Multivariate functional principal component analysis (MFPCA) is a powerful dimension reduction technique for analyzing multiple functional variables simultaneously. However, existing MFPCA methods assume that all functional observations are recorded over a common, fixed domain. This assumption is often violated in practical applications where the observation period varies across subjects, leading to what is known as variable domain functional data. We propose a novel approach for MFPCA that explicitly accommodates variable domains by extending existing multivariate functional principal component analysis to the variable domain setting. Our methodology involves performing univariate variable domain FPCA for each functional variable separately, stacking the resulting univariate scores, and then smoothing the empirical covariance matrix of these stacked scores over the domain length. This allows us to estimate multivariate eigenfunctions and scores that properly account for varying observation periods. We demonstrate through extensive simulation studies that our proposed method outperforms approaches that ignore the variable domain structure and rely on binning strategies. The practical utility of our method is illustrated through an application analyzing body temperature and capillary oxygen saturation (SpO$_2$) trajectories in COVID-19 hospital admitted patients, where patients experienced varying lengths of stay and monitoring periods.

翻译：多变量函数主成分分析（MFPCA）是一种同时分析多个函数变量的强大降维技术。然而，现有的MFPCA方法假设所有函数观测值均在共同固定域上记录。在实际应用中，当观测周期因受试者而异时，这一假设常被违反，从而产生所谓的变域函数数据。我们提出了一种新的MFPCA方法，通过将现有的多变量函数主成分分析扩展到变域设定，明确适应可变域。我们的方法包括：分别对每个函数变量进行单变量变域FPCA，堆叠所得的单变量得分，然后对这些堆叠得分的经验协方差矩阵在域长度上进行平滑处理。这使得我们能够估计正确考虑不同观测周期的多变量特征函数和得分。通过广泛的模拟研究，我们证明所提出的方法优于忽略变域结构并依赖分箱策略的现有方法。我们通过分析COVID-19住院患者的体温和毛细血管血氧饱和度（SpO$_2$）轨迹的应用实例，展示了该方法的实用价值——这些患者经历了不同的住院时间和监测周期。