Graphical Model with Ordinal Variables

• Activity Number: 451
• Type: Topic Contributed
• Date/Time: Wednesday, August 3, 2011 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Jian Guo*+ and Liza Levina and George Michailidis and Ji Zhu
• Companies: University of Michigan and University of Michigan and University of Michigan and University of Michigan
• Keywords: Graphical model ; Lasso ; Ordinal variable ; Probit model

Abstract

Existing graphical models mainly consider the variables either in numerical scale or in nominal scale. In this paper, we propose a new graphical model characterizing another important type of variables---ordinal variables, which consist of a limited number of levels with a natural order. Examples of ordinal variables include the user rating records for online movies or music. In the proposed model, namely probit graphical model, we assume these ordinal variables are discretized from the corresponding latent numeric variables, which jointly follows a multivariate Gaussian distribution and whose partial correlations can be used to characterize the dependence relationship between the original ordinal variables. Under this modeling framework, we developed an EM-like algorithm to recover the underlying Gaussian graphical structures. The proposed model exhibits its superior performance over the Gaussian graphical model on a few synthetic ordinal data sets. It was also applied to exploring the graphical structures between a number of movies based on their ratings by users and some interesting patterns were discovered.