Measure invariance of ergodic symbolic systems for low-delay detection of anomalous events

This paper investigates the spectral properties of ergodic measure of symbolic systems for applications to signal processing, pattern recognition, and anomaly detection in uncertain dynamical systems. The underlying algorithm is built upon the concept of ergodic sequences of measure-preserving transformations (MPT) on probability spaces, where non-stationary probabilistic finite state automata (PFSA) are constructed using short-length windows of time-series measurements. The resulting PFSA are non-homogeneous Markov, in which spectral properties of the MPT depend on several parameters that include the window length. The paper also develops an MPT-based metric to quantify the divergence of the evolving PFSA from that of the nominal PFSA; this information is then used for pattern classification and anomaly detection with low-delay tolerance. The MPT-based methodology has been validated with experimental data generated from a laboratory apparatus that deals with detection of thermoacoustic instabilities in combustion processes, which are known to have chaotic characteristics on a time scale of milliseconds. In this application, the concepts of MPT and ergodicity have been used to develop a novel symbolic time series analysis (STSA)-based detection method, whose performance is validated by comparison with two well-known techniques, namely, hidden Markov model (HMM) and cumulative sum (CUSUM), on the same experimental data. The results consistently show superior performance of the proposed MPT-based STSA.