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Joint Posterior Inference for Latent Gaussian Models and extended strategies using INLAChiuchiolo, Cristian 06 June 2022 (has links)
Bayesian inference is particularly challenging on hierarchical statistical models as computational complexity becomes a significant issue. Sampling-based methods like the popular Markov Chain Monte Carlo (MCMC) can provide accurate solutions, but they likely suffer a high computational burden. An attractive alternative is the Integrated Nested Laplace Approximations (INLA) approach, which is faster when applied to the broad class of Latent Gaussian Models (LGMs). The method computes fast and empirically accurate deterministic posterior marginal approximations of the model's unknown parameters. In the first part of this thesis, we discuss how to extend the software's applicability to a joint posterior inference by constructing a new class of joint posterior approximations, which also add marginal corrections for location and skewness. As these approximations result from a combination of a Gaussian Copula and internally pre-computed accurate Gaussian Approximations, we name this class Skew Gaussian Copula (SGC). By computing moments and correlation structure of a mixture representation of these distributions, we achieve new fast and accurate deterministic approximations for linear combinations in a subset of the model's latent field. The same mixture approximates a full joint posterior density through a Monte Carlo sampling on the hyperparameter set. We set highly skewed examples based on Poisson and Binomial hierarchical models and verify these new approximations using INLA and MCMC. The new skewness correction from the Skew Gaussian Copula is more consistent with the outcomes provided by the default INLA strategies. In the last part, we propose an extension of the parametric fit employed by the Simplified Laplace Approximation strategy in INLA when approximating posterior marginals. By default, the strategy matches log derivatives from a third-order Taylor expansion of each Laplace Approximation marginal with those derived from Skew Normal distributions. We consider a fourth-order term and adapt an Extended Skew Normal distribution to produce a more accurate approximation fit when skewness is large. We set similarly skewed data simulations with Poisson and Binomial likelihoods and show that the posterior marginal results from the new extended strategy are more accurate and coherent with the MCMC ones than its original version.
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COVID-19 Disease Mapping Based on Poisson Kriging Model and Bayesian Spatial Statistical ModelMu, Jingrui 25 January 2022 (has links)
Since the start of the COVID-19 pandemic in December 2019, much research has
been done to develop the spatial-temporal methods to track it and to predict the
spread of the virus. In this thesis, a COVID-19 dataset containing the number of biweekly infected cases registered in Ontario since the start of the pandemic to the end
of June 2021 is analysed using Bayesian Spatial-temporal models and Area-to-area
(Area-to-point) Poisson Kriging models. With the Bayesian models, spatial-temporal
effects on infected risk will be checked and ATP Poisson Kriging models will show
how the virus spreads over the space and the spatial clustering feature. According
to these models, a Shinyapp website https://mujingrui.shinyapps.io/covid19 is
developed to present the results.
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Matematické modely spolehlivosti v technické praxi / Mathematical Models of Reliability in Technical ApplicationsSchwarzenegger, Rafael January 2017 (has links)
Tato práce popisuje a aplikuje parametrické a neparametrické modely spolehlivosti na cenzorovaná data. Ukazuje implementaci spolehlivosti v metodologii Six Sigma. Metody jsou využity pro přežití/spolehlivost reálných technických dat.
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