{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T18:07:43Z","timestamp":1785434863808,"version":"3.56.0"},"reference-count":60,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T00:00:00Z","timestamp":1778112000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62573349"],"award-info":[{"award-number":["62573349"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62203461"],"award-info":[{"award-number":["62203461"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["62573349"],"award-info":[{"award-number":["62573349"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"award":["62203461"],"award-info":[{"award-number":["62203461"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Health-state assessment is a critical component of prognostics and health management (PHM) for complex equipment. Previous studies on assessing the health state of complex equipment have overlooked the statistical dependence arising from causal coupling relationships between subsystems, which is defined as causality-informed correlation in this study. This correlation introduces redundancy in health information, leading to assessment bias. To address these limitations, this study proposes a health-state assessment model based on the evidential reasoning rule considering causality-informed correlation (ERr-CIC). First, the causal coupling relationships in dynamics and their effects on health-assessment results are analyzed. Based on this analysis, the convergent cross-mapping (CCM) method is employed to examine causal coupling between subsystems. Subsequently, a health-assessment model based on ERr-CIC is developed. This model incorporates a discount factor to quantify the causality-informed correlation among indicators, realized using a conditionally hybrid correlation coefficient (CHCC), and a fusion order derived from signaling sequences. Furthermore, a sensitivity and robustness analysis of the model output to the CHCC is conducted to identify the key parameters governing system behavior and to assess the reliability of the model results under parameter perturbations. Finally, experiments are performed on the PAMD simulation device for validation, and the proposed model is compared with three other typical health-state assessment models. The results show that the ERr-CIC model proposed in this paper achieves relatively balanced performance in terms of stability and interpretability while maintaining competitive model accuracy.<\/jats:p>","DOI":"10.3390\/e28050533","type":"journal-article","created":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T14:05:46Z","timestamp":1778162746000},"page":"533","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Causality-Informed Correlation-Aware Health-State Assessment for Complex Equipment"],"prefix":"10.3390","volume":"28","author":[{"given":"Wenbo","family":"Li","sequence":"first","affiliation":[{"name":"Graduate School of Rocket Force University of Engineering, Xi\u2019an 710025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhichao","family":"Feng","sequence":"additional","affiliation":[{"name":"Graduate School of Rocket Force University of Engineering, Xi\u2019an 710025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yijie","family":"Sun","sequence":"additional","affiliation":[{"name":"Graduate School of Rocket Force University of Engineering, Xi\u2019an 710025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinyi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Graduate School of Rocket Force University of Engineering, Xi\u2019an 710025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Zhang, A., Tang, L., and Hu, G. (2026). Micro Fault Diagnosis of Driving Motor Bearings Based on Multi-Residual Neural Networks and Evidence Reasoning Rule. Entropy, 28.","DOI":"10.3390\/e28010053"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Feng, Q., Liu, Y., Li, Y., Chang, G., Liang, X., Su, Y., and Cao, G. (2025). Study on a Fault Diagnosis Method for Heterogeneous Chiller Units Based on Transfer Learning. Entropy, 27.","DOI":"10.3390\/e27101049"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Zhang, Y., Qin, J., Duan, J., and Zhou, Y. (2025). Dynamic MAML with Efficient Multi-Scale Attention for Cross-Load Few-Shot Bearing Fault Diagnosis. Entropy, 27.","DOI":"10.3390\/e27101063"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"111388","DOI":"10.1016\/j.engappai.2025.111388","article-title":"Distributed Fault-Tolerant for Leader-Following Multi-Unmanned Aerial Vehicle Systems with Faulty Sensors based on Belief Rule Base","volume":"157","author":"Yang","year":"2025","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1020","DOI":"10.1109\/TR.2026.3660916","article-title":"Reliability Analysis Based on Evidential Likelihood for Uncertain Mixed Weibull Distribution","volume":"75","author":"Zhou","year":"2026","journal-title":"IEEE Trans. Reliab."