A dependent circular-linear model for multivariate biomechanical data: Ilizarov ring fixator study

TL;DR

This paper introduces a novel vine copula-based framework for accurately modeling multivariate biomechanical data with circular and linear variables, capturing dependencies and cyclicity often neglected in traditional methods.

Contribution

It develops a new joint distribution model for circular-linear data using vine copulas, specifically tailored for biomechanical applications like Ilizarov ring fixator analysis.

Findings

Effective modeling of dependencies between circular and linear variables.

Incorporation of cyclicity improves model accuracy.

Framework applicable to other biomechanical datasets.

Abstract

Biomechanical and orthopaedic studies frequently encounter complex datasets that encompass both circular and linear variables. In most cases the circular and linear variables are (i) considered in isolation with dependency between variables neglected and (ii) the cyclicity of the circular variables disregarded resulting in erroneous decision making. Given the inherent characteristics of circular variables, it is imperative to adopt methods that integrate directional statistics to achieve precise modelling. This paper is motivated by the modelling of biomechanical data, i.e., the fracture displacements, that is used as a measure in external fixator comparisons. We focus on a data set, based on an Ilizarov ring fixator, comprising of six variables. A modelling framework applicable to the 6D joint distribution of circular-linear data based on vine copulas is proposed. The pair-copula decomposition concept of vine copulas represents the dependence structure as a combination of circular-linear, circular-circular and linear-linear pairs modelled by their respective copulas. This framework allows us to assess the dependencies in the joint distribution as well as account for the cyclicity of the circular variables. Thus, a new approach for accurate modelling of mechanical behaviour for Ilizarov ring fixators and other data of this nature is imparted.