In this dissertation, we consider estimations with regularization in three statistical problems relating to linear time system, stochastic block model and matrix variate regression. In the first part of the dissertation, we specifically investigate Mixture Multilayer Stochastic Block Model, where layers can be partitioned into groups of similar networks, and networks in each group are equipped with a distinct Stochastic Block Model. The goal is to partition the multilayer network into clusters of similar layers, and to identify communities in those layers. The Mixture Multilayer Stochastic Block Model was introduced by Bing-yi Jing  and a clustering methodology, TWIST, is developed based on regularized tensor decomposition. We propose a different technique, an alternating minimization algorithm (ALMA), that aims at simultaneous recovery of the layer partition, together with estimation of the matrices of connection probabilities of the distinct layers. Compared to TWIST, ALMA achieves higher accuracy both theoretically and numerically. The second part of the dissertation studies the estimation problems in discrete linear time invariant system in collaborative sensor networks, especially the sensor responses are maliciously manipulated. A series of distributed estimating schemes based on Kalman filtering are proposed to recovery the true states of the linear system when the system/sensor noises present, and the sensor measurements are falsified. In this dissertation, we propose various robust methods against the sensor attacks and provide their performance analysis theoretically and numerically. In matrix regressions, it is often the case that the true signal is of a low rank structure, or can be well approximated by a low rank structure. As such, sparsity is in terms of the rank of the matrix parameters, which is intrinsically different from sparsity in the number of nonzero entries. In this study, we proposed a new regularized matrix regression with rank constraint. Structural information of the matrix variate and the high-dimensionality are two key issues for solving the matrix regression. By introducing the rank constraint and regularization, our proposed method, bilinear matrix regression estimator, are able to incorporate high-dimensional matrix-valued predictor and vector-valued predictor.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Yu, Feng, "Regularized Estimations in Some Statistical Problems" (2021). Electronic Theses and Dissertations, 2020-. 1165.