On Sequential Estimation of Multivariate Associations
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In multivariate analysis, one of the major problems of interest is to model multivariate responses using observed predictors. We often encounter responses of either continuous, binary or count types, or may be of mixed types. Multivariate linear regression (MLR) models the relationship in case of Gaussian outcomes. When outcomes are non-Gaussian or mixed types, i.e., generalized, a possible modeling approach proceeds via maximization of likelihood obtained after assuming conditionally independent observed outcomes are from exponential dispersion family. In high-dimensional setting, responses maybe interrelated and predictors maybe correlated or unimportant. Such dependency can be induced through a low-rank and sparse coefficient matrix which also facilitates model interpretation. Specifically, the structure translates into having co-sparse left and right singular vectors in the singular value decomposition of the coefficient matrix. In this thesis, we have proposed algorithms to recover such matrices. In MLR, we reformulate the problem as a supervised co-sparse factor analysis, and develop an efficient computational procedure, named sequential factor extraction via co-sparse unit-rank estimation (SeCURE). A unit-step in SeCURE extracts a sparse and unit-rank coefficient matrix leading to co-sparsity in corresponding singular vectors. In the generalized setting, motivated by SeCURE, we propose a sequential procedure to recover the desired coefficient matrices, named as generalized sequential factor extraction via co-sparse unit-rank estimation (GSeCURE). Because of the complicated likelihood structure, a unit-step of GSeCURE estimates co-sparse singular vectors via iteratively optimizing a surrogate of the objective function. Efficacy of both SeCURE and GSeCURE are demonstrated by simulation studies and various applications.