A Riemannian manifold-valued trajectory modeling and elastic alignment framework: A cable degradation case study

Zigang Liu; Tianyao Ji; Zeineb Klai; Saleh M. Altowaijri; Zigang Liu; Tianyao Ji; Zeineb Klai; Saleh M. Altowaijri

doi:10.3934/math.20261022

AIMS Mathematics

2026, Volume 11, Issue 8: 25470-25505. doi: 10.3934/math.20261022

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Research article

A Riemannian manifold-valued trajectory modeling and elastic alignment framework: A cable degradation case study

1.
School of Electric Power Engineering, South China University of Technology, Guangzhou 510641, China
2.
School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore
3.
Department of Computer Sciences, Faculty of Computing and Information Technology, Northern Border University, Rafha 91911, Saudi Arabia
4.
Department of Information Systems, Faculty of Computing and Information Technology, Northern Border University, Rafha 91911, Saudi Arabia

Received: 14 June 2026 Revised: 31 July 2026 Accepted: 05 August 2026 Published: 17 August 2026
MSC : 15B48; 53B21; 65D10; 68W40; 90C39

Cable degradation is a nonlinear evolution process affected by structural, thermal, and spatial factors, which may introduce complex distortions into measured cable responses. Conventional Euclidean indicators based on fixed points, empirical sub-bands, or direct vector distances are often insufficient to represent such coupled variations. To address this issue, this paper proposes a Riemannian manifold-valued trajectory modeling and elastic alignment framework for analyzing cable degradation. The measured cable responses are first converted into localized symmetric positive definite (SPD) covariance matrices using a sliding-window strategy, representing each cable state as a continuous trajectory on the SPD manifold. These trajectories are then elastically aligned using the proposed log-Euclidean dynamic time warping (LE-DTW) framework, yielding two geometric descriptors: A warping path for horizontal parameter distortion and a residual shape distance for intrinsic geometric variation. Based on the residual distance, cumulative Riemannian shape energy (CRSE) is constructed and normalized into a monotonic health index (HI) for retrospective within-trajectory analysis. A causal cumulative degradation feature, which does not require terminal failure information, is further combined with the warping energy for independent cross-temperature predictions of remaining useful life. In the retrospective analysis, the proposed HI + CRSE representation reduces root mean squared error (RMSE) from 23.4370% to 0.9404% and mean absolute error from 15.5850% to 0.6768%, while increasing R² from 0.3808 to 0.9990. In the independent cross-temperature experiment, Gaussian process regression models trained exclusively on the 160 ℃ trajectories are tested on independent 140 ℃ cables with lengths of 1 m, 2 m, and 3 m, obtaining RMSE values of 4.7237%, 5.5727%, and 1.8697%, respectively.
- Riemannian manifold,
- SPD matrix,
- manifold-valued trajectory,
- elastic alignment,
- dynamic time warping,
- geometric modeling,
- non-Euclidean representation cable degradation
Citation: Zigang Liu, Tianyao Ji, Zeineb Klai, Saleh M. Altowaijri. A Riemannian manifold-valued trajectory modeling and elastic alignment framework: A cable degradation case study[J]. AIMS Mathematics, 2026, 11(8): 25470-25505. doi: 10.3934/math.20261022

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Abstract

Cable degradation is a nonlinear evolution process affected by structural, thermal, and spatial factors, which may introduce complex distortions into measured cable responses. Conventional Euclidean indicators based on fixed points, empirical sub-bands, or direct vector distances are often insufficient to represent such coupled variations. To address this issue, this paper proposes a Riemannian manifold-valued trajectory modeling and elastic alignment framework for analyzing cable degradation. The measured cable responses are first converted into localized symmetric positive definite (SPD) covariance matrices using a sliding-window strategy, representing each cable state as a continuous trajectory on the SPD manifold. These trajectories are then elastically aligned using the proposed log-Euclidean dynamic time warping (LE-DTW) framework, yielding two geometric descriptors: A warping path for horizontal parameter distortion and a residual shape distance for intrinsic geometric variation. Based on the residual distance, cumulative Riemannian shape energy (CRSE) is constructed and normalized into a monotonic health index (HI) for retrospective within-trajectory analysis. A causal cumulative degradation feature, which does not require terminal failure information, is further combined with the warping energy for independent cross-temperature predictions of remaining useful life. In the retrospective analysis, the proposed HI + CRSE representation reduces root mean squared error (RMSE) from 23.4370% to 0.9404% and mean absolute error from 15.5850% to 0.6768%, while increasing R² from 0.3808 to 0.9990. In the independent cross-temperature experiment, Gaussian process regression models trained exclusively on the 160 ℃ trajectories are tested on independent 140 ℃ cables with lengths of 1 m, 2 m, and 3 m, obtaining RMSE values of 4.7237%, 5.5727%, and 1.8697%, respectively.

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