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Entropy
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Multivariate Entropy-Informed Fault Diagnosis and Structural Health Monitoring
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Multivariate Entropy-Informed Fault Diagnosis and Structural Health Monitoring

A Special Issue of Entropy (ISSN 1099-4300) belonging to the section "Signal and Data Analysis".

Deadline for manuscript submissions: 28 February 2027 | Viewed by 1871

Editors

Dr. Rui Yuan

E-Mail Website
Guest Editor
Department of Mechanical Engineering, Wuhan University of Science and Technology, Wuhan 430081, China
Interests: multivariate signal processing; nonlinear dynamics; mechanical fault diagnosis; RUL prediction; structural health monitoring
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Weihang Zhu

E-Mail Website
Guest Editor
Department of Engineering Technology, University of Houston, Houston, TX 77204, USA
Interests: smart manufacturing; 3D printing; augmented reality; brain–computer interface; energy manufacturing and production scheduling optimization
Special Issues, Collections and Topics in MDPI journals
Dr. Xingkai Yang

E-Mail Website
Guest Editor
College of Mechanical and Vehicle Engineering, Hunan University, Changsha, China
Interests: digital twin; artificial intelligence; intelligent operation and health management; mechanical friction dynamics; life prediction and health evaluation methods
Special Issues, Collections and Topics in MDPI journals
Dr. Zhuo Long

E-Mail Website
Guest Editor
School of Electrical and Information Engineering, Changsha University of Science and Technology, Changsha 410114, China
Interests: intelligent perception and information processing; data fusion and artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Modern engineering systems—such as rotating machinery, power transmission components, energy storage devices, and large-scale civil or aerospace structures—operate under dynamic and uncertain conditions. Variations in load, speed, and environment generate nonstationary and multichannel signals whose statistical properties evolve over time, posing persistent challenges for reliable fault diagnosis and structural health monitoring. Traditional scalar indicators or single-sensor features are often distorted by noise, transient fluctuations, and cross-coupled fault mechanisms, resulting in reduced accuracy and interpretability.

Entropy-based complexity analysis provides a powerful information-theoretic tool to quantify irregularity, uncertainty, and dependence within measured signals. Extending conventional univariate formulations, multivariate entropy constructs joint embedding vectors across multiple sensors to reveal spatial correlations, phase relationships, and lagged dependencies. By quantifying joint uncertainty rather than independent variability, MvE captures intrinsic cross-channel dynamics while mitigating the effects of redundant or condition-induced variations. These advantages make it particularly suited for analyzing heterogeneous, multi-sensor data from complex engineering systems.

When combined with modern learning and fusion frameworks, multivariate entropy-based indicators enable interpretable and physics-guided fault diagnosis. They provide compact yet expressive representations that enhance classifier generalization, improve sensitivity to incipient degradations, and reduce dependence on extensive labeled datasets.

Ultimately, multivariate entropy-informed fault diagnosis and SHM frameworks offer a unified paradigm for integrating heterogeneous sensing data, improving diagnostic robustness, and enabling intelligent, condition-aware monitoring of mechanical and structural systems throughout their service life.

Dr. Rui Yuan
Prof. Dr. Weihang Zhu
Dr. Xingkai Yang
Dr. Zhuo Long
Guest Editors

Hongan Wu
Guest Editor Assistant

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • multivariate entropy
  • multivariate signal processing
  • fault diagnosis
  • structural health monitoring
  • prognostics and health management
  • remaining useful life prediction
  • multi-sensor data fusion
  • information-theoretic measures
  • nonstationary heterogeneous signals
  • intelligent condition monitoring

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Published Papers (3 papers)

