# The Pseudo-Pascal Triangle of Maximum Deng Entropy

• Xiaozhuan Gao
• Yong Deng

### Abstract

PPascal triangle (known as Yang Hui Triangle in Chinese) is an important model in mathematics while the entropy has been heavily studied in physics or as uncertainty measure in information science. How to construct the the connection between Pascal triangle and uncertainty measure is an interesting topic. One of the most used entropy, Tasllis entropy, has been modelled with Pascal triangle. But the relationship of the other entropy functions with Pascal triangle is still an open issue. Dempster-Shafer evidence theory takes the advantage to deal with uncertainty than probability theory since the probability distribution is generalized as basic probability assignment, which is more efficient to model and handle uncertain information. Given a basic probability assignment, its corresponding uncertainty measure can be determined by Deng entropy, which is the generalization of Shannon entropy. In this paper, a Pseudo-Pascal triangle based the maximum Deng entropy is constructed. Similar to the Pascal triangle modelling of Tasllis entropy, this work provides the a possible way of Deng entropy in physics and information theory.

### References

[1] Abellán, J.; Mantas, C.J.; Bossé, E.(2019). Basic Properties for Total Uncertainty Measures in the Theory of Evidence, Information Quality in Information Fusion and Decision Making, 99-108, 2019.
https://doi.org/10.1007/978-3-030-03643-0_5

[2] Abellán, J.; Mantas, C.J.; Castellano, J. G. (2017). A Random Forest approach using imprecise probabilities, Knowledge-Based Systems, 134, 72-84, 2017.
https://doi.org/10.1016/j.knosys.2017.07.019

[3] Abellán, J. (2017). Analyzing properties of Deng entropy in the theory of evidence, Chaos Solitons & Fractals, 95, 195-199, 2017.
https://doi.org/10.1016/j.chaos.2016.12.024

[4] Ahmia, M.; Belbachir, H.. (2012). Preserving log-convexity for generalized Pascal triangles, the electronic journal of combinatorics, 19(2), 16, 2012.
https://doi.org/10.37236/2255

[5] Becher, V.; Carton, O. (2019). Normal numbers and nested perfect necklaces, Journal of Complexity, 54, 101403, 2019.
https://doi.org/10.1016/j.jco.2019.03.003

[6] Blyth, M.G.; Pozrikidis, C. (2006). A lobatto interpolation grid over the triangle, IMA journal of applied mathematics, 71(1), 153-169, 2006.
https://doi.org/10.1093/imamat/hxh077

[7] Cao, X.; Deng, Y. (2019). A lobatto interpolation grid over the triangle, IEEE ACCESS, 7(1), 95547-95554, 2019.
https://doi.org/10.1109/ACCESS.2019.2928581

[8] Cao, Z; Ding, W.; Wang, Y.-K.; Hussain F., Al-Jumaily, A. Lin, C.-T. (2019). Effects of Repetitive SSVEPs on EEG Complexity using Multiscale Inherent Fuzzy Entropy, Neurocomputing, DOI: 10.1016/j.neucom.2018.08.091, 2019.
https://doi.org/10.1016/j.neucom.2018.08.091

[9] Cao, Z.; Lin, C.-T. (2018). Inherent fuzzy entropy for the improvement of EEG complexity evaluation, IEEE Transactions on Fuzzy Systems, 26(2), 1032-1035, 2018.
https://doi.org/10.1109/TFUZZ.2017.2666789

[10] Dempster, A.P. (1967). Upper and Lower Probabilities Induced by a Multivalued Mapping, Annals of Mathematical Statistics, 38(2), 325-339, 1967.
https://doi.org/10.1214/aoms/1177698950

[11] Deng, W.; Deng, Y. (2018). Entropic methodology for entanglement measures, Physica A: Statistical Mechanics and its Applications, 512, 693-697, 2018.
https://doi.org/10.1016/j.physa.2018.07.044

