On Time-Series Topological Data Analysis: New Data and Opportunities

This work introduces a new dataset and framework for the exploration of topological data analysis (TDA) techniques applied to time-series data. We examine the end-toend TDA processing pipeline for persistent homology applied to time-delay embeddings of time series – embeddings that capture the underlying system dynamics from which time series data is acquired. In particular, we consider stability with respect to time series length, the approximation accuracy of sparse filtration methods, and the discriminating ability of persistence diagrams as a feature for learning. We explore these properties across a wide range of time-series datasets spanning multiple domains for single source multi-segment signals as well as multi-source single segment signals. Our analysis and dataset captures the entire TDA processing pipeline and includes time-delay embeddings, persistence diagrams, topological distance measures, as well as kernels for similarity learning and classification tasks for a broad set of time-series data sources. We outline the TDA framework and rationale behind the dataset and provide insights into the role of TDA for time-series analysis as well as opportunities for new work.