Traditional Functional Principal Component Analysis typically focuses on densely observed univariate functional data, yet many applications, particularly in longitudinal studies, involve multivariate functional data observed sparsely and irregularly across subjects. A common approach for extracting multivariate functional principal components in such settings relies on an eigen decomposition of univariate functional principal component scores to capture cross-component correlations. We propose a new approach for the estimation of multivariate functional principal components by improving the univariate eigenanalysis through maximum likelihood estimation combined with a modified Gram-Schmidt orthonormalization. The performance of the proposed approach is evaluated against two established methods, and its practical utility is demonstrated through an application to longitudinal cognitive biomarker data from an Alzheimer's disease study and a collection of data on dairy milk yield and milk compositions from research dairy farms in Ireland.

翻译：传统函数主成分分析通常聚焦于密集观测的单变量函数数据，然而许多实际应用，尤其在纵向研究中，涉及跨受试者稀疏且不规则观测的多元函数数据。在此类情境下，提取多元函数主成分的常见方法依赖于对单变量函数主成分得分进行特征分解，以捕获跨成分相关性。我们提出一种估计多元函数主成分的新方法，该方法通过最大似然估计结合改进的Gram-Schmidt正交规范化，优化单变量特征分析。所提方法的性能与两种既有方法进行了对比评估，并通过其在阿尔茨海默病研究的纵向认知生物标志物数据以及爱尔兰研究型乳品农场乳产量与乳成分数据集中的应用，展示了其实用价值。