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

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 scale), pain levels (truncated scale), mood states (ordinal scale), and the occurrence of daily life events (binary scale). These assessments, when indexed by time of day, can be treated and analyzed as functional data corresponding to their respective types: continuous, truncated, ordinal, and binary. Motivated by these examples, 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 generating 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 in the bridging process. The approach is then extended with methods for handling both dense and sparse sampling designs, calculating subject-specific latent representations of observed data, latent principal components and principal component scores. Simulation studies demonstrate the method's competitive performance under both dense and sparse sampling designs. The method is applied to data from 497 participants in the National Institute of Mental Health Family Study of Mood Spectrum Disorders to characterize differences in within-day temporal patterns of mood in individuals with the major mood disorder subtypes, including Major Depressive Disorder and Type 1 and 2 Bipolar Disorder. Software implementation of the proposed method is provided in an R-package.