  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.
