Principal component analysis of hybrid functional and vector data

We propose a practical principal component analysis (PCA) framework that provides a nonparametric means of simultaneously reducing the dimensions of and modeling functional and vector (multivariate) data. We first introduce a Hilbert space that combines functional and vector objects as a single hybrid object. The framework, termed a PCA of hybrid functional and vector data (HFV-PCA), is then based on the eigen-decomposition of a covariance operator that captures simultaneous variations of functional and vector data in the new space. This approach leads to interpretable principal components that have the same structure as each observation and a single set of scores that serves well as a low-dimensional proxy for hybrid functional and vector data. To support practical application of HFV-PCA, the explicit relationship between the hybrid PC decomposition and the functional and vector PC decompositions is established, leading to a simple and robust estimation scheme where components of HFV-PCA are calculated using the components estimated from the existing functional and classical PCA methods. This estimation strategy allows flexible incorporation of sparse and irregular functional data as well as multivariate functional data. We derive the consistency results and asymptotic convergence rates for the proposed estimators. We demonstrate the efficacy of the method through simulations and analysis of renal imaging data.