[12] Deng, X.; Jiang, W. (2019). Evaluating green supply chain management practices under fuzzy environment: a novel method based on D number theory, International Journal of Fuzzy Systems, 21, 1389-1402, 2019.
https://doi.org/10.1007/s40815-019-00639-5

[13] Deng, X.; Jiang, W. (2019). A total uncertainty measure for D numbers based on belief intervals, International Journal of Intelligent Systems, 34(12), 3302-3316, 2019.
https://doi.org/10.1002/int.22195

[14] Deng, Y. (2016). Deng Entropy, Chaos, Solitons & Fractals, 91, 549-553, 2016.
https://doi.org/10.1016/j.chaos.2016.07.014

[15] Dragan, I.-M.; Isaic-Maniu, A. (2019). An Innovative Model of Reliability-The Pseudo-Entropic Model, Entropy, 21(9), 846, 2019.
https://doi.org/10.3390/e21090846

[16] Elmore, P. A.; Petry F.E. Yager, R.R. (2017). Dempster-Shafer Approach to Temporal Uncertainty, IEEE Transactions on Emerging Topics in Computational Intelligence, 1(5), 316-325, 2017.
https://doi.org/10.1109/TETCI.2017.2719711

[17] Fang, R.; Liao, H.; Yang, J.-B., Xu, D.-L. (2019). Generalised probabilistic linguistic evidential reasoning approach for multi-criteria decision-making under uncertainty, Journal of the Operational Research Society, DOI:10.1080/01605682.2019.1654415, 2019.
https://doi.org/10.1080/01605682.2019.1654415

[18] Gao, S.; Deng, Y. (2019). An evidential evaluation of nuclear safeguards, International Journal of Distributed Sensor Networks, 15(12), DOI:10.1177/1550147719894550, 2019.
https://doi.org/10.1177/1550147719894550

[19] Hacène, B.; Nèmeth, L.; Szalay, L.. (2016). Hyperbolic pascal triangles, Applied Mathematics and Computation, 273, 453-464, 2016.
https://doi.org/10.1016/j.amc.2015.10.001

[20] Huang, Z.; Yang, L.; Jiang, W. (2019). Uncertainty measurement with belief entropy on the interference effect in the quantum-like Bayesian Networks, Applied Mathematics and Computation, 347, 417-428, 2019.
https://doi.org/10.1016/j.amc.2018.11.036

[21] Hurley, J.; Johnson, C.; Dunham, J.; Simmons, J. (2019). Nonlinear Algorithms for Combining Conflicting Identification Information in Multisensor Fusion, 2019 IEEE Aerospace Conference, 1-7, 2019.
https://doi.org/10.1109/AERO.2019.8741967

[22] Jafferis, D. L.; Lewkowycz, A.; Maldacena, J.; Suh, S. J. (2016). Relative entropy equals bulk relative entropy, Journal of High Energy Physics, 2016(6), 4, 2016.
https://doi.org/10.1007/JHEP06(2016)004

[23] Jiang, W.; Cao, Y.; Deng, X. (2019). A Novel Z-network Model Based on Bayesian Network and Z-number, IEEE Transactions on Fuzzy Systems, DOI:10.1109/TFUZZ.2019.2918999, 2019.
https://doi.org/10.1109/TFUZZ.2019.2918999

[24] Kang, B.; Deng, Y. (2019). The maximum Deng entropy, IEEE ACCESS, 7(1), 120758-120765, 2019.
https://doi.org/10.1109/ACCESS.2019.2937679

[25] Karci, A. (2016). Fractional order entropy: New perspectives, Optik, 127(20), 9172-9177, 2016.
https://doi.org/10.1016/j.ijleo.2016.06.119

[26] Khan, N.; Anwar, S. (2019). Time-Domain Data Fusion Using Weighted Evidence and Dempster- Shafer Combination Rule: Application in Object Classification, Sensors, 19(23), 5187, 2019.
https://doi.org/10.3390/s19235187

[27] Kuzemsky, A. L. (2018). Temporal evolution, directionality of time and irreversibility, Rivista Del Nuovo Cimento, 41(10), 513-574, 2018.

