Structure Information and Dimension Reduction of Data

Abstract

We not only need to consider data, but also need to take account of the mechanism that generating the data in statistical modeling. The description of the mechanism is called structure information of data. To show the structure information of data distinctly, we usually use a graph or a network. How to perform statistical inference more effectively by using the structure information of data? In this report, we present some results obtained recently about the above problem. Firstly, we give some sufficient and necessary conditions for decomposing a complex and high dimensional structure into some simple and low dimensional structures, and obtain the corresponding algorithm. Secondly, for a given variable set A, we propose a algorithm to look for the smallest variable set B, which must contain set A, such that the high variable set is collapsible onto the marginal model B, and we can find the maximum likelihood estimates on the marginal model B, where a especial example was presented by Madigan & Mosurski (Biometrika, 1990, 1999). Keywords: dimension reduction; collapsibility; structure; graphical decomposition