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"6754","DOI":"10.1109\/TNSE.2026.3658546","article-title":"Importance Ranking in Complex Networks via Influence-Aware Causal Node Embedding","volume":"13","author":"Gao","year":"2026","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1016\/j.renene.2018.04.033","article-title":"A prediction method for the real-time remaining useful life of wind turbine bearings based on the Wiener process","volume":"127","author":"Hu","year":"2018","journal-title":"Renew. Energy"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"123890","DOI":"10.1016\/j.energy.2022.123890","article-title":"Weight optimized unscented Kalman filter for degradation trend prediction of lithium-ion battery with error compensation strategy","volume":"251","author":"Zhang","year":"2022","journal-title":"Energy"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"111502","DOI":"10.1016\/j.measurement.2022.111502","article-title":"Measurement of health evolution tendency for aircraft engine using a data-driven method based on multi-scale series reconstruction and adaptive hybrid model","volume":"199","author":"Jiang","year":"2022","journal-title":"Measurement"},{"key":"ref_10","first-page":"12","article-title":"Bearing performance degradation trend prediction sparrow search algorithm optimization bidirectional gating cycle unit","volume":"42","author":"Chen","year":"2023","journal-title":"J. Vib. Shock."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"110387","DOI":"10.1016\/j.ress.2024.110387","article-title":"A health assessment method with attribute importance modeling for complex systems using belief rule base","volume":"251","author":"Lian","year":"2024","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1016\/j.fss.2007.10.011","article-title":"A new hybrid artificial neural networks and fuzzy regression model for time series forecasting","volume":"159","author":"Khashei","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1458","DOI":"10.1109\/21.310529","article-title":"A general multi-level evaluation process for hybrid MADM with uncertainty","volume":"24","author":"Yang","year":"1994","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1109\/TSMCA.2002.802746","article-title":"On the Evidential Reasoning Algorithm for Multiple Attribute Decision Analysis Under Uncertainty","volume":"32","author":"Yang","year":"2002","journal-title":"IEEE Trans. Syst. Man Cybern. Part A Syst. Hum."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/j.ejor.2004.09.017","article-title":"The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties","volume":"171","author":"Yang","year":"2006","journal-title":"Eur. J. Oper. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.artint.2013.09.003","article-title":"Evidential reasoning rule for evidence combination","volume":"205","author":"Yang","year":"2013","journal-title":"Artif. Intell."},{"key":"ref_17","first-page":"1004","article-title":"A Method for Combining Correlated Evidence","volume":"22","author":"Sun","year":"1999","journal-title":"Chin. J. Comput."},{"key":"ref_18","unstructured":"Ferson, S., Hajagos, J., Berleant, D.J., Zhang, J., Tucker, W.T., Ginzburg, L.R., and Oberkampf, W.L. (2004). Dependence in Probabilistic Modeling, Dempster-Shafer Theory, and Probability Bounds Analysis, Sandia National Laboratories."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1142\/S021848850700439X","article-title":"Modeling Dependence in Dempster-Shafer Theory","volume":"15","author":"Monney","year":"2007","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_20","first-page":"773","article-title":"A Combination Rule for Correlated Evidence","volume":"26","author":"Xiao","year":"2011","journal-title":"Control Decis."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/0020-0255(95)00226-X","article-title":"On the Evidence Inference Theory","volume":"89","author":"Wu","year":"1996","journal-title":"Inf. Sci."},{"key":"ref_22","first-page":"970","article-title":"Fusion of Correlated Information Using Evidence Theory","volume":"23","author":"Luo","year":"2001","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/0020-0255(87)90007-7","article-title":"On the Dempster-Shafer Framework and New Combination Rules","volume":"41","author":"Yager","year":"1987","journal-title":"Inf. