Research On Basic Theory Of Graphical Models

For decades, with the rapid development of computer technology, large dimen-sional data analysis plays more and more important role in modern scientific research. Especially, in the fields of microarray data in biology, stock market analysis in finance and wireless communication networks, etc. Unfortunately, the traditional methods of statistical inference modelling can not catch up the development of the data analysis, there is much limits in large dimensional data analysis.Graphical models, which have their origin in statistical physics, are increasingly valuable in problems of higher dimensional data and greater complex system, and have been an hot research domain in modern statistic science. graphical models use graphs, either undirected, directed, or mixed, to represent multivariate dependence in a visual and computationally efficient manner. The assessment of dependence among stochas-tic variables is a central of statistical science. The familiar concepts of correlation regression, and prediction are special ases, and identification of causal relationships ultimately rests on representations of multivariate dependence. A graphical model is formally a family of distributions which satisfy a set of conditional independence re-lations encoded by a graph. The method of graphical models, which combine the qualitative and quantitative methods, is an increasing popular method in modern sta-tistical modeling. In fact, graphical models are traditional statistical models, which efficiently use conditional independencies among stochastic variables. Thus, it is apt to say graphical models are typical representative of modern statistical theory. Accord-ing to the graph used in the model, there are four kinds of graphical models. They are undirected graph models (Markov networks), directed acyclic graph models (Bayesian networks), chain graph models, and ancestral graph models.The representation of conditional independence knowledge, i.e., Markov theory is the basic theory of graphical models. It is the original resource from which graphical models have usefulness, influence and limit the development of graphical models. It has always been a very difficult subject in graphical models research. For this basic theory, five contributions have been conducted in this paper. First, we study the global Markov property in chain graph models, and propose the directed graph criterion-i separation. I-separation criterion is equivalent to the moral criterion, very simple and manipulated easily. Thus, it will be constructive for many difficult theoretical prob-lems to be solved and the models to be used widely. Second, we study the faithfulness and completeness of chain graph models, which are two difficult theoretical problems. By i-separation criterion, we prove that for any chian graph, there is a faithful distribu-tion (discrete or Gaussian)and i-separation is not only atomic complete but also strong complete. This shows that the application of chain graph models is justifiacation in logic and eliminates many people’s doubt in using chain graph models to statistical modeling. Third, we study the separation tree of ancestral graph models, which is called m-separation tree. We get the beautiful properties of m-separation tree, and give the construction method for m-separation tree. Thus, Markov knowledge expressed by ancestral graphs can also be described by decomposition. This will certainly facilitate statistical reasoning and structure learning in ancestral graph models. Fourth, we study the chromatic parameters of graphs, since graph coloring has played an important role in graphical models. The complete chromatic number and the incidence chromatic number of3-Halin graphs are determined. It is also proved that pseudo-Halin graphs has an incidence (Δ+2,2) coloring, where Δ is the maximal degree of graphs.This thesis further highlights the important position of the graphical models in modern statistics, is bound to have a positive impact on the development of graphical models and statistical science.