Uncertainty measure in evidence theory

As an extension of probability theory, evidence theory is able to better handle unknown and imprecise information. Owing to its advantages, evidence theory has more flexibility and effectiveness for modeling and processing uncertain information. Uncertainty measure plays an essential role both in evidence theory and probability theory. In probability theory, Shannon entropy provides a novel perspective for measuring uncertainty. Various entropies exist for measuring the uncertainty of basic probability assignment (BPA) in evidence theory. However, from the standpoint of the requirements of uncertainty measurement and physics, these entropies are controversial. Therefore, the process for measuring BPA uncertainty currently remains an open issue in the literature. Firstly, this paper reviews the measures of uncertainty in evidence theory followed by an analysis of some related controversies. Secondly, we discuss the development of Deng entropy as an effective way to measure uncertainty, including introducing its definition, analyzing its properties, and comparing it to other measures. We also examine the concept of maximum Deng entropy, the pseudo-Pascal triangle of maximum Deng entropy, generalized belief entropy, and measures of divergence. In addition, we conduct an analysis of the application of Deng entropy and further examine the challenges for future studies on uncertainty measurement in evidence theory. Finally, a conclusion is provided to summarize this study.

[1]  Yong Deng,et al.  The Pseudo-Pascal Triangle of Maximum Deng Entropy , 2020, Int. J. Comput. Commun. Control.

[2]  Qing Liu,et al.  An Improved Deng Entropy and Its Application in Pattern Recognition , 2019, IEEE Access.

[3]  Jian-Bo Yang,et al.  On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty , 2002, IEEE Trans. Syst. Man Cybern. Part A.

[4]  Dong-Ling Xu,et al.  Evidential reasoning rule for evidence combination , 2013, Artif. Intell..

[5]  Fuyuan Xiao,et al.  An Improved Method for Combining Conflicting Evidences Based on the Similarity Measure and Belief Function Entropy , 2018, Int. J. Fuzzy Syst..

[6]  Luning Liu,et al.  An Evidential Reliability Indicator-Based Fusion Rule for Dempster-Shafer Theory and its Applications in Classification , 2018, IEEE Access.

[7]  Yong Deng,et al.  The Maximum Deng Entropy , 2015, IEEE Access.

[8]  Xinyang Deng,et al.  Analyzing the monotonicity of belief interval based uncertainty measures in belief function theory , 2017, Int. J. Intell. Syst..

[9]  Zhen Li,et al.  Emergency alternative evaluation under group decision makers: a new method based on entropy weight and DEMATEL , 2020, Int. J. Syst. Sci..

[10]  Fuyuan Xiao,et al.  EFMCDM: Evidential Fuzzy Multicriteria Decision Making Based on Belief Entropy , 2020, IEEE Transactions on Fuzzy Systems.

[11]  Yong Deng,et al.  Generalized Ordered Propositions Fusion Based on Belief Entropy , 2018, Int. J. Comput. Commun. Control.

[12]  Yong Deng,et al.  A New Belief Entropy to Measure Uncertainty of Basic Probability Assignments Based on Belief Function and Plausibility Function , 2018, Entropy.

[13]  Aihua Zhu,et al.  Bearing Fault Diagnosis Based on a Hybrid Classifier Ensemble Approach and the Improved Dempster-Shafer Theory , 2019, Sensors.

[14]  Fuyuan Xiao,et al.  Generalized belief function in complex evidence theory , 2020, J. Intell. Fuzzy Syst..

[15]  Kürşad Özkan.  Comparing Shannon entropy with Deng entropy and improved Deng entropy for measuring biodiversity when a priori data is not clear , 2018 .

[16]  G. Klir.  Uncertainty and Information: Foundations of Generalized Information Theory , 2005 .

[17]  Shanlin Yang,et al.  Multiple criteria group decision making with belief distributions and distributed preference relations , 2019, Eur. J. Oper. Res..

[18]  James C. Bezdek,et al.  Uncertainty measures for evidential reasoning I: A review , 1992, Int. J. Approx. Reason..

[19]  Maria Longobardi,et al.  A Dual Measure of Uncertainty: The Deng Extropy , 2020, Entropy.

[20]  Lotfi A. Zadeh,et al.  A Note on Z-numbers , 2011, Inf. Sci..

[21]  Quan Pan,et al.  Classifier Fusion With Contextual Reliability Evaluation , 2018, IEEE Transactions on Cybernetics.

[22]  Wen Jiang,et al.  An evidential Markov decision making model , 2017, Inf. Sci..

[23]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[24]  Souleymane Oumtanaga,et al.  A New Uncertainty Measure in Belief Entropy Framework , 2018 .

[25]  Lipeng Pan,et al.  Probability Transform Based on the Ordered Weighted Averaging and Entropy Difference , 2020, Int. J. Comput. Commun. Control.

