Interpretable Analysis of Multivariate Functional DataZhang, Jun (2020) Interpretable Analysis of Multivariate Functional Data. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractMultivariate functional data have become increasingly popular in medical studies. Existing approaches to conducting multivariate functional data analysis (FDA) are limited in that they are difficult to interpret, since the estimates are nontrivial function of each variate of all the time points. This dissertation proposes novel interpretable multivariate FDA in two specific aspects: principal component analysis (PCA) and linear discriminant analysis (LDA) that provide scientifically interpretable components or classifiers, which are both sparse among variates and localized in time within variates. In the first part of the dissertation we develop a novel approach to conducting interpretable PCA on multivariate multilevel functional data. We decompose the total variation into subject-level and replicate-within-subject-level variation. The sparsity and localization of components for each level is obtained through a novel localized sparse-variate functional PCA (LVPCA) achieved by solving an innovative rank-one based convex optimization problem with block Frobenius and matrix L1-norm based penalties. We apply the proposed method to the Blunted and Discordant Affect study to summarize the joint variation across multiple frequency bands for both subject-level whole-brain response as well as electrode-level response and identify correlates with blunted affect. In the second part of the dissertation we develop a novel approach to conducting interpretable LDA on multivariate functional data. The proposed approach is a two-step procedure that first projects the high-dimensional data along localized functional basis through LVPCA proposed before, then performs sparse LDA on the low-dimensional space to select basis relevant to the between-class covariance. We apply the proposed methods to the AgeWise study to uncover physiological discriminants between older adults with and without poor self-reported sleep quality using theta EEG, melatonin and temperature. Public health significance: Psychiatric disorders such as blunted affect and disturbed sleep present a large public health burden. As of yet, the physiological and neurophysiological mechanisms behind blunted affect and disturbed sleep are not well understood. Interpretable PCA and LDA using multivariate functional processes can describe variability in physiological activity and reveal discriminants of clinical manifestations, which can help scientists uncover the underlying biological mechanisms and potentially guide treatment. Share
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