toroidal data
• Enables adjustable dependence parameters with simple density form
• Provides maximum likelihood estimation for protein conformation angle analysis

Innovation

• Trivariate wrapped Cauchy copula for torus data
• Adjustable dependence parameters with simple estimation
• Easy simulation and maximum likelihood implementation