[26]  Yingshun Li,et al.  Fire Control System Operation Status Assessment Based on Information Fusion: Case Study † , 2019, Sensors.

[27]  阿部 純義,et al.  Nonextensive statistical mechanics and its applications , 2001 .

[28]  George J. Klir,et al.  Uncertainty-Based Information , 1999 .

[29]  Xiaoyang Li,et al.  A Novel Belief Entropy for Measuring Uncertainty in Dempster-Shafer Evidence Theory Framework Based on Plausibility Transformation and Weighted Hartley Entropy , 2019, Entropy.

[30]  Yong Deng,et al.  DS-VIKOR: A New Multi-criteria Decision-Making Method for Supplier Selection , 2018, International Journal of Fuzzy Systems.

[31]  You He,et al.  New method for measuring the degree of conflict among general basic probability assignments , 2011, Science China Information Sciences.

[32]  Jun Sang,et al.  A novel weighted evidence combination rule based on improved entropy function with a diagnosis application , 2019, Int. J. Distributed Sens. Networks.

[33]  Yong Deng,et al.  An Improved Belief Entropy in Evidence Theory , 2020, IEEE Access.

[34]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[35]  A. L. Kuzemsky.  Temporal evolution, directionality of time and irreversibility , 2018 .

[36]  Fuyuan Xiao,et al.  A Novel Evidence Theory and Fuzzy Preference Approach-Based Multi-Sensor Data Fusion Technique for Fault Diagnosis , 2017, Sensors.

[37]  Xianguo Wu,et al.  Multi-classifier information fusion in risk analysis , 2020, Inf. Fusion.

[38]  Yongchuan Tang,et al.  Deng Entropy Weighted Risk Priority Number Model for Failure Mode and Effects Analysis , 2020, Entropy.

[39]  D. Dubois,et al.  Properties of measures of information in evidence and possibility theories , 1987 .

[40]  Fang Liu,et al.  A Novel Method of DS Evidence Theory for Multi-Sensor Conflicting Information , 2018 .

[41]  Richard D. Gill,et al.  Pearle’s Hidden-Variable Model Revisited , 2015, Entropy.

[42]  A. Dempster.  Upper and Lower Probabilities Generated by a Random Closed Interval , 1968 .

[43]  Yu Liu,et al.  Evidence Combination Based on Credal Belief Redistribution for Pattern Classification , 2020, IEEE Transactions on Fuzzy Systems.

[44]  Yafei Song,et al.  Uncertainty measure in evidence theory with its applications , 2017, Applied Intelligence.

[45]  Fuyuan Xiao,et al.  Negation of Belief Function Based on the Total Uncertainty Measure , 2019, Entropy.

[46]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[47]  N. Pal,et al.  QUANTIFICATION OF CONFLICT IN DEMPSTER-SHAFER FRAMEWORK: A NEW APPROACH , 1996 .

[48]  You He,et al.  Two adaptive detectors for range-spread targets in non-Gaussian clutter , 2010, Science China Information Sciences.

[49]  Wen Jiang,et al.  A new method to evaluate risk in failure mode and effects analysis under fuzzy information , 2018, Soft Comput..

[50]  Liguo Fei,et al.  An Improved Belief Entropy to Measure Uncertainty of Basic Probability Assignments Based on Deng Entropy and Belief Interval , 2019, Entropy.

[51]  Fuyuan Xiao,et al.  An improved distance-based total uncertainty measure in belief function theory , 2017, Applied Intelligence.

[52]  Yong Deng,et al.  Generalized Belief Entropy and Its Application in Identifying Conflict Evidence , 2019, IEEE Access.

[53]  Lipeng Pan,et al.  Uncertainty measure based on Tsallis entropy in evidence theory , 2019, Int. J. Intell. Syst..

[54]  Luning Liu,et al.  Evidence combination using OWA‐based soft likelihood functions , 2019, Int. J. Intell. Syst..

[55]  Fuyuan Xiao,et al.  Generalization of Dempster–Shafer theory: A complex mass function , 2020, Applied Intelligence.

[56]  Ronald R. Yager,et al.  Pythagorean fuzzy subsets , 2013, 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS).

[57]  Yong Deng,et al.  Combination of Evidential Sensor Reports with Distance Function and Belief Entropy in Fault Diagnosis , 2019, Int. J. Comput. Commun. Control.

[58]  Souleymane Oumtanaga,et al.  A belief entropy-based approach for conflict resolution in IoT applications , 2018, 2018 1st International Conference on Smart Cities and Communities (SCCIC).

[59]  Fuyuan Xiao,et al.  A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion , 2020, Inf. Sci..

