Covariance Estimation and Principal Component Analysis for Mixed-Type Functional Data with application to mHealth in Mood Disorders
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Mobile digital health (mHealth) studies often collect multiple within-day self-reported assessments of participants' behaviour and health. Indexed by time of day, these assessments can be treated as functional observations of continuous, truncated, ordinal, and binary type. We develop covariance estimation and principal component analysis for mixed-type functional data like that. We propose a semiparametric Gaussian copula model that assumes a generalized latent non-paranormal process generating observed mixed-type functional data and defining temporal dependence via a latent covariance. The smooth estimate of latent covariance is constructed via Kendall's Tau bridging method that incorporates smoothness within the bridging step. 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. Importantly, the proposed framework handles all four mixed types in a unified way. Simulation studies show a competitive performance of the proposed method under both dense and sparse sampling designs. The method is applied to data from 497 participants of National Institute of Mental Health Family Study of the Mood Disorder Spectrum to characterize the 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.
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