Recent advances in sensor technology and computer hardware has led to a shift towards data-driven analysis and modeling of engineered and natural systems. The datasets are obtained through either numerical simulations or experiments and they often contain complex dynamics hidden in some high-dimensional structure. Current literature on time series analysis aims to study this structure by searching for a lower dimensional representation; however, the need for user-defined inputs, the sensitivity of these inputs to error, and the expensive computational effort limit the usability of available knowledge, especially for in-situ signal analysis. This project is a collaborative effort that seeks to investigate an innovative framework for time series analysis through advancing and linking signal processing, dynamical systems, and applied topology. It pursues a topological approach that utilizes persistent homology, a new approach in the field of topological data analysis, for studying the time series of dynamical systems. The theoretical underpinning of the proposed research is especially suitable for detecting and describing dynamical signatures such as underlying attractors, chaos, and self-similarity using lower dimensional descriptors, rather than lower dimensional representation; therefore, it is capable of providing a new perspective into our understanding of time series analysis particularly for dynamical systems with complex behavior.

This project aims to research the underpinnings of machine learning on the information rich representations from topological data analysis (TDA). Specifically, We seek to understand and formulate the foundations of machine learning when the important features of a dynamical system are summarized by descriptors generated with topological data analysis (TDA). One of the current impediments to further exploring the relationship between TDA and dynamical systems is the lack of machine learning theory that can operate on the resulting, complex TDA structures. Therefore, we are pursuing a better understanding of the relationship between TDA and dynamical systems that will lead to a novel, general, and robust machine learning framework for studying dynamic signals.