[28] Lee, S.; Jin, M.; Koo, B.; Sin, C.; Kim, S. (2016). Pascal's triangle-based range-free localization for anisotropic wireless networks, Wireless Networks, 22(7), 2221-2238, 2016.
https://doi.org/10.1007/s11276-015-1095-9

[29] Li, D.; Deng, Y. (2019). A new correlation coefficient based on generalized information quality, IEEE ACCESS, 7(1), 175411-175419, 2019.
https://doi.org/10.1109/ACCESS.2019.2957796

[30] Li, D.; Deng, Y.; Gao, X. (2019). A generalized expression for information quality of basic probability assignment, IEEE ACCESS, 7(1), 174734-174739, 2019.
https://doi.org/10.1109/ACCESS.2019.2956956

[31] Li, H.; He, Y.; Nie, X. (2018). Structural reliability calculation method based on the dual neural network and direct integration method, Neural Computing and Applications, 29(7), 425-433, 2018.
https://doi.org/10.1007/s00521-016-2554-7

[32] Li, H.; Yuan, R.; Fu, J. (2019). A reliability modeling for multi-component systems considering random shocks and multistate degradation, IEEE ACCESS, 7(1), 168805-168814, 2019.
https://doi.org/10.1109/ACCESS.2019.2953483

[33] Li, M.; Deng, Y. (2019). Evidential Decision Tree Based on Belief Entropy, Entropy, 21(9), 897, 2019.
https://doi.org/10.3390/e21090897

[34] Li, Y.; Deng, Y. (2019). Intuitionistic evidence sets, IEEE ACCESS, 7(1), 106417-106426, 2019.
https://doi.org/10.1109/ACCESS.2019.2932763

[35] Liu, F.; Gao, X.; Zhao, J.; Deng, Y. (2019). Generalized belief entropy and its application in identifying conflict evidence, IEEE ACCESS, 7(1), 126625-126633, 2019.
https://doi.org/10.1109/ACCESS.2019.2939332

[36] Liu, P.; Zhang, X. (2019). A Multicriteria Decision-Making Approach with Linguistic D Numbers Based on the Choquet Integral, Cognitive Computation, DOI: 10.1007/s12559-019-09641-3, 2019.
https://doi.org/10.1007/s12559-019-09641-3

[37] Liu, P.; Zhang, X.; Wang, Z. (2019). An Extended VIKOR Method for Multiple Attribute Decision Making with Linguistic D Numbers Based on Fuzzy Entropy, International Journal of Information Technology & Decision Making, DOI: 10.1142/S0219622019500433, 2019.
https://doi.org/10.1142/S0219622019500433

[38] Liu, W.; Wang, T.; Zang, T.; Huang, Z.; Wang, J.; Huang, T.; Wei, X.; Li, C. (2020). A fault diagnosis method for power transmission networks based on spiking neural P systems with self-updating rules considering biological apoptosis mechanism, Complexity, DOI: 10.1155/2020/2462647, 2020.
https://doi.org/10.1155/2020/2462647

[39] Liu, Y.; Jiang, W. (2019). A new distance measure of interval-valued intuitionistic fuzzy sets and its application in decision making, Complexity, 23, DOI:10.1007/s00500-019-04332-5, 2019.
https://doi.org/10.1007/s00500-019-04332-5

[40] Liu, Z.; Deng, Y. (2019). A matrix method of basic belief assignment's negation in Dempster- Shafer theory, IEEE Transactions on Fuzzy Systems, 27, DOI:10.1109/TFUZZ.2019.2930027, 2019.
https://doi.org/10.1109/TFUZZ.2019.2930027

[41] Mamb, M. D.; N'Takpe, T.; Anoh, N. G.; Oumtanaga, S. (2018). A New Uncertainty Measure in Belief Entropy Framework, International Journal of Advanced Computer Science and Applications, 9(11), 600-606, 2018.
https://doi.org/10.14569/IJACSA.2018.091184

