Clustering Tree-Structured Data on Manifold

• Activity Number: 693
• Type: Contributed
• Date/Time: Thursday, August 8, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistics in Imaging
• Author(s): Hongyu Miao*+
• Companies: Department of Biostatistics, University of Rochester
• Keywords: Tree-Structured Data ; Topology-Geometry Matrix ; Non-Negative Matrix Factorization ; Geodesic Distance ; Clustering on Manifold

Abstract

Tree-structured data have long been exploited for many biomedical problems such as brain or tumor angiogenesis. A distinct feature of such data is that manifolds are necessarily needed for appropriate data parameterization and analysis. However, the majority of existing statistical methods and theories make the assumption of data parameterization in a Euclidean vector space, and are thus have to be re-invented for trees. Specifically, tree-structured data contain both topological and geometrical information so the distance between two trees is not a Euclidean vector norm but a geodesic on manifold. We propose to parameterize tree-structured data using a Topology-Geometry (T-G) matrix such that the problem can be investigated on a manifold. Through non-negative matrix factorization (NMF), we determine the meta-trees from the T-G matrix and then obtain the quotient matrices. A new metric is proposed in the quotient matrix space to quantify the distance between trees. Combining the idea of Fréchet mean, clustering tree data thus becomes feasible. Simulation studies suggest that the proposed method can efficiently distinguish the treatment sample from the control one.