Incorporating covariates into functional principal component analysis (PCA) can substantially improve the representation efficiency of the principal components and predictive performance. However, many existing functional PCA methods do not make use of covariates, and those that do often have high computational cost or make overly simplistic assumptions that are violated in practice. In this article, we propose a new framework, called Covariate Dependent Functional Principal Component Analysis (CD-FPCA), in which both the mean and covariance structure depend on covariates. We propose a corresponding estimation algorithm, which makes use of spline basis representations and roughness penalties, and is substantially more computationally efficient than competing approaches of adequate estimation and prediction accuracy. A key aspect of our work is our novel approach for modeling the covariance function and ensuring that it is symmetric positive semi-definite. We demonstrate the advantages of our methodology through a simulation study and an astronomical data analysis.

翻译：将协变量纳入函数主成分分析可以显著提升主成分的表达效率和预测性能。然而，现有许多函数主成分分析方法未利用协变量信息，而利用协变量的方法往往计算成本高昂或做出过于简化且在实际中难以成立的假设。本文提出一种名为协变量依赖函数主成分分析（CD-FPCA）的新框架，其中均值函数和协方差结构均依赖于协变量。我们设计了一种相应的估计算法，该算法利用样条基表示和粗糙度惩罚，在保证估计与预测精度的前提下，计算效率显著优于同类方法。本研究的关键创新在于提出一种协方差函数建模方法，并确保其对称半正定性。通过模拟研究与天文数据分析，我们验证了所提方法的优势。