ANOMALY DETECTION IN DYNAMICAL SYSTEMS FROM MEASURED TIME SERIES

Abstract

The paper addresses a problem of abnormalities detection in nonlinear processes represented by measured time series. Anomaly detection problem is usually formulated as finding outlier data points relative to some usual signals such as unexpected spikes, drops, or trend changes. In nonlinear dynamical systems, there are cases where a time series does not contain statistical outliers while the process corresponds to an abnormal configuration of the dynamical system. Since the polynomial neural architecture has a strong connection with the theory of differential equations, we use it for the feature extraction that describes the dynamical system itself. The paper considers both simulations and a practical example with real measurements. The applicability of the proposed approach and it's benchmarking with the existing methods is discussed.

1. INTRODUCTION

Most of the works related to anomaly detection in time series data are referred to the detection of the "observation which deviates so much from other observations as to arouse suspicions that it was generated by a different mechanism" (see Hawkins (1980) ). Anomaly detection problem is usually formulated as finding outlier data points relative to some usual signals such as unexpected spikes, drops, trend changes, and level shifts. Blázquez-García et al. (2020) provides a literature review that deals exclusively with time series data and provides a taxonomy for the classification of outlier detection techniques according to their main characteristics. In contrast to these methods, we address the problem of anomaly detection in dynamical systems from measured time series. In this case, we are interested in detecting the anomalies which do not deviate so much from other observations while they were still generated by a different configuration of the dynamical system. To better explain this issue, let us consider a dynamical system in the form of the ordinary differential equation (ODE): d dt X = F (t, X, a 1 , a 2 ), where X = (x 1 , x 2 , . . . , x n ) is a state vector, F is nonlinear function depending on two scalar parameters a 1 and a 2 . The dynamical system (1) generates a trajectory in the form of the multivariate time series as a particular solution for a given initial condition X 0 . Let us also assume for simplicity that the initial conditions X(0) = X 0 are always the same for different trajectories but only parameters are varied and belong to the normal distribution. Since the system (1) is nonlinear, the distribution of the trajectories is unknown in advance and may differ from the normal one. This may cause that trajectories corresponding to abnormal system parameters are located somewhere among the other trajectories. Fig. 1 demonstrates that parameters of the system (1) taken from the tail of the normal distribution can correspond to centered trajectories in the time-space. This example formulates the problem of anomalies detection in the dynamical systems represented only by the measured trajectories. We are interested in the unsupervised methods for recovering the representative features of the dynamical system from the time series. Such a method should calculate features that are correlated with the dynamical system itself but not just with a time series that is generated by the dynamical system. Also, collecting massive training sets in industrial applications