[42] Mi, J.; Li, Y. F.; Beer, M.; Broggi, M.; Cheng, Y. (2020). Importance measure of probabilistic common cause failures under system hybrid uncertainty based on Bayesian network, Eksploatacja i Niezawodnosc-Maintenance and Reliability, 13(22), 112-120, 2020.
https://doi.org/10.17531/ein.2020.1.13

[43] Millard, P.; Massou, S.; Portais, J.-C.; Letise, F. (2014). Isotopic studies of metabolic systems by mass spectrometry: using Pascal's triangle to produce biological standards with fully controlled labeling patterns, Analytical chemistry, 86(20), 10288-10295, 2014.
https://doi.org/10.1021/ac502490g

[44] Mo, H.; Deng, Y. (2019). Identifying node importance based on evidence theory in complex networks, Physica A: Statistical Mechanics & Its Applications, DOI:10.1016/j.physa.2019.121538, 2019.
https://doi.org/10.1016/j.physa.2019.121538

[45] Moussa, A.; Hacène, B. (2012). Preserving log-convexity for generalized Pascal triangles, The electronic journal of combinatorics, 19(2), 16, 2012.
https://doi.org/10.37236/2255

[46] Nemeth, L.; Szalay, L. (2018). Power sums in hyperbolic Pascal triangles, Analele Stiintifice Ale Universitatii Ovidius Constanta-Seria Matematica, 26(1), 189-203, 2018.
https://doi.org/10.2478/auom-2018-0012

[47] Ozkan, K. (2018). Comparing Shannon entropy with Deng entropy and improved Deng entropy for measuring biodiversity when a priori data is not clear, Journal of the faculty of forestry- Istanbul University, 68(2), 136-140, 2018.

[48] Pan, L.; Deng, Y. (2020). An association coefficient of belief function and its application in target recognition system, International Journal of Intelligent Systems, 35(1), 85-104, 2020.
https://doi.org/10.1002/int.22200

[49] Pan, Y.; Zhang, L.; Li, Z.; Ding, L. (2019). Improved Fuzzy Bayesian Network-Based Risk Analysis With Interval-Valued Fuzzy Sets and D-S Evidence Theory, IEEE Transactions on Fuzzy Systems, DOI:10.1109/TFUZZ.2019.2929024, 2019.
https://doi.org/10.1109/TFUZZ.2019.2929024

[50] Qian, H.-M.; Huang, H.-Z.; Li, Y.-F. (2019). A novel single-loop procedure for time-variant reliability analysis based on Kriging model, Applied Mathematical Modelling, 75, 735-748, 2019.
https://doi.org/10.1016/j.apm.2019.07.006

[51] Rényi, A. (1961). On measures of entropy and information, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, The Regents of the University of California 1961.

[52] Ristic, B.; Smets, P. (2005). Target classification approach based on the belief function theory, IEEE Transactions on Aerospace and Electronic Systems, 41(2), 574-583, 2005.
https://doi.org/10.1109/TAES.2005.1468749

[53] Robledo, A. (2013). Generalized Statistical Mechanics at the Onset of Chaos, IEEE Transactions on Aerospace and Electronic Systems, 15(12), 5178-5222, 2013.
https://doi.org/10.3390/e15125178

[54] Romagnoli, S. (2019). A vague multidimensional dependency structure: Conditional versus Unconditional fuzzy copula models, IEEE Transactions on Aerospace and Electronic Systems, 512, 1202-1213, 2019.
https://doi.org/10.1016/j.ins.2019.10.052

[55] Schubert, J. (2011). Conflict management in Dempster-Shafer theory using the degree of falsity, International Journal of Approximate Reasoning, 52(3), 449-460, 2011.
https://doi.org/10.1016/j.ijar.2010.10.004

[56] Seiti, H.; Hafezalkotob, A.; Najafi, S.E.; Khalaj, M. (2018). A risk-based fuzzy evidential framework for FMEA analysis under uncertainty: An interval-valued DS approach, Journal of Intelligent & Fuzzy Systems, 35(2), 1419-1430, 2018.
https://doi.org/10.3233/JIFS-169684

[57] Shafer, G. (1967). A mathematical theory of evidence, Princeton university press, 42, 1967.

