Biomechanics and human movement research often involves measuring multiple kinematic or kinetic variables regularly throughout a movement, yielding data that present as smooth, multivariate, time-varying curves and are naturally amenable to functional data analysis. It is now increasingly common to record the same movement repeatedly for each individual, resulting in curves that are serially correlated and can be viewed as longitudinal functional data. We present a new approach for modelling multivariate multilevel longitudinal functional data, with application to kinematic data from recreational runners collected during a treadmill run. For each stride, the runners' hip, knee and ankle angles are modelled jointly as smooth multivariate functions that depend on subject-specific covariates. Longitudinally varying multivariate functional random effects are used to capture the dependence among adjacent strides and changes in the multivariate functions over the course of the treadmill run. A basis modelling approach is adopted to fit the model -- we represent each observation using a multivariate functional principal components basis and model the basis coefficients using scalar longitudinal mixed effects models. The predicted random effects are used to understand and visualise changes in the multivariate functional data over the course of the treadmill run. In our application, our method quantifies the effects of scalar covariates on the multivariate functional data, revealing a statistically significant effect of running speed at the hip, knee and ankle joints. Analysis of the predicted random effects reveals that individuals' kinematics are generally stable but certain individuals who exhibit strong changes during the run can also be identified. A simulation study is presented to demonstrate the efficacy of the proposed methodology under realistic data-generating scenarios.

翻译：生物力学与人体运动研究通常涉及在运动过程中定期测量多个运动学或动力学变量，所产生的数据表现为平滑、多元、时变的曲线，天然适用于函数型数据分析。如今，对每个个体重复记录同一运动的情况日益普遍，这导致曲线存在序列相关性，可被视为纵向函数型数据。本文提出一种建模多元多层级纵向函数型数据的新方法，并将其应用于在跑步机跑步期间采集的休闲跑者运动学数据。针对每个步态周期，将跑者的髋、膝和踝关节角度联合建模为依赖于个体特异性协变量的平滑多元函数。采用纵向变化的多元函数型随机效应来捕捉相邻步态周期之间的依赖关系以及多元函数在跑步机跑步过程中的变化。模型拟合采用基函数建模方法——我们使用多元函数型主成分基表示每个观测值，并利用标量纵向混合效应模型对基系数进行建模。预测的随机效应用于理解和可视化多元函数型数据在跑步机跑步过程中的变化。在本研究中，我们的方法量化了标量协变量对多元函数型数据的影响，揭示了跑步速度对髋、膝和踝关节具有统计学显著效应。对预测随机效应的分析表明，个体的运动学特征总体稳定，但也能识别出某些在跑步过程中表现出显著变化的个体。本文还通过模拟研究验证了所提方法在现实数据生成场景下的有效性。