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    Sparse summaries of complex covariance structures : a thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics, School of Natural & Computational Sciences, Massey University, Auckland, New Zealand
    (Massey University, 2020) Bashir, Amir
    A matrix that has most of its elements equal to zero is called a sparse matrix. The zero elements in a sparse matrix reduce the number of parameters for its potential interpretability. Bayesians desiring a sparse model frequently formulate priors that enhance sparsity. However, in most settings, this leads to sparse posterior samples, not to a sparse posterior mean. A decoupled shrinkage and selection posterior - variable selection approach was proposed by (Hahn & Carvalho, 2015) to address this problem in a regression setting to set some of the elements of the regression coefficients matrix to exact zeros. Hahn & Carvallho (2015) suggested to work on a decoupled shrinkage and selection approach in a Gaussian graphical models setting to set some of the elements of a precision matrix (graph) to exact zeros. In this thesis, I have filled this gap and proposed decoupled shrinkage and selection approaches to sparsify the precision matrix and the factor loading matrix that is an extension of Hahn & Carvallho’s (2015) decoupled shrinkage and selection approach. The decoupled shrinkage and selection approach proposed by me uses samples from the posterior over the parameter, sets a penalization criteria to produce progressively sparser estimates of the desired parameter, and then sets a rule to pick the final desired parameter from the generated parameters, based on the posterior distribution of fit. My proposed decoupled approach generally produced sparser graphs than a range of existing sparsification strategies such as thresholding the partial correlations, credible interval, adaptive graphical Lasso, and ratio selection, while maintaining a good fit based on the log-likelihood. In simulation studies, my decoupled shrinkage and selection approach had better sensitivity and specificity than the other strategies as the dimension p and sample size n grew. For low-dimensional data, my decoupled shrinkage and selection approach was comparable with the other strategies. Further, I have extended my proposed decoupled shrinkage and selection approach for one population to two populations by modifying the ADMM (alternating directions method of multipliers) algorithm in the JGL (joint graphical Lasso) R – package (Danaher et al, 2013) to find sparse sets of differences between two inverse covariance matrices. The simulation studies showed that my decoupled shrinkage and selection approach for two populations for the sparse case had better sensitivity and specificity than the sensitivity and specificity using JGL. However, sparse sets of differences were challenging for the dense case and moderate sample sizes. My decoupled shrinkage and selection approach for two populations was also applied to find sparse sets of differences between the precision matrices for cases and controls in a metabolomics dataset. Finally, decoupled shrinkage and selection is used to post-process the posterior mean covariance matrix to produce a factor model with a sparse factor loading matrix whose expected fit lies within the upper 95% of the posterior over fits. In the Gaussian setting, simulation studies showed that my proposed DSS sparse factor model approach performed better than fanc (factor analysis using non-convex penalties) (Hirose and Yamamoto, 2015) in terms of sensitivity, specificity, and picking the correct number of factors. Decoupled shrinkage and selection is also easily applied to models where a latent multivariate normal underlies non-Gaussian marginals, e.g., multivariate probit models. I illustrate my findings with moderate dimensional data examples from modelling of food frequency questionnaires and fish abundance.
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    Computationally tractable fitting of graphical models : the cost and benefits of decomposable Bayesian and penalized likelihood approaches : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Albany, New Zealand
    (Massey University, 2012) Fitch, Anne Marie
    Gaussian graphical models are a useful tool for eliciting information about relationships in data with a multivariate normal distribution. In the rst part of this thesis we demonstrate that partial correlation graphs facilitate di erent and better insight into high-dimensional data than sample correlations. This raises the question of which method one should use to model and estimate the parameters. In the second, and major part, we take a more theoretical focus examining the costs and bene ts of two popular approaches to model selection and parameter estimation (penalized likelihood and decomposable Bayesian) when the true graph is non-decomposable. We rst consider the e ect a restriction to decomposable models has on the estimation of both the inverse covariance matrix and the covariance matrix. Using the variance as a measure of variability we compare non-decomposable and decomposable models. Here we nd that, if the true model is non-decomposable, the variance of estimates is demonstrably larger when a decomposable model is used. Although the cost in terms of accuracy is fairly small when estimating the inverse covariance matrix, this is not the case when estimation of the covariance matrix is the goal. In this case using a decomposable model caused up to 200-fold increases in the variance of estimates. Finally we compare the latest decomposable Bayesian method (the feature-inclusion stochastic search) with penalized likelihood methods (graphical lasso and adaptive graphical lasso) on measures of model selection and prediction performance. Here we nd that graphical lasso is clearly outclassed on all measures by both adaptive graphical lasso and feature-inclusion stochastic search. The sample size and the ultimate goal of the estimation will determine whether adaptive graphical lasso or feature-inclusion stochastic search is better.