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Ordinal network-based affine invariant Riemannian measure and its expansion: powerful similarity measure tools for complex systems

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Abstract

The ordinal network is an effective method to characterize the dynamic features of time series. Meanwhile, Riemannian geometry is an important mathematical tool widely used in pattern recognition and machine learning. In this article, we propose an ordinal network-based affine invariant Riemannian measure (ONAIRM), which can measure the shape-based similarity of dynamic evolution between systems. The robustness and effectiveness of ONAIRM are verified by simulation data. Then, we propose the classical multidimensional scaling method based on ONAIRM and validate that it can distinguish the stochastic processes, periodic data, and chaotic systems. Finally, we extend the ONAIRM with the purpose of classification and prediction of complex systems, the ONAIRM-based agglomerative hierarchical clustering algorithm and ONAIRM-based Riemannian mean classification algorithm are proposed and applied to the research of physiological data and rail corrugation. The empirical results show that they have better performance than other similarity calculation methods.

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Data Availability

The rail corrugation data used in the current study are provided by China academy of railway sciences corporation limited. The physiological datasets are publicly available in the Physionet, [https://physionet.org/].

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Funding

This study is supported by the funds of the Fundamental Research Funds for the Central Universities (2021YJS166), China Academy of Railway Science Cooperation Limited (2019YJ153) and the National Natural Science Foundation of China (62171018).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, China

    Zhuo Wang & Pengjian Shang

  2. Infrastructure Inspection Research Institute, China Academy of Railway Sciences Corporation Limited, Beijing, 100081, China

    Xuegeng Mao

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  1. Zhuo Wang
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  2. Pengjian Shang
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Correspondence to Zhuo Wang.

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Wang, Z., Shang, P. & Mao, X. Ordinal network-based affine invariant Riemannian measure and its expansion: powerful similarity measure tools for complex systems. Nonlinear Dyn 111, 3587–3603 (2023). https://doi.org/10.1007/s11071-022-07991-6

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