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A NOVEL FUZZYBASED SIMILARITY MEASURE FOR COLLABORATIVE FILTERING TO ALLEVIATE THE SPARSITY PROBLEM
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Memorybased collaborative filtering is the most popular approach to build recommender systems. Despite its success in many applications, it still suffers from several major limitations, including data sparsity. Sparse data affect the quality of the user similarity measurement and consequently the quality of the recommender system. In this paper, we propose a novel user similarity measure based on fuzzy set theory along with default voting technique aimed to provide a valid similarity measurement between users wherever the available ratings are relatively rare. The main idea of this research is to model the rating behaviour of each user by a fuzzy set, and use this model to determine the user's degree of interest on items. Experimental results on the MovieLens and Netflix datasets show the effectiveness of the proposed algorithm in handling data sparsity problem. It also outperforms some stateoftheart collaborative filtering algorithms in terms of prediction quality.
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1
18


Masoud
Saeed
School of Electrical and Computer Engineering, Shiraz University,
Shiraz, Iran
School of Electrical and Computer Engineering,
Iran
msaeedmz@gmail.com


Eghbal G
Mansoori
School of Electrical and Computer Engineering, Shiraz University,
Shiraz, Iran
School of Electrical and Computer Engineering,
Iran
Recommender system
Collaborative filtering
Similarity measure
Data sparsity
[[1] G. Adomavicius and A. Tuzhilin, Toward the next generation of recommender systems: A survey of##the stateoftheart and possible extensions, Knowledge and Data Engineering, IEEE Transactions,##17(6) (2005), 734{749.##[2] C. C. Aggarwal, Recommender Systems: The Textbook, Springer, 2016.##[3] D. Anand and K. K. Bharadwaj, Utilizing various sparsity measures for enhancing accuracy of##collaborative recommender systems based on local and global similarities, Expert Systems with##Applications, 38(5) (2011), 5101{5109.##[4] J. Bobadilla, F. Ortega, A. Hernando and A. Gutirrez, Recommender systems survey, Knowledge##Based Systems, 46 (2013), 109{132.##[5] J. S. Breese, D. Heckerman and C. Kadie, Empirical analysis of predictive algorithms for##collaborative ltering, In Proceedings of the 14th Conference on Uncertainty in Articial##Intelligence, (1998), 43{52.##[6] L. Chen, G. Chen and F. Wang, Recommender systems based on user reviews: the state of the art,##User Modeling and UserAdapted Interaction, 25(2) (2015), 99{154.##[7] C. Cornelis, X. Guo, J. Lu and G. Zhang, A Fuzzy Relational Approach to Event Recommendation,##In Proceedings of the 2nd Indian International Conference on Articial Intelligence, 5 (2005), 2231{##[8] C. Desrosiers and G. Karypis, A novel approach to compute similarities and its application to item##recommendation, In Pacic Rim International Conference on Articial Intelligence, (2010), 39{51.##[9] C. Desrosiers and G. Karypis, A comprehensive survey of neighborhoodbased recommendation##methods, In Recommender Systems Handbook, (2011), 107{144.##[10] M. D. Ekstrand, J. T. Riedl and J. A. Konstan, Collaborative ltering recommender systems,##Foundations and Trends in HumanComputer Interaction, 4(2) (2011), 81{173.##[11] F. Fouss, A. Pirotte, J. M. Renders and M. Saerens, Randomwalk computation of similarities##between nodes of a graph with application to collaborative recommendation, IEEE Transactions on##Knowledge and Data Engineering, 19(3) (2007), 355{369.##[12] M. A. Ghazanfar and A. PrugelBennett, Leveraging clustering approaches to solve the graysheep##users problem in recommender systems, Expert Systems with Applications, 41(7) (2014), 3261{##[13] K. Goldberg, T. Roeder, D. Gupta and C. Perkins, Eigentaste: A constant time collaborative##ltering algorithm, Information Retrieval, 4(2) (2001), 133{151.##[14] J. L. Herlocker, J. A. Konstan, A. Borchers and J. Riedl, An algorithmic framework for performing##collaborative ltering, In Proceedings of the 22nd Annual International ACM SIGIR Conference##on Research and Development in Information Retrieval, (1999), 230{237.##[15] J. L. Herlocker, J. A. Konstan, L. G. Terveen and J. T. Riedl, Evaluating collaborative ltering##recommender systems, ACM Transactions on Information Systems (TOIS), 22(1) (2004), 5{53.##[16] R. J. Hyndman and A. B. Koehler, Another look at measures of forecast accuracy, International##Journal of Forecasting, 22(4) (2006), 679688.