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1526","DOI":"10.1109\/TAES.2025.3632264","article-title":"An Evidential Reasoning Rule-Based Performance Evaluation Method for Complex Electronic System Under Off-Cycle Data","volume":"62","author":"Sun","year":"2026","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_25","first-page":"7926","article-title":"Performance Evaluation Method for Complex Systems Based on Outcome-Oriented Correlated Evidence Reasoning","volume":"72","author":"Sun","year":"2025","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"107515","DOI":"10.1016\/j.engfailanal.2023.107515","article-title":"Complex physical-model based dynamic system safety analysis of Aviation Piston Engine considering hybrid uncertainty of fault","volume":"152","author":"Li","year":"2023","journal-title":"Eng. Fail. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Fu, X., Yang, M., Liu, H., Wang, L., and Li, Q. (2024). Risk Analysis and Simulation of Large Bridge Construction Based on System Dynamics. Buildings, 14.","DOI":"10.3390\/buildings14051488"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3160","DOI":"10.30574\/ijsra.2024.13.1.2047","article-title":"Integrating machine learning with causal inference for enhanced system dynamics modeling: A framework for predicting complex interactions","volume":"13","author":"Kamdem","year":"2024","journal-title":"Int. J. Sci. Res. Arch."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Chen, H., Wang, J.-G., Ding, P., Ye, X.-Y., Yao, Y., and Chen, H.-L. (2023). A Granger causality analysis method based on GRBF network. Proceedings of the 2023 IEEE 12th Data Driven Control and Learning Systems Conference (DDCLS), Xiangtan, China, 12\u201314 May 2023, IEEE.","DOI":"10.1109\/DDCLS58216.2023.10166901"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"122827","DOI":"10.1016\/j.eswa.2023.122827","article-title":"Graph neural network-based bearing fault diagnosis using Granger causality test","volume":"242","author":"Zhang","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Qu, H., Liu, Y., Liu, H., and Wang, B. (2025). TF-Based Causal Inference for Industrial Alarm Overload Mitigation. Electronics, 14.","DOI":"10.3390\/electronics14204066"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Liu, X., Liu, J., Yang, X., Wu, Z., Wei, Y., Xu, Z., and Wen, J. (2025). Fault Root Cause Analysis Based on Liang\u2013Kleeman Information Flow and Graphical Lasso. Entropy, 27.","DOI":"10.3390\/e27020213"},{"key":"ref_33","unstructured":"Sharma, S., Lakshminarayanan, S., Karimi, I., and Srinivasan, R. (2021). Convergent Cross-mapping based Fault Detection and Diagnosis for Non-linear Dynamic Systems. Proceedings of the 2021 60th Annual Conference of the Society of Instrument and Control Engineers (SICE), Tokyo, Japan, 8\u201310 September 2021, IEEE."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"13619","DOI":"10.1016\/j.ifacol.2020.12.858","article-title":"Causal network construction based on convergent cross mapping (CCM) for alarm system root cause tracing of nonlinear industrial process","volume":"53","author":"Tian","year":"2020","journal-title":"IFAC-PapersOnLine"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"104124","DOI":"10.1016\/j.irfa.2025.104124","article-title":"Forecasting Crude Oil Prices: A Gated Recurrent Unit-Based Nonlinear Granger Causality Model","volume":"102","author":"Liang","year":"2025","journal-title":"Int. Rev. Financ. Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2595","DOI":"10.1109\/TSP.2023.3286529","article-title":"On Causal Discovery with Convergent Cross Mapping","volume":"71","author":"Butler","year":"2023","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1137\/130946344","article-title":"On the Efficacy of State Space Reconstruction Methods in Determining Causality","volume":"14","author":"Cummins","year":"2015","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s003329900072","article-title":"Delay Embeddings for Forced Systems. I. Deterministic Forcing","volume":"9","author":"Stark","year":"1999","journal-title":"J. Nonlinear Sci."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Wallot, S., and M\u00f8nster, D. (2018). Calculation of Average Mutual Information (AMI) and False-Nearest Neighbors (FNN) for the Estimation of Embedding Parameters of Multidimensional Time Series in Matlab. Front. Psychol., 9.","DOI":"10.3389\/fpsyg.2018.01679"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s10462-023-10626-w","article-title":"Feature subset selection for multi-scale neighborhood decision information system via mutual information","volume":"57","author":"Zhang","year":"2024","journal-title":"Artif. Intell. Rev."},{"key":"ref_41","first-page":"2769","article-title":"Measuring and testing dependence by correlation of distances","volume":"35","author":"Rizzo","year":"2007","journal-title":"Ann. Stat."