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- ItemBayesian Modelling of Direct and Indirect Effects of Marine Reserves on Fishes : A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Albany, New Zealand.(Massey University, 2016) Smith, Adam Nicholas Howard
Show more This thesis reviews and develops modern advanced statistical methodology for sampling and modelling count data from marine ecological studies, with specific applications to quantifying potential direct and indirect effects of marine reserves on fishes in north eastern New Zealand. Counts of snapper (Pagrus auratus: Sparidae) from baited underwater video surveys from an unbalanced, multi-year, hierarchical sampling programme were analysed using a Bayesian Generalised Linear Mixed Model (GLMM) approach, which allowed the integer counts to be explicitly modelled while incorporating multiple fixed and random effects. Overdispersion was modelled using a zero-inflated negative-binomial error distribution. A parsimonious method for zero inflation was developed, where the mean of the count distribution is explicitly linked to the probability of an excess zero. Comparisons of variance components identified marine reserve status as the greatest source of variation in counts of snapper above the legal size limit. Relative densities inside reserves were, on average, 13-times greater than outside reserves. Small benthic reef fishes inside and outside the same three reserves were surveyed to evaluate evidence for potential indirect effects of marine reserves via restored populations of fishery-targeted predators such as snapper. Sites for sampling were obtained randomly from populations of interest using spatial data and geo-referencing tools in R—a rarely used approach that is recommended here more generally to improve field-based ecological surveys. Resultant multispecies count data were analysed with multivariate GLMMs implemented in the R package MCMCglmm, based on a multivariate Poisson lognormal error distribution. Posterior distributions for hypothesised effects of interest were calculated directly for each species. While reserves did not appear to affect densities of small fishes, reserve-habitat interactions indicated that some endemic species of triplefin (Tripterygiidae) had different associations with small-scale habitat gradients inside vs outside reserves. These patterns were consistent with a behavioural risk effect, where small fishes may be more strongly attracted to refuge habitats to avoid predators inside vs outside reserves. The approaches developed and implemented in this thesis respond to some of the major current statistical and logistic challenges inherent in the analysis of counts of organisms. This work provides useful exemplar pathways for rigorous study design, modelling and inference in ecological systems.Show more - ItemSparse 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
Show more 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.Show more