Abstract We introduce a novel method of principal component analysis for data with varying domain lengths for each functional observation. We refer to this technique as variable-domain functional principal component analysis, or vd-FPCA. We fit a trivariate smoother using penalized thin plate splines to estimate the covariance as a function of the domain length. Principal components are then calculated through eigen-decomposition of the estimated covariance matrix, conditional on the domain length. We apply vd-FPCA in two functional data settings, first to daily measures of patient wellness during a stay in the ICU, and second, to accelerometer recordings of repeated in-lab movements. In each example, vd-FPCA uses fewer principal components than typical FPCA methods to explain a greater proportion of the variation in the data. We also find the principal components provide greater flexibility in interpretation with respect to domain length than traditional approaches. These methods are easily implementable through standard statistical software and applicable to a wide variety of datasets involving continuous observations over a variable domain. Supplementary materials for this article are available online.

中文翻译：

---

变域泛函主成分分析

摘要 我们介绍了一种新的主​​成分分析方法，用于针对每个功能观察具有不同域长度的数据。我们将这种技术称为可变域函数主成分分析，或 vd-FPCA。我们使用惩罚薄板样条拟合三变量平滑器来估计作为域长度函数的协方差。然后通过估计协方差矩阵的特征分解计算主成分，条件是域长度。我们在两个功能数据设置中应用 vd-FPCA，首先是在 ICU 住院期间患者健康的日常测量，其次是实验室重复运动的加速度计记录。在每个示例中，vd-FPCA 使用比典型 FPCA 方法更少的主成分来解释数据中更大比例的变化。我们还发现主成分在解释域长度方面比传统方法提供了更大的灵活性。这些方法可以通过标准统计软件轻松实现，并且适用于涉及对可变域的连续观察的各种数据集。本文的补充材料可在线获取。