Periodic measures and Wasserstein distance for analysing periodicity of time series datasets

Feng, Chunrong; Liu, Yujia; Zhao, Huaizhong

Abstract

In this article, we establish the probability foundation of the periodic measure approach in analysing periodicity of a dataset. It is based on recent work of random periodic processes. While random periodic paths provide a pathwise model for time series datasets with a periodic pattern, their law is a periodic measure and gives a statistical description and the ergodic theory offers a scope of statistical analysis. The connection of a sample path and the periodic measure is revealed in the law of large numbers (LLN). We prove first the period is actually a deterministic number and then for discrete processes, Bézout’s identity comes in naturally in the LLN along an arithmetic sequence of an arbitrary increment. The limit is a periodic measure whose period is equal to the greatest common divisor between the test period and the true period of the random periodic process. This leads to a new scheme of detecting random periodicity of a dataset and finding its period, as an alternative to the Discrete Fourier Transformation (DFT) and periodogram approach. We find that in some situations, the classical method does not work robustly, but the new one can work efficiently. We prove that the periodicity is quantified by the Wasserstein distance, in which the convergence of empirical distributions is established.

Citation

Feng, C., Liu, Y., & Zhao, H. (2023). Periodic measures and Wasserstein distance for analysing periodicity of time series datasets. Communications in Nonlinear Science and Numerical Simulation, 120, Article 107166. https://doi.org/10.1016/j.cnsns.2023.107166

• Journal Article Type: Article
• Acceptance Date: Feb 8, 2023
• Online Publication Date: Feb 11, 2023
• Publication Date: 2023-06
• Deposit Date: Feb 17, 2023
• Publicly Available Date: Feb 17, 2023
• Journal: Communications in Nonlinear Science and Numerical Simulation
• Print ISSN: 1007-5704
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 120
• Article Number: 107166
• DOI: https://doi.org/10.1016/j.cnsns.2023.107166
• Public URL: https://durham-repository.worktribe.com/output/1179292