Fractal-based belief entropy

The total uncertainty measurement of basic probability assignment (BPA) in evidence theory has always been an open issue. Although many scholars have put forward various measures and requirements of bodies of evidence (BoE), none of them are widely recognized. So in order to express the uncertainty in evidence theory, transforming basic probability assignment (BPA) into probability distribution is a widely used method, but all the previous methods of probability transformation are directly allocating focal elements in evidence theory to their elements without specific transformation process. Based on above, this paper simulates the pignistic probability transformation (PPT) process based on the idea of fractal, making the PPT process and the information volume lost during transformation more intuitive. Then apply this idea to the total uncertainty measure in evidence theory. A new belief entropy called Fractal-based (FB) entropy is proposed, which is the first time to apply fractal idea in belief entropy. After verification, the new entropy is superior to all existing total uncertainty measurements.

[1]  Jian-Bo Yang,et al.  Data classification using evidence reasoning rule , 2017, Knowl. Based Syst..

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

[3]  Liguo Fei,et al.  An Attitudinal Nonlinear Integral and Applications in Decision Making , 2020, International Journal of Fuzzy Systems.

[4]  Fabio Cuzzolin,et al.  On the relative belief transform , 2012, Int. J. Approx. Reason..

[5]  George J. Klir,et al.  Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems - Selected Papers by Lotfi A Zadeh , 1996, Advances in Fuzzy Systems - Applications and Theory.

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

[7]  Bingyou Liu,et al.  A Method for Recognizing Fatigue Driving Based on Dempster-Shafer Theory and Fuzzy Neural Network , 2017 .

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

[9]  Yong Deng,et al.  A Matrix Method of Basic Belief Assignment's Negation in Dempster–Shafer Theory , 2020, IEEE Transactions on Fuzzy Systems.

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

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

[12]  Claude E. Shannon,et al.  The mathematical theory of communication , 1950 .

[13]  G. Klir,et al.  MEASURING TOTAL UNCERTAINTY IN DEMPSTER-SHAFER THEORY: A NOVEL APPROACH , 1994 .

[14]  Yong DENG,et al.  Uncertainty measure in evidence theory , 2020, Science China Information Sciences.

[15]  Joaquín Abellán,et al.  Critique of modified Deng entropies under the evidence theory , 2020 .

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

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

[18]  Jin Wang,et al.  An Efficient Algorithm for Influence Maximization Based on Propagation Path Analysis , 2017, CSA/CUTE.

[19]  Kang Hao Cheong,et al.  Parrondo’s paradox from classical to quantum: A review , 2020 .

[21]  Eloi Bosse,et al.  Critique of Recent Uncertainty Measures Developed Under the Evidence Theory and Belief Intervals , 2020, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[22]  Limao Zhang,et al.  Improved Fuzzy Bayesian Network-Based Risk Analysis With Interval-Valued Fuzzy Sets and D–S Evidence Theory , 2020, IEEE Transactions on Fuzzy Systems.

[23]  Hamido Fujita,et al.  Enhancing PROMETHEE method with intuitionistic fuzzy soft sets , 2020, Int. J. Intell. Syst..

[24]  Fuyuan Xiao.  CEQD: A Complex Mass Function to Predict Interference Effects , 2021, IEEE Transactions on Cybernetics.

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

[26]  Edward N. Lorenz,et al.  Available Potential Energy and the Maintenance of the General Circulation , 1955 .

[28]  Yong Deng.  Information Volume of Mass Function , 2020, Int. J. Comput. Commun. Control.

[29]  Asghar Shahpari,et al.  Using Mutual Aggregate Uncertainty Measures in a Threat Assessment Problem Constructed by Dempster–Shafer Network , 2015, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[30]  G. Klir,et al.  MEASURES OF UNCERTAINTY AND INFORMATION BASED ON POSSIBILITY DISTRIBUTIONS , 1982 .

[31]  Andrés R. Masegosa,et al.  Requirements for total uncertainty measures in Dempster–Shafer theory of evidence , 2008, Int. J. Gen. Syst..

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

[33]  Alex Pentland,et al.  Fractal-Based Description of Natural Scenes , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[34]  Xinyang Deng,et al.  A new probability transformation method based on a correlation coefficient of belief functions , 2019, Int. J. Intell. Syst..

[35]  Philippe Smets,et al.  The Transferable Belief Model , 1991, Artif. Intell..

[36]  Fuyuan Xiao,et al.  On the Maximum Entropy Negation of a Complex-Valued Distribution , 2019, IEEE Transactions on Fuzzy Systems.

[37]  Prakash P. Shenoy,et al.  On the plausibility transformation method for translating belief function models to probability models , 2006, Int. J. Approx. Reason..

[38]  Philippe Smets,et al.  Decision making in the TBM: the necessity of the pignistic transformation , 2005, Int. J. Approx. Reason..

[39]  Zhijie Zhou,et al.  A BRB-Based Effective Fault Diagnosis Model for High-Speed Trains Running Gear Systems , 2022, IEEE Transactions on Intelligent Transportation Systems.

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

[41]  Ali Karci,et al.  Fractional order entropy: New perspectives , 2016 .

[42]  Yongchuan Tang,et al.  A modified belief entropy in Dempster-Shafer framework , 2017, PloS one.

[43]  Éloi Bossé,et al.  A new distance between two bodies of evidence , 2001, Inf. Fusion.

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

[45]  B. Mandelbrot.  How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , 1967, Science.

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

[47]  Éloi Bossé,et al.  Drawbacks of Uncertainty Measures Based on the Pignistic Transformation , 2018, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

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

[49]  Wen Jiang,et al.  A Novel Z-Network Model Based on Bayesian Network and Z-Number , 2020, IEEE Transactions on Fuzzy Systems.

[50]  Shuai Xu,et al.  A Weighted Belief Entropy-Based Uncertainty Measure for Multi-Sensor Data Fusion , 2017, Sensors.

[51]  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.

[52]  Chin-Teng Lin,et al.  Multi-channel EEG recordings during a sustained-attention driving task , 2018, Scientific Data.

[53]  Milan Daniel.  On transformations of belief functions to probabilities , 2006, Int. J. Intell. Syst..

[54]  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.

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

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

[57]  Yong Hu,et al.  A new information dimension of complex networks , 2013, ArXiv.

[58]  Frank Lad,et al.  Extropy: a complementary dual of entropy , 2011, ArXiv.

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

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

[61]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

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

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

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

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

[66]  Quan Pan,et al.  Combination of Classifiers With Optimal Weight Based on Evidential Reasoning , 2018, IEEE Transactions on Fuzzy Systems.

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

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

[69]  Yafei SONG,et al.  Self-adaptive combination method for temporal evidence based on negotiation strategy , 2020, Science China Information Sciences.

[70]  Oldrich Zmeskal,et al.  Entropy of fractal systems , 2013, Comput. Math. Appl..

[71]  Liguo Fei,et al.  An extended best-worst multi-criteria decision-making method by belief functions and its applications in hospital service evaluation , 2020, Comput. Ind. Eng..

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

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

[74]  Huchang Liao,et al.  A Deng-Entropy-Based Evidential Reasoning Approach for Multi-expert Multi-criterion Decision-Making with Uncertainty , 2020, Int. J. Comput. Intell. Syst..

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

[76]  Yong Deng,et al.  A New Probability Transformation Based on the Ordered Visibility Graph , 2016, Int. J. Intell. Syst..

[77]  Yu-Wang Chen,et al.  Weight assignment method for multiple attribute decision making with dissimilarity and conflict of belief distributions , 2020, Comput. Ind. Eng..

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