{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T11:32:30Z","timestamp":1782300750419,"version":"3.54.5"},"reference-count":50,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,2]],"date-time":"2022-11-02T00:00:00Z","timestamp":1667347200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NWPU Research Fund for Young Scholars","award":["G2022WD01010"],"award-info":[{"award-number":["G2022WD01010"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Dempster\u2013Shafer evidence theory is widely used in modeling and reasoning uncertain information in real applications. Recently, a new perspective of modeling uncertain information with the negation of evidence was proposed and has attracted a lot of attention. Both the basic probability assignment (BPA) and the negation of BPA in the evidence theory framework can model and reason uncertain information. However, how to address the uncertainty in the negation information modeled as the negation of BPA is still an open issue. Inspired by the uncertainty measures in Dempster\u2013Shafer evidence theory, a method of measuring the uncertainty in the negation evidence is proposed. The belief entropy named Deng entropy, which has attracted a lot of attention among researchers, is adopted and improved for measuring the uncertainty of negation evidence. The proposed measure is defined based on the negation function of BPA and can quantify the uncertainty of the negation evidence. In addition, an improved method of multi-source information fusion considering uncertainty quantification in the negation evidence with the new measure is proposed. Experimental results on a numerical example and a fault diagnosis problem verify the rationality and effectiveness of the proposed method in measuring and fusing uncertain information.<\/jats:p>","DOI":"10.3390\/e24111596","type":"journal-article","created":{"date-parts":[[2022,11,3]],"date-time":"2022-11-03T03:11:21Z","timestamp":1667445081000},"page":"1596","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":47,"title":["Measuring Uncertainty in the Negation Evidence for Multi-Source Information Fusion"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2568-9628","authenticated-orcid":false,"given":"Yongchuan","family":"Tang","sequence":"first","affiliation":[{"name":"School of Microelectronics, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yong","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Big Data and Software Engineering, Chongqing University, Chongqing 401331, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Deyun","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Microelectronics, Northwestern Polytechnical University, Xi\u2019an 710072, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"113246","DOI":"10.1016\/j.dss.2020.113246","article-title":"Bayesian neural networks for flight trajectory prediction and safety assessment","volume":"131","author":"Zhang","year":"2020","journal-title":"Decis. Support Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"108835","DOI":"10.1016\/j.ress.2022.108835","article-title":"A generic physics-informed neural network-based framework for reliability assessment of multi-state systems","volume":"229","author":"Zhou","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1214\/aoms\/1177698950","article-title":"Upper and Lower Probabilities Induced by a Multi-valued Mapping","volume":"38","author":"Dempster","year":"1967","journal-title":"Ann. Math. Stat."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Shafer, G. (1976). A Mathematical Theory of Evidence, Princeton University Press.","DOI":"10.1515\/9780691214696"},{"key":"ref_5","first-page":"1","article-title":"Unsupervised Change Detection From Heterogeneous Data Based on Image Translation","volume":"60","author":"Liu","year":"2022","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1109\/TFUZZ.2018.2869125","article-title":"BPEC: Belief-peaks evidential clustering","volume":"27","author":"Su","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"109937","DOI":"10.1016\/j.knosys.2022.109937","article-title":"TECM: Transfer learning-based evidential c-means clustering","volume":"257","author":"Jiao","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"103985","DOI":"10.1016\/j.engappai.2020.103985","article-title":"A new classification method based on the negation of a basic probability assignment in the evidence theory","volume":"96","author":"Wu","year":"2020","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1016\/j.ins.2022.08.077","article-title":"Interpretable fuzzy clustering using unsupervised fuzzy decision trees","volume":"611","author":"Jiao","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.ijar.2017.09.008","article-title":"SELP: Semi\u2013supervised evidential label propagation algorithm for graph data clustering","volume":"92","author":"Zhou","year":"2018","journal-title":"Int. J. Approx. Reason."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2015","DOI":"10.1109\/TNNLS.2020.2995862","article-title":"Combination of transferable classification with multisource domain adaptation based on evidential reasoning","volume":"32","author":"Liu","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1256","DOI":"10.1007\/s40815-017-0436-5","article-title":"An improved method for combining conflicting evidences based on the similarity measure and belief function entropy","volume":"20","author":"Xiao","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1007\/s40815-019-00746-3","article-title":"Comparison of Evidential Reasoning Algorithm with Linear Combination in Decision Making","volume":"22","author":"Fu","year":"2020","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1016\/j.ejor.2018.08.012","article-title":"Multiple criteria group decision making with belief distributions and distributed preference relations","volume":"273","author":"Fu","year":"2019","journal-title":"Eur. J. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"106355","DOI":"10.1016\/j.cie.2020.106355","article-title":"An extended best-worst multi-criteria decision-making method by belief functions and its applications in hospital service evaluation","volume":"142","author":"Fei","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.knosys.2017.06.014","article-title":"An improved Dempster\u2013Shafer approach to construction safety risk perception","volume":"132","author":"Zhang","year":"2017","journal-title":"Knowl.-Based Syst."},{"key":"ref_17","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_18","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1016\/j.anucene.2017.10.045","article-title":"Dependence assessment in human reliability analysis based on evidence credibility decay model and IOWA operator","volume":"112","author":"Zheng","year":"2018","journal-title":"Ann. Nucl. Energy"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1296","DOI":"10.1111\/risa.12347","article-title":"Dependence Assessment in Human Reliability Analysis Using Evidence Theory and AHP","volume":"35","author":"Su","year":"2015","journal-title":"Risk Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1321","DOI":"10.1007\/s40815-017-0400-4","article-title":"Evidential supplier selection based on DEMATEL and game theory","volume":"20","author":"Liu","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.engappai.2018.08.010","article-title":"A new failure mode and effects analysis model using Dempster\u2013Shafer evidence theory and grey relational projection method","volume":"76","author":"Chen","year":"2018","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1786","DOI":"10.1002\/qre.2660","article-title":"An improved failure mode and effects analysis method based on uncertainty measure in the evidence theory","volume":"36","author":"Wu","year":"2020","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1550147718823990","DOI":"10.1177\/1550147718823990","article-title":"A novel weighted evidence combination rule based on improved entropy function with a diagnosis application","volume":"15","author":"Chen","year":"2019","journal-title":"Int. J. Distrib. Sens. Netw."