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/j.engfailanal.2013.04.002","article-title":"Reliability analysis of stiffened tank-roof stability with multiple random variables using minimum distance and lagrange methods","volume":"32","author":"He","year":"2013","journal-title":"Eng. Fail. Anal."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"954","DOI":"10.1109\/TSMC.2016.2630800","article-title":"A new evidential reasoning-based method for online safety assessment of complex systems","volume":"48","author":"Zhao","year":"2018","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"4027","DOI":"10.1109\/TCYB.2020.3015664","article-title":"A New Evidential Reasoning Rule-Based Safety Assessment Method With Sensor Reliability for Complex Systems","volume":"52","author":"Tang","year":"2022","journal-title":"IEEE Trans. Cybern."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIM.2021.3135317","article-title":"A Concurrent Fault Diagnosis Model via the Evidential Reasoning Rule","volume":"71","author":"Ning","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Yang, J.-B., and Xu, D.-L. (2017). Inferential modelling and decision making with data. Proceedings of the 2017 23rd International Conference on Automation and Computing (ICAC), Huddersfield, UK, 7\u20138 September 2017, IEEE.","DOI":"10.23919\/IConAC.2017.8082048"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"5142","DOI":"10.1103\/PhysRevE.61.5142","article-title":"Learning Driver-Response Relationships from Synchronization Patterns","volume":"61","author":"Quiroga","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Wang, Z., Yang, X., and Cai, H. (2025). Analysis of Influencing Factors of Soil Erosion Changes Based on Structural Equation Model. Land, 14.","DOI":"10.3390\/land14020304"},{"key":"ref_49","first-page":"116","article-title":"Measuring principle of electronic theodolite and its applications in spacecraft inspection","volume":"24","author":"Liu","year":"2007","journal-title":"Spacecr. Environ. Eng."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Mu, Y., Hou, N., Wang, C., Zhao, Y., Chen, K., and Chi, Y. (2022). An Optoelectronic Detector with High Precision for Compact Grating Encoder Application. Electronics, 11.","DOI":"10.3390\/electronics11213486"},{"key":"ref_51","first-page":"2076","article-title":"Simulation and Analysis of the Electromechanical Coupling Dynamic Model of Photoelectric Theodolite","volume":"37","author":"Xu","year":"2008","journal-title":"Acta Photonica Sin."},{"key":"ref_52","first-page":"1577","article-title":"Electromechanical Dynamics Modeling and Coupling of Photoelectric Theodolite","volume":"15","author":"Li","year":"2007","journal-title":"Opt. Precis. Eng."},{"key":"ref_53","unstructured":"Li, K., Yuan, F., Ding, Z., and Qiu, Z. (2011). Vision measurement error compensation research of double-theodolite based on neural network approaching. Proceedings of the 30th Chinese Control Conference, Yantai, China, 22\u201324 July 2011, IEEE."},{"key":"ref_54","first-page":"952","article-title":"Calibration of theodolites","volume":"Volume 2","author":"Jarvis","year":"1988","journal-title":"Proceedings of the 1988 IEEE International Conference on Robotics and Automation, Philadelphia, PA, USA, 24\u201329 April 1988"},{"key":"ref_55","unstructured":"(2012). Reliability Testing\u2014Part 1: Test Conditions and Statistical Test Principles (Standard No. GB\/T 5080.1-2012)."},{"key":"ref_56","unstructured":"(2018). Electronic Theodolite (Standard No. GB\/T 36537-2018)."},{"key":"ref_57","unstructured":"(2018). General Requirement for Photoelectric Detection Instruments Reliability (Standard No. GB\/T 37084-2018)."},{"key":"ref_58","first-page":"1446","article-title":"Rolling bearing fault diagnosis method based on MFMD and Transformer-CNN","volume":"38","author":"Junfeng","year":"2023","journal-title":"J. Aerosp. Power"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"110537","DOI":"10.1016\/j.patcog.2024.110537","article-title":"Self-attention empowered graph convolutional network for structure learning and node embedding","volume":"153","author":"Jiang","year":"2024","journal-title":"Pattern Recognit."},{"key":"ref_60","first-page":"7","article-title":"Fault diagnosis based on fuzzy Bayesian risk and T-S fuzzy model","volume":"38","author":"Tao","year":"2019","journal-title":"Meas. Control Technol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/5\/533\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T13:56:32Z","timestamp":1779371792000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/5\/533"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,7]]},"references-count":60,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["e28050533"],"URL":"https:\/\/doi.org\/10.3390\/e28050533","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,7]]}}}