Estimation of Multiple High-Dimensional Directed Acyclic Graphs

• Activity Number: 546
• Type: Invited
• Date/Time: Wednesday, August 3, 2011 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Yiping Yuan*+
• Companies: University of Minnesota at Twin Cities

Abstract

Directed acyclic graphs(DAGs) are a popular tool to represent causal relationship among random variables. Besides estimation of a DAG, detection of changes in a graph is also of our interest when we have more than one data set from the same random variables. These changes usually reveal important causality facts in the regarding ¯eld. In addition, when the ordering of nodes is known, the problem of estimating DAGs reduces to the problem of estimating the structure of the network. In this article, we consider two adjacent DAGs from the same Gaussian random variables with a natural ordering. Sparsity is assumed in the context of high-dimensionality. We propose a method to estimate the adjacency matrices of two DAGs and in theory multiple DAGs with an approximate L0 constraint on the elements of adjacency matrices of DAGs to encourage sparsity and with another approximate L0 constraint on the element-wise di®erence of adjacency matrices to detect changes in the graph. Computationally, we introduce an e±cient algorithm that is based on the augmented Lagrange multipliers, di®erence convex method and a fast algorithm for solving the original LASSO problem. Compared to the Lagrange version of the problem, this constrained setting gives better theoretical properties as well as easier tuning for the tuning parameters. We also demonstrate the method for simulated data and real data.