The ratio R of two random quantities is frequently encountered in probability and statistics. But while for unidimensional statistical variables the distribution of R can be computed relatively easily, for symmetric p...The ratio R of two random quantities is frequently encountered in probability and statistics. But while for unidimensional statistical variables the distribution of R can be computed relatively easily, for symmetric positive definite random matrices, this ratio can take various forms and its distribution, and even its definition, can offer many challenges. However, for the distribution of its determinant, Meijer G-function often provides an effective analytic and computational tool, applicable at any division level, because of its reproductive property.展开更多
The conditional autoregressive model is a routinely used statistical model for areal data thatarise from, for instances, epidemiological, socio-economic or ecological studies. Various multivariate conditional autoregr...The conditional autoregressive model is a routinely used statistical model for areal data thatarise from, for instances, epidemiological, socio-economic or ecological studies. Various multivariate conditional autoregressive models have also been extensively studied in the literatureand it has been shown that extending from the univariate case to the multivariate case is nottrivial. The difficulties lie in many aspects, including validity, interpretability, flexibility and computational feasibility of the model. In this paper, we approach the multivariate modelling froman element-based perspective instead of the traditional vector-based perspective. We focus onthe joint adjacency structure of elements and discuss graphical structures for both the spatialand non-spatial domains. We assume that the graph for the spatial domain is generally knownand fixed while the graph for the non-spatial domain can be unknown and random. We proposea very general specification for the multivariate conditional modelling and then focus on threespecial cases, which are linked to well-known models in the literature. Bayesian inference forparameter learning and graph learning is provided for the focused cases, and finally, an examplewith public health data is illustrated.展开更多
文摘The ratio R of two random quantities is frequently encountered in probability and statistics. But while for unidimensional statistical variables the distribution of R can be computed relatively easily, for symmetric positive definite random matrices, this ratio can take various forms and its distribution, and even its definition, can offer many challenges. However, for the distribution of its determinant, Meijer G-function often provides an effective analytic and computational tool, applicable at any division level, because of its reproductive property.
文摘The conditional autoregressive model is a routinely used statistical model for areal data thatarise from, for instances, epidemiological, socio-economic or ecological studies. Various multivariate conditional autoregressive models have also been extensively studied in the literatureand it has been shown that extending from the univariate case to the multivariate case is nottrivial. The difficulties lie in many aspects, including validity, interpretability, flexibility and computational feasibility of the model. In this paper, we approach the multivariate modelling froman element-based perspective instead of the traditional vector-based perspective. We focus onthe joint adjacency structure of elements and discuss graphical structures for both the spatialand non-spatial domains. We assume that the graph for the spatial domain is generally knownand fixed while the graph for the non-spatial domain can be unknown and random. We proposea very general specification for the multivariate conditional modelling and then focus on threespecial cases, which are linked to well-known models in the literature. Bayesian inference forparameter learning and graph learning is provided for the focused cases, and finally, an examplewith public health data is illustrated.