[58] Sheikholeslami, M.; Jafaryar, M.; Shafee, A.; Li, Z.; Haq, R. (2019). Heat transfer of nanoparticles employing innovative turbulator considering entropy generation, International Journal of Heat and Mass Transfer, 136, 1233-1240, 2019.
https://doi.org/10.15837/3735/ijccc.2020.1.3735 9
https://doi.org/10.1016/j.ijheatmasstransfer.2019.03.091

[59] Song, Y.; Deng, Y. (2019). A new soft likelihood function based on power ordered weighted average operator, International Journal of Intelligent Systems, 34(11), 2988-2999, 2019.
https://doi.org/10.1002/int.22182

[60] Song, Y.; Deng, Y. (2019). Divergence measure of belief function and its application in data fusion, IEEE ACCESS, 7, 107465-107472, 2019.
https://doi.org/10.1109/ACCESS.2019.2932390

[61] Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics, Journal of statistical physics, 52(1-2), 479-487, 1988.
https://doi.org/10.1007/BF01016429

[62] Tsallis, C.; Gellmann, M.; Sato, Y. (2005). Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive, Proceedings of the National Academy of Sciences of the United States of America, 102(43), 15377-15382, 2005.
https://doi.org/10.1073/pnas.0503807102

[63] Tugal, I. (2019). Karcı and Shannon entropies and their effects on centrality of social networks, Physica A: Statistical Mechanics and its Applications, 523, 352-363, 2019.
https://doi.org/10.1016/j.physa.2019.02.026

[64] Velarde, C.; Robledo, A. (2015). Pascal (Yang Hui) triangles and power laws in the logistic map, Journal of Physics Conference Series, 604, 012018, 2015.
https://doi.org/10.1088/1742-6596/604/1/012018

[65] Wang, H.; Fang, Y.-P.; Zio, E. (2019). Risk Assessment of an Electrical Power System Considering the Influence of Traffic Congestion on a Hypothetical Scenario of Electrified Transportation System in New York Stat, IEEE Transactions on Intelligent Transportation Systems, doi:10.1109/TITS.2019.2955359, 2019.
https://doi.org/10.1109/ICSRS.2018.8688718

[66] Wang, D.; Gao, J.; Wei, D. (2019). A New Belief Entropy Based on Deng Entropy, Entropy, 21(10), doi:10.3390/e21100987, 2019.
https://doi.org/10.3390/e21100987

[67] Wang, T.; Wang, J.; Ming, J.; Sun, Z.; Wei, C.; Lu, C.; Pérez-Jiménez, M.J. (2018). Application of neural-like P systems with state values for power coordination of photovoltaic/battery microgrids, IEEE ACCESS, 6, 46630-46642, 2018.
https://doi.org/10.1109/ACCESS.2018.2865122

[68] Wang, T.; Wei, X.; Huang, T.; Wang, J.; Peng, H.; Pérez-Jiménez, M. J.; Valencia-Cabrera, L. (2019). Modeling fault propagation paths in power systems: A new framework based on event SNP systems with neurotransmitter concentration, IEEE ACCESS, 7, 12798-12808, 2019.
https://doi.org/10.1109/ACCESS.2019.2892797

[69] Wang, T.; Wei, X.; Huang, T.; Wang, J.; Valencia-Cabrera, L.; Fan, Z.; Pérez-Jiménez, M. J. (2019). Cascading Failures Analysis Considering Extreme Virus Propagation of Cyber-Physical Systems in Smart Grids, Complexity, 2019, 7428458, 2019.
https://doi.org/10.1155/2019/7428458

