Power Law and Dimension of the Maximum Value for Belief Distribution With the Maximum Deng Entropy

Deng entropy is a novel and efficient uncertainty measure to deal with imprecise phenomenon, which is an extension of Shannon entropy. In this paper, power law and dimension of the maximum value for belief distribution with the max Deng entropy are presented, which partially uncover the inherent physical meanings of Deng entropy from the perspective of statistics. This indicated some work related to power law or scale-free can be analyzed using Deng entropy. The results of some numerical simulations are used to support the new views.

[1]  Jiming Liu,et al.  A New Uncertainty Measure for Belief Networks with Applications to Optimal Evidential Inferencing , 2001, IEEE Trans. Knowl. Data Eng..

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

[3]  Daniel A. Lidar,et al.  Is the Geometry of Nature Fractal? , 1998, Science.

[4]  Josip Pecaric,et al.  On Zipf-Mandelbrot entropy , 2019, J. Comput. Appl. Math..

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

[6]  Wen Jiang,et al.  An evidential sensor fusion method in fault diagnosis , 2016 .

[7]  R.A. Aliev,et al.  The arithmetic of continuous Z-numbers , 2016, Inf. Sci..

[8]  Liguo Fei,et al.  Meausre divergence degree of basic probability assignment based on Deng relative entropy , 2015, 2016 Chinese Control and Decision Conference (CCDC).

[9]  Yong Deng,et al.  Identifying influential nodes in complex networks: A node information dimension approach. , 2018, Chaos.

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

[11]  Y. Chu,et al.  Association of Jensen’s inequality for s-convex function with Csiszár divergence , 2019, Journal of Inequalities and Applications.

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

[13]  Fuyuan Xiao,et al.  Conflict management based on belief function entropy in sensor fusion , 2016, SpringerPlus.

[14]  Adwait Ratnaparkhi,et al.  Learning to Parse Natural Language with Maximum Entropy Models , 1999, Machine Learning.

[15]  Mohammad R. Akbarzadeh-Totonchi,et al.  Introducing validity in fuzzy probability for judicial decision-making , 2014, Int. J. Approx. Reason..

[16]  Quentin L. Burrell,et al.  The 80/20 Rule: Library Lore or Statistical Law? , 1985, J. Documentation.

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

[18]  Y. Chu,et al.  Converses of the Jensen inequality derived from the Green functions with applications in information theory , 2019, Mathematical Methods in the Applied Sciences.

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

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

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

[22]  Constantino Tsallis,et al.  Approach of Complexity in Nature: Entropic Nonuniqueness , 2016, Axioms.

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

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

[25]  Karmele López de Ipiña,et al.  Application of Entropy and Fractal Dimension Analyses to the Pattern Recognition of Contaminated Fish Responses in Aquaculture , 2014, Entropy.

[26]  M. Tribus,et al.  Probability theory: the logic of science , 2003 .

[27]  George J. Klir,et al.  Fuzzy Logic and Mathematics: A Historical Perspective , 2017 .

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

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

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

[31]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.

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

[33]  Wei Deng,et al.  Entropic methodology for entanglement measures , 2018, Physica A: Statistical Mechanics and its Applications.

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

[35]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .

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

[37]  Xinyang Deng,et al.  The Negation of a Basic Probability Assignment , 2019, IEEE Transactions on Fuzzy Systems.

[38]  Yafei Song,et al.  Uncertainty measure for interval-valued belief structures , 2016 .

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

[40]  Xianguo Wu,et al.  Perceiving safety risk of buildings adjacent to tunneling excavation: An information fusion approach , 2017 .

[41]  James Theiler,et al.  Estimating fractal dimension , 1990 .

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

[43]  Yong Deng,et al.  Evaluation method based on fuzzy relations between Dempster–Shafer belief structure , 2018, Int. J. Intell. Syst..

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

[45]  Rafik A. Aliev,et al.  The arithmetic of discrete Z-numbers , 2015, Inf. Sci..

[46]  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.

[47]  J. Pečarić,et al.  Bounds for Shannon and Zipf‐Mandelbrot entropies , 2017 .

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

[49]  Hui Li,et al.  Structural damage identification based on integration of information fusion and shannon entropy , 2008 .

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

[51]  Sankaran Mahadevan,et al.  A new rule to combine dependent bodies of evidence , 2019, Soft Comput..

[52]  Wen Jiang,et al.  Intuitionistic Fuzzy Power Aggregation Operator Based on Entropy and Its Application in Decision Making , 2018, Int. J. Intell. Syst..

[53]  Xu Jian-hua Mei An-xin Wu Jian-ping Zhou Jian-hua Zhao Jing.  A study on the information entropy and fractal dimension of land use structure and form in Shanghai , 2004 .

[54]  Constantino Tsallis,et al.  Nonadditive entropy: The concept and its use , 2008, 0812.4370.

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

[56]  J. Pečarić,et al.  Bounds for Csiszár divergence and hybrid Zipf‐Mandelbrot entropy , 2019, Mathematical Methods in the Applied Sciences.