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.

翻译：生物力学与骨科研究常遇到包含圆形变量和线性变量的复杂数据集。多数情况下，圆形变量与线性变量（i）被孤立考虑，忽略了变量间的依赖关系；（ii）圆形变量的周期性被忽视，导致决策错误。鉴于圆形变量的内在特性，必须采用整合方向统计学的方法以实现精确建模。本文以生物力学数据（即骨折位移数据）的建模为动机，这类数据常用于外固定架对比研究。我们聚焦于基于Ilizarov环式外固定架的数据集，该数据集包含六个变量。提出一种基于藤copula的框架，适用于圆环-线性数据的六维联合分布建模。藤copula的配对copula分解概念将依赖结构表示为圆环-线性、圆环-圆环和线性-线性三组配对变量的组合，每组配对分别由其对应的copula建模。该框架既能评估联合分布中的依赖关系，又能考虑圆形变量的周期性。由此，为Ilizarov环式外固定架及其他此类数据的力学行为精确建模提供了新方法。