The Shape of Data: A New Way to Detect Critical Shifts in System Performance

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

This project is about a new way to detect and classify shifts in the patterns of measurements taken by sensors. Subtle shifts in the output from a vibration sensor on a bridge abutment, for instance, can indicate that a crack is developing in that structure. Identifying these shifts can be a real challenge because modern sensors can generate so much information, and so quickly. Existing approaches to this use the mathematics of statistics: averages, variability, and the like. This project uses techniques from topology, the branch of mathematics that is concerned with shape, to detect these kinds of shifts. This project will develop techniques to assess and characterize temporal patterns in nonstationary time-series data. The project will compute aspects of the homology---e.g., the first few Betti numbers---of the stream "on the fly" to obtain a signature of its regime. To mitigate the computational burden of traditional techniques, this project uses a simplicial complex based on a small, representative set of "landmarks," with the remaining data, the "witnesses," defining the connections. New techniques include a generalization of the oft-used false near-neighbor method to evaluate the fidelity of the topology of the reconstruction, efficient selection of landmarks and choice of witness relation, classification of structure through multi-parameter persistent homology of embedded data, and development of a map on the witness complex to obtain a dynamical signature of each regime. The goals include efficient detection of regime shifts, the development of a catalog of signatures within regimes, and the detection and mitigation of noise. The classification techniques will be applicable, for example, to the detection of failure modes in manufacturing systems, malware attacks on a network, incipient heart problems, or an impending containment failure.