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.

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