Topological Data Analysis for Time Series Data

Music City Center 

We study a popular tool in topological data analysis (TDA) called sublevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly ordered and cyclically ordered domains. We prove duality of filtrations of sublevel sets that undergirds a range of duality results of sublevel set persistent homology without the need to invoke complications of continuous functions or classical Morse theory. We show that Morse-like behavior can be achieved for flat extrema without assuming genericity. Furthermore, we discuss aspects of barcode construction rules, surgery of circular and linearly ordered sets. We end by discussing ideas of future work that integrate this framework with more traditional statistical techniques for analyzing time series.