Recent advances in sensor technology and computer hardware have led to a shift towards data-driven analysis and modeling of engineered and natural systems. The study of the resulting streams of data currently requires extensive expertise and often utilizes methods that are not guaranteed to be optimal. This leads to an overly qualitative analysis that can miss a significant portion of the information hidden within the signal. This project aims to utilize and advance tools from topological data analysis, an emergent field focused on providing quantitative measurements of the shape of data, and to elucidate invisible features and structure of signals which current methods cannot detect. Therefore, the resulting framework will be able to provide insight into the systems that generated the data while providing a strong mathematical footing for automating the analysis of systems and processes. The knowledge gained from the work will benefit a wide variety of scientific fields where analysis from sensor feedback signals is needed such as additive manufacturing and smart drug delivery systems. Furthermore, the interdisciplinary nature of this project will offer students rich opportunities for collaboration and cross-training in a variety of areas in engineering, applied math, and signal analysis. The PIs will also continue and expand their efforts to identify and recruit graduate students from under-represented groups with relevant interests by working with the Association for Women in Mathematics and the Society for Women Engineers.
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 topological data analysis. The research team will pursue a topological approach that utilizes the recently developed persistent homology to study the time series of dynamical systems. Specifically, they will (1) derive a solid mathematical foundation based on applied topology which enables a more robust and quantitative investigation of dynamical systems, (2) research innovative methods for studying the topology of the underlying manifold in dynamical systems from collected time series using topological data analysis, and (3) demonstrate and assess the validity of the theoretical results using numerical and physical experiments. The theoretical underpinning of the 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, this work is capable of providing a new perspective into our understanding of time series analysis particularly for dynamical systems with complex behavior.
