Time irreversibility analysis and abnormality detection based on Riemannian geometry for complex time series

Abstract

Time irreversibility (TIR) is one of the basic properties of nonlinear systems, and it can be used as an effective way to explore the nonlinear complexity of dynamic systems. In this paper, we propose the TIR measure algorithm based on Riemannian geometry and apply this algorithm to abnormality detection tasks. The core idea is to calculate the positive definite covariance matrices of the forward sequence and its inverse sequence based on the theories of sliding window and phase space reconstruction, and compute the affine invariant Riemannian metric (AIRM) between them to characterize the TIR of the time series. The advantage of this algorithm is that the framework based on Riemannian geometry can more effectively use the spatial information of time series. In addition, compared with the traditional methods, the new method is suitable for nonstationary complex systems, which introduces a sliding window to make subsequences satisfy weak stationary conditions to measure local time irreversibility. The simulated and surrogate data are used to verify the effectiveness of the algorithm. Finally, we apply the algorithm to empirical analysis. The results show that, the algorithm detects the financial crisis period and stable period of the stock markets in the analysis of the financial system, and the loss degree of TIR of physiologic signals of different types of cardiac patients is detected in the study of physiologic signals. In the research of Parkinson’s gait, we verify that the walker is an effective tool to improve the gait of early Parkinson’s patients through statistical tests.