Anomaly detection and prediction evaluation for discrete nonlinear dynamical systems

Abstract

Anomalies in dynamical systems mostly occur as deviations between measurement and prediction. Current anomaly detection methods in multivariate time series often require prior clustering, training data, or cannot distinguish local and global anomalies. Furthermore, no generalized metric exists to evaluate and compare different prediction functions regarding their amount of anomalous behavior. We propose a novel methodology to detect local and global anomalies in time series data of dynamical systems. For this purpose, a theoretical density distribution is derived assuming that only noise conceals the time series. If the theoretical and the empirical density distribution yield significantly different entropies, an anomaly is assumed. For a local anomaly detection, the Mahalanobis distance using the theoretical noise distribution’s covariance is applied to evaluate sequences of predictions and measurements. In addition, the Wasserstein metric enables a comparison of predictions using the distance between the noise and empirical distribution as a measure for selecting the best prediction function. The proposed method performs well on nonlinear time series such as logistic growth and enables a useful selection of a prediction model for satellite orbits. Thus, the proposed method improves anomaly detection in time series and model selection for nonlinear systems.