[70] Wei, B.; Feng, X.; Yang, S. (2019). Fully Distributed Synchronization of Dynamic Networked Systems with Adaptive Nonlinear Couplings, IEEE Transactions on Cybernetics, DOI:10.1109/TCYB.2019.2944971, 2019.
https://doi.org/10.1109/TCYB.2019.2944971

[71] Wei, B.; Feng, X.; Yang, S. (2019). Synchronization in Kuramoto Oscillator Networks With Sampled-Data Updating Law, IEEE Transactions on Cybernetics, DOI:10.1109/TCYB.2019.2940987, 2019.
https://doi.org/10.1109/TCYB.2019.2940987

[72] Wen, T.; Deng, Y. (2020). The vulnerability of communities in complex networks: An entropy approach, Reliability Engineering & System Safety, 196, 106782, 2020.
https://doi.org/10.1016/j.ress.2019.106782

[73] Xiao, F. (2019). EFMCDM: Evidential fuzzy multicriteria decision making based on belief entropy, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2936368, 2019.
https://doi.org/10.1109/TFUZZ.2019.2936368

[74] Xiao, F. (2019). Generalization of Dempster-Shafer theory: A complex mass function, Applied Intelligence, DOI: 10.1007/s10489-019-01617-y, 2019.

[75] Xiao, F. (2020). A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion, Information Sciences, 514, 462-483, 2020.
https://doi.org/10.15837/3735/ijccc.2020.1.3735 10
https://doi.org/10.1016/j.ins.2019.11.022

[76] Yager, R.R. (2014). On the maximum entropy negation of a probability distribution, IEEE Transactions on Fuzzy Systems, 23(5), 1899-1902, 2014.
https://doi.org/10.1109/TFUZZ.2014.2374211

[77] Yager, R. R. (2019). Generalized Dempster-Shafer Structures, IEEE Transactions on Fuzzy Systems, 27(3), 428-435, 2019.
https://doi.org/10.1109/TFUZZ.2018.2859899

[78] Yuan, R.; Tang, M.; Wang, H.; Li, H. (2019). A Reliability Analysis Method of Accelerated Performance Degradation Based on Bayesian Strategy, IEEE Access, 7, 169047-169054, 2019.
https://doi.org/10.1109/ACCESS.2019.2952337

[79] Zhao, H.; Xie, Z. (2014). Preliminary study of cellular automat on mobile computing application, Applied Mechanics and Materials, 519, 838-841, 2014.
https://doi.org/10.4028/www.scientific.net/AMM.519-520.838

[80] Zhou, M.; Liu, X.;, Chen, Y.; Yang, J. (2018). Evidential reasoning rule for MADM with both weights and reliabilities in group decision making, Knowledge-Based Systems, 143, 142-161, 2018.
https://doi.org/10.1016/j.knosys.2017.12.013

[81] Zhou, M.; Liu, X.; Yang, J.; Chen, Y.; Wu, J. (2019). Evidential reasoning approach with multiple kinds of attributes and entropy-based weight assignment, Knowledge-Based Systems, 163, 358- 375, 2019.
https://doi.org/10.1016/j.knosys.2018.08.037

[82] Zurek, W. H. (2018). Complexity, entropy and the physics of information, CRC Press, 2018.
Published
2020-02-03
How to Cite
GAO, Xiaozhuan; DENG, Yong. The Pseudo-Pascal Triangle of Maximum Deng Entropy. INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL, [S.l.], v. 15, n. 1, feb. 2020. ISSN 1841-9844. Available at: <http://univagora.ro/jour/index.php/ijccc/article/view/1006>. Date accessed: 23 oct. 2020. doi: https://doi.org/10.15837/ijccc.2020.1.3735.
Citation Formats
Section
Articles

### Keywords

Deng entropy, Maximum Deng Entropy, Pascal triangle, Dempster-Shafer evidence theory, basic probability assignment.