[60]  Xiaoyang Li,et al.  A Novel Antagonistic Weapon-Target Assignment Model Considering Uncertainty and its Solution Using Decomposition Co-Evolution Algorithm , 2019, IEEE Access.

[61]  Bingyi Kang.  Construction of Stable Hierarchy Organization from the Perspective of the Maximum Deng Entropy , 2019, IUKM.

[62]  Constantino Tsallis,et al.  Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[63]  Wen Jiang,et al.  An improved evidential DEMATEL identify critical success factors under uncertain environment , 2019, Journal of Ambient Intelligence and Humanized Computing.

[64]  Fuyuan Xiao,et al.  Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy , 2019, Inf. Fusion.

[65]  Philippe Smets,et al.  Information Content of an Evidence , 1983, Int. J. Man Mach. Stud..

[66]  Éloi Bossé,et al.  Measuring ambiguity in the evidence theory , 2006, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[67]  Yong Deng,et al.  A new method to measure the divergence in evidential sensor data fusion , 2019, Int. J. Distributed Sens. Networks.

[68]  Yinjing Guo,et al.  Multisensor Fusion Method Based on the Belief Entropy and DS Evidence Theory , 2020, J. Sensors.

[69]  Lipeng Pan,et al.  An association coefficient of a belief function and its application in a target recognition system , 2019, Int. J. Intell. Syst..

[70]  Fuyuan Xiao,et al.  A Multiple-Criteria Decision-Making Method Based on D Numbers and Belief Entropy , 2019, International Journal of Fuzzy Systems.

[71]  Fuyuan Xiao,et al.  A Weighted Combination Method for Conflicting Evidence in Multi-Sensor Data Fusion , 2018, Sensors.

[72]  Ronald R. Yager,et al.  Entropy and Specificity in a Mathematical Theory of Evidence , 2008, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[73]  Ronald R. Yager,et al.  Interval valued entropies for Dempster-Shafer structu97res , 2018, Knowl. Based Syst..

[74]  Fuyuan Xiao,et al.  An Improved Multi-Source Data Fusion Method Based on the Belief Entropy and Divergence Measure , 2019, Entropy.

[75]  Wen Jiang,et al.  A new method to air target threat evaluation based on Dempster-Shafer evidence theory , 2018, 2018 Chinese Control And Decision Conference (CCDC).

[76]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[77]  Shanlin Yang,et al.  Multiple criteria group decision making based on group satisfaction , 2020, Inf. Sci..

[78]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[79]  Fuyuan Xiao,et al.  Combine Conflicting Evidence Based on the Belief Entropy and IOWA Operator , 2019, IEEE Access.

[80]  George J. Klir,et al.  A Note on the Measure of Discord , 1992, UAI.

[81]  Fuyuan Xiao,et al.  An Evidential Aggregation Method of Intuitionistic Fuzzy Sets Based on Belief Entropy , 2019, IEEE Access.

[82]  Zhiyong Gao,et al.  Failure mode and effects analysis using Dempster-Shafer theory and TOPSIS method: Application to the gas insulated metal enclosed transmission line (GIL) , 2018, Appl. Soft Comput..

[83]  Gend Lal Prajapati,et al.  REEDS: Relevance and enhanced entropy based Dempster Shafer approach for next word prediction using language model , 2019, J. Comput. Sci..

[84]  Fuyuan Xiao,et al.  GIQ: A Generalized Intelligent Quality-Based Approach for Fusing Multisource Information , 2019, IEEE Transactions on Fuzzy Systems.

[85]  Yun Liu,et al.  Collaborative Fusion for Distributed Target Classification Using Evidence Theory in IOT Environment , 2018, IEEE Access.

[86]  Emanuel Aldea,et al.  Evidential query-by-committee active learning for pedestrian detection in high-density crowds , 2019, Int. J. Approx. Reason..

[87]  Xinping Yan,et al.  An Evidential Reasoning‐Based CREAM to Human Reliability Analysis in Maritime Accident Process , 2017, Risk analysis : an official publication of the Society for Risk Analysis.

[88]  Joaquín Abellán,et al.  Maximum of Entropy for Belief Intervals Under Evidence Theory , 2020, IEEE Access.

[89]  Felix T. S. Chan,et al.  An Extension to Deng’s Entropy in the Open World Assumption with an Application in Sensor Data Fusion , 2018, Sensors.

[90]  Dan Wang,et al.  A New Belief Entropy Based on Deng Entropy , 2019, Entropy.

[91]  Joaquín Abellán,et al.  Analyzing properties of Deng entropy in the theory of evidence , 2017 .

[92]  Jian Wang,et al.  An improvement for combination rule in evidence theory , 2019, Future Gener. Comput. Syst..

[93]  Wen Jiang,et al.  An Uncertainty Measure for Interval-valued Evidences , 2017, Int. J. Comput. Commun. Control.

