Symbolic Time Series Analysis for Anomaly Detection in Measure-Invariant Ergodic Systems

Abstract

This paper presents a novel framework of symbolic time series analysis (STSA) for anomaly detection in dynamical systems. The core concept is built upon a property of measure-preserving transformation (MPT) sequence, acting on a probability space with ergodic measure, that the eigenfunctions of these transformations would be time-invariant. As a result, unlike a standard STSA that is required to generate time-homogeneous Markov chains, the proposed MPT-based STSA is allowed to have time-inhomogeneous Markov chains, where the (possibly time-varying) state transition probability matrices have time-invariant eigenvectors. Such a time-invariance facilitates analysis of the dynamical system by using short-length time series of measurements. This is particularly important in applications, where the underlying dynamics and process anomalies need fast monitoring and control actions in order to mitigate any potential structural damage and/or to avoid catastrophic failures. The MPT-based STSA has been applied for low-delay detection of fatigue damage, which is a common source of failures in mechanical structures and which is known to have uncertain dynamical characteristics. The underlying algorithm has been validated with experimental data generated from a laboratory apparatus that uses ultrasonic sensors to detect fatigue damage in polycrystalline–alloy specimens. The performance of the proposed MPT-based STSA is evaluated by comparison with those of a standard STSA and a hidden Markov model (HMM) on the same experimental data. The results consistently show superior performance of the MPT-based STSA.