Functional Principal Component Analysis for Continuous non-Gaussian, Truncated, and Discrete Functional Data

Abstract: Mobile health studies often collect multiple within-day self-reported assessments of participants' behavior and well-being on different scales such as physical activity (continuous), pain levels (truncated), mood states (ordinal), and life events (binary). These assessments, when indexed by time of day, can be treated as functional data of different types - continuous, truncated, ordinal, and binary. We develop a functional principal component analysis that deals with all four types of functional data in a unified manner. It employs a semiparametric Gaussian copula model, assuming a generalized latent non-paranormal process as the underlying mechanism for these four types of functional data. We specify latent temporal dependence using a covariance estimated through Kendall's tau bridging method, incorporating smoothness during the bridging process. Simulation studies demonstrate the method's competitive performance under both dense and sparse sampling conditions. We then apply this approach to data from 497 participants in the National Institute of Mental Health Family Study of the Mood Disorder Spectrum to characterize within-day temporal patterns of mood differences among individuals with major mood disorder subtypes, including Major Depressive Disorder, Type 1, and Type 2 Bipolar Disorder.