##[17] M. Jamali and M. Ester, Trustwalker: a random walk model for combining trustbased and item##based recommendation, In Proceedings of the 15th ACM SIGKDD International Conference on##Knowledge Discovery and Data Mining, (2009), 397{406.##[18] D. Jannach, M. Zanker, A. Felfernig and G. Friedrich, Recommender systems: an introduction,##Cambridge University Press, 2010.##[19] G. Karypis, Evaluation of itembased topn recommendation algorithms, In Proceedings of the 10th##International Conference on Information and Knowledge Management, (2001), 247{254. ##[20] J. A. Konstan and J. Riedl, Recommender systems: from algorithms to user experience, User##Modeling and UserAdapted Interaction, 22(12) (2012), 101{123.##[21] G. Koutrika, B. Bercovitz and H. GarciaMolina, FlexRecs: expressing and combining ##recommendations, In Proceedings of the 2009 ACM SIGMOD International Conference on##Management of Data, (2009), 745{758.##[22] A. S. Lampropoulos and G. A. Tsihrintzis, Machine Learning Paradigms, Springer, 2015.##[23] C. W. Leung, S. C. Chan and F. Chung, A collaborative ltering framework based on fuzzy##association rules and multiplelevel similarity, Knowledge and Information Systems, 10(3) (2006),##[24] H. Luo, C. Niu, R. Shen and C. Ullrich, A collaborative ltering framework based on both local##user similarity and global user similarity, Machine Learning, 72(3) (2008), 231{245.##[25] H. Ma, I. King and M. R. Lyu, Eective missing data prediction for collaborative ltering,##In Proceedings of the 30th Annual International ACM SIGIR Conference on Research and##Development in Information Retrieval, (2007), 39{46.##[26] P. Massa and P. Avesani, Trust metrics in recommender systems, In Computing with Social Trust,##(2009), 259{285.##[27] B. K. Patra, R. Launonen, V. Ollikainen and S. Nandi, A new similarity measure using##Bhattacharyya coecient for collaborative ltering in sparse data, KnowledgeBased Systems, 82##(2015), 163{177.##[28] P. Resnick, N. Iacovou, M. Suchak, P. Bergstrom and J. Riedl, Grouplens: an open architecture##for collaborative ltering of netnews, In Proceedings of the 1994 ACM Conference on Computer##Supported Cooperative Work, (1994), 175186.##[29] F. Ricci, L. Rokach and B. Shapira, Introduction to recommender systems handbook, Springer,##(2011), 1{35.##[30] B. Sarwar, G. Karypis, J. Konstan and J. Riedl, Analysis of recommendation algorithms for e##commerce, In Proceedings of the 2nd ACM Conference on Electronic Commerce, (2000), 158{167.##[31] B. Sarwar, G. Karypis, J. Konstan and J. Riedl, Itembased collaborative ltering recommendation##algorithms, In Proceedings of the 10th International Conference on World Wide Web, (2001),##[32] J. B. Schafer, J. Konstan and J. Riedl, Recommender systems in ecommerce, In Proceedings of##the 1st ACM Conference on Electronic commerce, (1999), 158{166.##[33] G. Shani and A. Gunawardana, Evaluating recommendation systems, In Recommender Systems##Handbook, Springer US, (2011), 257{297.##[34] U. Shardanand and P. Maes, Social information ltering: Algorithms for automating word of##mouth, In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, (1995),##[35] Y. Shi, M. Larson and A. Hanjalic, Collaborative ltering beyond the useritem matrix: A survey##of the state of the art and future challenges, ACM Computing Surveys (CSUR), 47(1) (2014), 3.##[36] X. Su and T. M. Khoshgoftaar, A survey of collaborative ltering techniques, Advances in Articial##Intelligence, (2009), 2{19.##[37] R. Yera, J. Castro and L. Martnez, A fuzzy model for managing natural noise in recommender##systems, Applied Soft Computing, 40 (2016), 187{198.##[38] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.##[39] Z. Zhang, X. Tang and D. Chen, Applying userfavoriteitembased similarity into slope one scheme##for collaborative ltering, Computing and Communication Technologies (WCCCT), 2014 World##Congress on, (2014), 5{7.##]
DISTINGUISHABILITY AND COMPLETENESS OF CRISP DETERMINISTIC FUZZY AUTOMATA
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In this paper, we introduce and study notions like state\linebreak distinguishability, inputdistinguishability and output completeness of states of a crisp deterministic fuzzy automaton. We show that for each crisp deterministic fuzzy automaton there corresponds a unique (up to isomorphism), equivalent distinguished crisp deterministic fuzzy automaton. Finally, we introduce two axioms related to output completeness of states and discuss the interrelationship between them.