[94]  Li Fu,et al.  A Novel Fuzzy Approach for Combining Uncertain Conflict Evidences in the Dempster-Shafer Theory , 2019, IEEE Access.

[95]  Zhuo Zhang,et al.  A New Failure Mode and Effects Analysis Method Based on Dempster–Shafer Theory by Integrating Evidential Network , 2019, IEEE Access.

[96]  Qian Pan,et al.  A New Belief Entropy in Dempster–Shafer Theory Based on Basic Probability Assignment and the Frame of Discernment , 2020, Entropy.

[97]  Yong Deng.  The Information Volume of Uncertain Informaion: (1) Mass Function , 2020 .

[98]  Xinyang Deng,et al.  A total uncertainty measure for D numbers based on belief intervals , 2017, Int. J. Intell. Syst..

[99]  Luning Liu,et al.  On entropy function and reliability indicator for D numbers , 2019, Applied Intelligence.

[100]  Shanlin Yang,et al.  An evidential reasoning approach based on criterion reliability and solution reliability , 2019, Comput. Ind. Eng..

[101]  Yun Liu,et al.  A Weighted Evidence Combination Approach for Target Identification in Wireless Sensor Networks , 2017, IEEE Access.

[102]  Yi Yang,et al.  A new distance-based total uncertainty measure in the theory of belief functions , 2016, Knowl. Based Syst..

[103]  Jian-Bo Yang,et al.  Rule and utility based evidential reasoning approach for multiattribute decision analysis under uncertainties , 2001, Eur. J. Oper. Res..

[104]  Wang Zezhou,et al.  Evidence combination method in time domain based on reliability and importance , 2018 .

[105]  Yong Deng,et al.  Evidential Decision Tree Based on Belief Entropy , 2019, Entropy.

[106]  Rong Huang,et al.  Accurate solutions of product linear systems associated with rank-structured matrices , 2019, J. Comput. Appl. Math..

[107]  Yafei Song,et al.  A Novel Measure of Uncertainty in the Dempster-Shafer Theory , 2020, IEEE Access.

[108]  Fuyuan Xiao,et al.  An Intuitionistic Evidential Method for Weight Determination in FMEA Based on Belief Entropy , 2019, Entropy.

[109]  Yong Deng,et al.  Divergence Measure of Belief Function and Its Application in Data Fusion , 2019, IEEE Access.

[110]  Yong Deng,et al.  A New Divergence Measure of Pythagorean Fuzzy Sets Based on Belief Function and Its Application in Medical Diagnosis , 2020, Mathematics.

[111]  James C. Bezdek,et al.  Uncertainty measures for evidential reasoning II: A new measure of total uncertainty , 1993, Int. J. Approx. Reason..

[112]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[113]  Joel Dunham,et al.  Nonlinear Algorithms for Combining Conflicting Identification Information in Multisensor Fusion , 2019, 2019 IEEE Aerospace Conference.

[114]  Jerzy W. Grzymala-Busse,et al.  Rough Sets , 1995, Commun. ACM.

[115]  Fuyuan Xiao,et al.  CED: A Distance for Complex Mass Functions , 2020, IEEE Transactions on Neural Networks and Learning Systems.

[116]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[117]  George J. Klir,et al.  Uncertainty in the dempster-shafer Theory - A Critical Re-examination , 1990 .

[118]  Nazmuzzaman Khan,et al.  Time-Domain Data Fusion Using Weighted Evidence and Dempster–Shafer Combination Rule: Application in Object Classification , 2019, Sensors.

[119]  G. Klir,et al.  Uncertainty-based information: Elements of generalized information theory (studies in fuzziness and soft computing). , 1998 .

[120]  Fuyuan Xiao,et al.  A Distance Measure for Intuitionistic Fuzzy Sets and Its Application to Pattern Classification Problems , 2021, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[121]  S. Moral,et al.  MEASURES OF ENTROPY IN THE THEORY OF EVIDENCE , 1988 .

[122]  Djamel Djenouri,et al.  DFIOT: Data Fusion for Internet of Things , 2020, Journal of Network and Systems Management.

[123]  Prakash P. Shenoy,et al.  A new definition of entropy of belief functions in the Dempster-Shafer theory , 2018, Int. J. Approx. Reason..

[124]  Zdzis?aw Pawlak,et al.  Rough sets , 2005, International Journal of Computer & Information Sciences.

[125]  Fuyuan Xiao,et al.  An Improved Multisensor Data Fusion Method and Its Application in Fault Diagnosis , 2019, IEEE Access.

[126]  S. Abe,et al.  Nonextensive Statistical Mechanics and Its Applications , 2010 .

[127]  Lin Yang,et al.  Uncertainty measurement with belief entropy on interference effect in Quantum-Like Bayesian Networks , 2017, Appl. Math. Comput..