Spatial models with random covariance structure

• Autor(a) principal: Danna Lesley Cruz Reyes
• Data de Publicação: 2021
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/38393; https://orcid.org/ 0000-0002-5977-8162
• topic: Spatial statistic; CAR model; Graph of edges; Shrinkage prior; Robust spatial models; Estatística - Teses; Análise espacial (Estatística) – teses; Autoregressão (Estatística) – Teses; Câncer – Região Sul – Métodos estatísticos. -Teses

The conditional autoregressive model (CAR model) is the most popular distribution for jointly modeling the a priori uncertainty over spatially correlated data. In general, it is used in hierarchical spatial models where it models the uncertainty about random spatial e ects. A limitation of the CAR model is its inability to produce high correlations between neighboring areas. We propose a robust model for area data that alleviates this problem. We represent the map by an undirected graph where nodes represent areas and edges connect neighboring nodes on the map. We assign distinct and random weights to the edges. The model is based on a spatially structured t≠Student multivariate distribution, in which the precision matrix is indirectly constructed assuming a multivariate distribution for the random weights of the edges. Such t≠ Student distribution spatially correlates the edge weights and induces another t-Student model for the spatial e ects of the areas that correlates them and is able to accommodate outliers and heavy tail behavior for these e ects . More importantly, the proposed model can produce a higher marginal correlation between spatial e ects than the CAR model, overcoming one of the main limitations of this model. We adjusted the proposed model to map the incidence of some types of cancer in southern Brazil and compared its performance with several alternative models proposed in the literature. The results show that the proposed model is competitive and provides similar and, in some cases, better results than those obtained by fitting models commonly used to analyze this type of data. In the second proposal, we approach the problem of dimensionality reduction in regression models. One of the most used methods to avoid overfitting and to select relevant variables in regression models with many predictors is the penalized regression technique. Under such approaches, variable selection is performed in a non-probabilistic way using some optimization criterion. Bayesian approaches to penalized regression have been proposed assuming an a priori distribution for the regression coe cients that plays a role similar to the penalty term in classical statistics: compressing towards zero non- significant coe cients and putting a probability mass significant in coe cients that can be grouped. Generally, such a priori distributions, called shrinkage priors (shrinkage a priori distributions), assume independence between the e ects of the covariates, which may not be an appropriate assumption in many cases. In this work, we focus on the dimensionality reduction of categorical variables with many levels. These variables are included in the model through variables dummy inducing sparsity in the design matrix, which can generate overfitting and di culties in interpreting the results. The e ect of the levels of these categorical variables are naturally correlated. To deal with this problem, we propose two a priori shrinkage distributions for the coe cients associated with the levels of categorical variables, correlating them. The proposed distributions are proper and, in addition to sparsity, they have the property of grouping similar e ects. We illustrate the use of these distributions by applying them to dimensionality reduction in a linear regression. Their performances are analyzed and compared to pre-existing methods through simulated data studies and considering housing price data available on Airbnb.