A Spectral Graph Theoretic Approach for Monitoring Multivariate Time Series Data From Complex Dynamical Processes

Abstract

The objective of this paper is to monitor complex process dynamics manifest in multivariate (multidimensional) time series data using a spectral (algebraic) graph theoretic approach. We test the hypothesis that the spectral graph-based topological invariants detect incipient process drifts earlier [lower average run length (ARL <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> )] and with higher fidelity (consistency of detection) when compared with the conventional statistics-based approaches. The presented approach maps a multidimensional sensor data stream X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N×d</sup> (visualize N as time and d as the number of sensors) as an unweighted and undirected network graph G(V, E), indexed by its vertices V and edges E, i.e., X → G(V, E). The rationale is that the graph-based topological invariants are surrogate representatives of the system state. We compare the monitoring performance of spectral graph theoretic invariants with conventional statistical features in an exponentially weighted moving average control chart setting. The practical utility of the approach is substantiated in the context of process monitoring in two advanced manufacturing scenarios, namely, ultraprecision machining (UPM) and semiconductor chemical mechanical planarization. These studies corroborate the hypothesis that graph theoretic invariants, when used as monitoring statistics, lead to lower ARL <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and more consistent detections in contrast to conventional statistical features. For instance, in the UPM case, the fault detection delay using graph theoretic invariants is less than 160 ms, compared with over 8 s of delay with statistical features.