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.isatra.2019.09.006","article-title":"Evidence updating with static and dynamical performance analyses for industrial alarm system design","volume":"99","author":"Xu","year":"2020","journal-title":"ISA Trans."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1016\/j.ins.2021.08.088","article-title":"Conflicting evidence combination from the perspective of networks","volume":"580","author":"Xiong","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"9793","DOI":"10.1007\/s00500-019-03804-y","article-title":"A new rule to combine dependent bodies of evidence","volume":"23","author":"Su","year":"2019","journal-title":"Soft Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.inffus.2018.04.003","article-title":"Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy","volume":"46","author":"Xiao","year":"2019","journal-title":"Inf. Fusion"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"934","DOI":"10.1007\/s10489-016-0877-9","article-title":"A method to determine basic probability assignment in the open world and its application in data fusion and classification","volume":"46","author":"Zhang","year":"2017","journal-title":"Appl. Intell."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1056","DOI":"10.1007\/s10489-020-01876-0","article-title":"A new base basic probability assignment approach for conflict data fusion in the evidence theory","volume":"51","author":"Jing","year":"2021","journal-title":"Appl. Intell."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.ijar.2017.10.010","article-title":"A new definition of entropy of belief functions in the Dempster\u2013Shafer theory","volume":"92","author":"Shenoy","year":"2018","journal-title":"Int. J. Approx. Reason."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.ijar.2020.01.004","article-title":"On properties of a new decomposable entropy of Dempster\u2013Shafer belief functions","volume":"119","author":"Shenoy","year":"2020","journal-title":"Int. J. Approx. Reason."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/j.physa.2018.02.036","article-title":"Information measures of a deformed harmonic oscillator in a static electric field","volume":"499","author":"Nascimento","year":"2018","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1002\/int.22094","article-title":"Uncertainty and negation-Information theoretic applications","volume":"34","author":"Srivastava","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_34","first-page":"107","article-title":"Evaluation of hotel websites using the multicriteria analysis of PROMETHEE and GAIA: Evidence from the five-star hotels of Mashhad","volume":"30","author":"Ostovare","year":"2019","journal-title":"Tour. Manag. Perspect."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1080\/03081079608945130","article-title":"Quantification of conflict in Dempster-Shafer framework: A new approach","volume":"24","author":"George","year":"1996","journal-title":"Int. J. Gen. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.measurement.2015.11.032","article-title":"Uncertainty measure for interval-valued belief structures","volume":"80","author":"Song","year":"2015","journal-title":"Measurement"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.chaos.2016.07.014","article-title":"Deng entropy","volume":"91","author":"Deng","year":"2016","journal-title":"Chaos Solitons Fractals"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1002\/int.21980","article-title":"An improved soft likelihood function for Dempster\u2013Shafer belief structures","volume":"33","author":"Jiang","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.ijar.2018.09.001","article-title":"A correlation coefficient for belief functions","volume":"103","author":"Jiang","year":"2018","journal-title":"Int. J. Approx. Reason."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.ins.2021.12.032","article-title":"Fractal-based belief entropy","volume":"587","author":"Zhou","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11432-020-3006-9","article-title":"Uncertainty measure in evidence theory","volume":"63","author":"Deng","year":"2020","journal-title":"Sci. China Inf. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1899","DOI":"10.1109\/TFUZZ.2014.2374211","article-title":"On the maximum entropy negation of a probability distribution","volume":"23","author":"Yager","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1109\/TFUZZ.2018.2871756","article-title":"The negation of a basic probability assignment","volume":"27","author":"Yin","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/j.ins.2019.12.080","article-title":"On the negation of a Dempster\u2013Shafer belief structure based on maximum uncertainty allocation","volume":"516","author":"Deng","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1007\/s10044-021-00966-0","article-title":"A new approach for generation of generalized basic probability assignment in the evidence theory","volume":"24","author":"Tang","year":"2021","journal-title":"Pattern Anal. Appl."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.dss.2004.04.015","article-title":"Combining belief functions based on distance of evidence","volume":"38","author":"Deng","year":"2004","journal-title":"Decis. Support Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"9562","DOI":"10.3390\/s140609562","article-title":"Novel algorithm for identifying and fusing conflicting data in wireless sensor networks","volume":"14","author":"Zhang","year":"2014","journal-title":"Sensors"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1186\/s40064-016-2205-6","article-title":"Conflict management based on belief function entropy in sensor fusion","volume":"5","author":"Yuan","year":"2016","journal-title":"SpringerPlus"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Jiang, W., Xie, C., Zhuang, M., Shou, Y., and Tang, Y. (2016). Sensor Data Fusion with Z-Numbers and Its Application in Fault Diagnosis. Sensors, 16.","DOI":"10.3390\/s16091509"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Tang, Y., Zhou, D., Xu, S., and He, Z. (2017). A weighted belief entropy-based uncertainty measure for multi-sensor data fusion. Sensors, 17.","DOI":"10.3390\/s17040928"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1596\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:09:35Z","timestamp":1760144975000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1596"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,2]]},"references-count":50,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["e24111596"],"URL":"https:\/\/doi.org\/10.3390\/e24111596","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,2]]}}}