On the Persistent Homology of Time-Delay Embeddings

• Activity Number: 3
• Type: Invited
• Date/Time: Sunday, August 4, 2013 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Jose Andres Perea*+ and John Harer
• Companies: Duke University and Mathematics Department, Duke University
• Keywords: Persistent homology ; Time-delay embedding ; Time series ; Periodicity ; Taken's theorem

Abstract

Persistent homology is a topological method for measuring the shapes of spaces and the features of functions. One of its most important applications is to point clouds, where shape is usually interpreted as the geometry of some implicit underlying object near which the point cloud is sampled.

Time-delay embeddings, on the other hand, have been used mostly in the study of time series and dynamical systems to understand the nature of their attractors. In this talk, we analyze the geometry and topology of time-delay embeddings through the lens of persistent homology. In particular, we propose maximum persistence as a measure of periodicity at the signal level, present structural theorems for the resulting diagrams, and derive estimates for their dependency on the window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic signals, and present comparisons with state-of-the-art methods in gene expression analysis. This is joint work with John Harer at Duke University, Mathematics Department.