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Research

24 pages, 18515 KB  
Article
Wind Turbine Blade Fault Diagnosis Integrating Multi-Scale Enhanced Hierarchical Fuzzy Entropy, Isolation Forest and GWO-GRU
by Min Wang, Xiao-Fei Zhang, Guo-Jun Qin and Ming Liu
Entropy 2026, 28(7), 810; https://doi.org/10.3390/e28070810 - 16 Jul 2026
Cited by 1 | Viewed by 381
Abstract
To effectively extract fault characteristics from complex vibration signals and improve the diagnostic performance of deep learning networks, this paper introduces a wind turbine blade fault diagnosis method that combines Multi-scale Enhanced Hierarchical Fuzzy Entropy (MEHFE), Isolation Forest, and the Grey Wolf Optimization [...] Read more.
To effectively extract fault characteristics from complex vibration signals and improve the diagnostic performance of deep learning networks, this paper introduces a wind turbine blade fault diagnosis method that combines Multi-scale Enhanced Hierarchical Fuzzy Entropy (MEHFE), Isolation Forest, and the Grey Wolf Optimization (GWO) algorithm for optimizing the Gated Recurrent Unit (GRU). Initially, the MEHFE algorithm is applied to decompose and reconstruct three-directional vibration signals at the blade root, thereby extracting “scale-frequency” dual-dimensional features that represent the evolution of fault frequency structure and complexity across multiple scales. Subsequently, Isolation Forest is employed to assess and filter feature importance, constructing an optimal feature subset to mitigate redundancy and noise interference. Finally, the optimal features are fed into the GRU network for fault pattern recognition, and the GWO algorithm is utilized to adaptively optimize network hyperparameters, thereby enhancing classification accuracy and noise resilience. Simulation experiments on typical wind turbine blade faults reveal that when GRU serves as the classifier, the diagnostic accuracy of MEHFE exceeds 76%. After feature optimization with Isolation Forest and network parameter optimization with GWO, the diagnostic accuracy surpasses 93%, demonstrating notable advantages in both classification capability and stability. Even under conditions of noise interference, the accuracy remains above 90%. The research substantiates that the proposed method can effectively extract pattern information indicative of blade structural damage from vibration data, achieving high fault recognition accuracy and robustness. Full article
(This article belongs to the Special Issue Multivariate Entropy-Informed Fault Diagnosis and Structural Health Monitoring)
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24 pages, 11738 KB  
Article
Balanced Adaptive Logit-Compensated Cross-Entropy and Quadratic Convolutional Network for Intelligent Fault Diagnosis Under Long-Tailed Data Distribution
by Wenbin Zhang, Zikang Cao, Haijian Wu, Dewei Guo and Yasong Pu
Entropy 2026, 28(7), 783; https://doi.org/10.3390/e28070783 - 10 Jul 2026
Cited by 1 | Viewed by 372
Abstract
Long-tailed data are very common in industrial scenarios because equipment failures occur with a low probability, resulting in far fewer faulty samples than normal ones. However, when facing long-tailed data distributions, existing deep learning methods suffer from a significant degradation in performance and [...] Read more.
Long-tailed data are very common in industrial scenarios because equipment failures occur with a low probability, resulting in far fewer faulty samples than normal ones. However, when facing long-tailed data distributions, existing deep learning methods suffer from a significant degradation in performance and exhibit high bias. To overcome this limitation, this paper proposes a network that combines balanced adaptive logit-compensated cross-entropy loss with quadratic convolution (BALQNet) to improve diagnostic performance under long-tailed data conditions. The proposed method mainly consists of a balanced adaptive logit-compensated cross-entropy loss (BAL) and a quadratic convolution backbone. By jointly incorporating logit compensation, label smoothing, and class reweighting, BAL enhances the optimization of minority-class samples, thereby improving the classifier’s ability to distinguish different categories without introducing additional architectural complexity. Meanwhile, quadratic convolution further improves the effectiveness of feature representation learning. Finally, experiments are conducted on self-built bearing, gear, and motor datasets. The results show that BALQNet maintains strong diagnostic performance when handling long-tailed data. In addition, the ablation results provide further evidence for the effectiveness of the proposed approach. Full article
(This article belongs to the Special Issue Multivariate Entropy-Informed Fault Diagnosis and Structural Health Monitoring)
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22 pages, 6270 KB  
Article
A Hybrid CNN-GRU-SE Forecasting Method for Short-Term Photovoltaic Power Considers AFD and Data Aggregation
by Keyan Liu, Dongli Jia, Huiyu Zhan, Jun Zhou, Zezhou Wang and Jianfei Bao
Entropy 2026, 28(5), 511; https://doi.org/10.3390/e28050511 - 1 May 2026
Viewed by 471
Abstract
To enhance the accuracy and robustness of short-term photovoltaic (PV) power forecasting, this paper proposes a novel forecasting method that integrates data aggregation, adaptive frequency decomposition (AFD), modified improved beluga whale optimization (MIBWO), and a CNN-GRU-SE hybrid model. First, the Pearson correlation coefficient [...] Read more.
To enhance the accuracy and robustness of short-term photovoltaic (PV) power forecasting, this paper proposes a novel forecasting method that integrates data aggregation, adaptive frequency decomposition (AFD), modified improved beluga whale optimization (MIBWO), and a CNN-GRU-SE hybrid model. First, the Pearson correlation coefficient and the entropy weight method are combined to screen meteorological features that are strongly correlated with PV power output. Considering the geographical distance, a spatial data aggregation strategy is proposed to exploit the spatial correlation among neighboring PV stations and suppress the output volatility of individual stations. Then, the AFD is adopted to adaptively decompose the PV power series into trend and seasonal components, and the MIBWO algorithm is utilized to optimize the cutoff frequency of AFD and key hyperparameters of the CNN-GRU-SE forecasting model simultaneously. Finally, the SHAP method is employed for model interpretability analysis to quantify the contribution of each feature to the prediction results. Simulation results verified the power forecasting accuracy and robustness of the proposed method. Compared with CNN-GRU and BWO-CNN-GRU-SE, the proposed method reduces MAE by 96.23% and 95.03%, respectively. The method maintains stable performance with sunny and cloudy conditions. Full article
(This article belongs to the Special Issue Multivariate Entropy-Informed Fault Diagnosis and Structural Health Monitoring)
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