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Renu
.
Indian School of Mines, Dhanbad, India
Indian School of Mines, Dhanbad, India
India
renuismmaths@gmail.com


S. P.
Tiwari
Department of Applied Mathematics, Indian Institute of Technology
(ISM), Dhanbad826004, India
Department of Applied Mathematics, Indian
India
Crisp deterministic fuzzy automaton
Indistinguishable states
Inputindistinguishable
Homomorphism
Output complete
[[1] Z. Bavel, Structure and transition preserving functions of nite automata, Journal of Asso##ciation for Computing machinery, 15 (1968), 135{158.##[2] Y. Cao and Y. Ezawa, Nondeterministic fuzzy automata, Information Sciences, 191 (2012),##[3] M. Ciric and J. Ignjatovic, Fuzziness in automata theory: why? how?, Studies in Fuzziness##and Soft Computing, 298 (2013), 109{114.##[4] M. Doostfatemeh and S. C. Kremer, New directions in fuzzy automata, International Journal##of Approximate Reasoning, 38 (2005), 175{214.##[5] S. Ginsburg, Some remark on abstract automata, Transactions of the American Mathematical##Society, 96 (1960), 400{444.##[6] X. Guo, Grammar theory based on latticeorder monoid, Fuzzy Sets and Systems, 160 (2009),##1152{1161.##[7] W. M. L. Holcombe, Algebraic automata theory, Cambridge University Press, Cambridge,##[8] J. Ignjatovic, M. Ciric and S. Bogdanovic, Determinization of fuzzy automata with member##ship values in complete residuated lattices, Information Sciences, 178 (2008), 164{180.##[9] J. Ignjatovic, M. Ciric, S. Bogdanovic and T. Petkovic, MyhillNerode type theory for fuzzy##languages and automata, Fuzzy Sets and Systems, 161 (2010), 1288{1324.##[10] M. Ito, Algebraic structures of automata, Theoretical Computer Science, 429 (2012), 164{##[11] J. Jin, Q. Li and Y. Li, Algebraic properties of Lfuzzy nite automata, Information Sciences,##234 (2013), 182{202.##[12] Y. B. Jun, Intuitionistic fuzzy nite state automata, Journal of Applied Mathematics and##Computing, 17 (2005), 109{120.##[13] Y. B. Jun, Intuitionistic fuzzy nite switchboard state automata, Journal of Applied Mathe##matics and Computing, 20 (2006), 315{325.##[14] Y. B. Jun, Quotient structures of intuitionistic fuzzy nite state automata, Information Sci##ences, 177 (2007), 4977{4986.##[15] Y. H. Kim, J. G. Kim and S. J. Cho, Products of Tgeneralized state automata and T##generalized transformation semigroups, Fuzzy Sets and Systems, 93 (1998), 87{97.##[16] H. V. Kumbhojkar and S. R. Chaudhari, On proper fuzzication of fuzzy nite state automata,##International Journal of Fuzzy Mathematics, 4 (2008), 1019{1027.##[17] Y. Li and W. Pedrycz, Fuzzy nite automata and fuzzy regular expressions with membership##values in latticeordered monoids, Fuzzy Sets and Systems, 156 (2005), 68{92.##[18] Y. Li and W. Pedrycz, The equivalence between fuzzy Mealy and fuzzy Moore automata, Soft##Computing, 10 (2006), 953{959. ##[19] Y. Li and Q. Wang, The universal fuzzy automaton, Fuzzy Sets and Systems, 249 (2014),##[20] D. S. Malik, J. N. Mordeson and M. K. Sen, Subautomata of fuzzy nite state automaton,##Journal of Fuzzy Mathematics, 2 (1994), 781{792.##[21] J. N. Mordeson and D. S. Malik, Fuzzy automata and languages: Theory and Applications,##Chapman and Hall/CRC. London/Boca Raton, 2002.##[22] K. Peeva and Zl. Zahariev, Computing behavior of nite fuzzy automataalgorithm and its##application to reduction and minimization, Information Sciences, 178 (2008), 4152{4165.##[23] D. Qiu, Automata theory based on complete residuated latticevalued logic (I), Science in##China, 44 (2001), 419{429.##[24] D. Qiu, Automata theory based on complete residuated latticevalued logic (II), Science in##China, 45 (2002), 442{452.##[25] E. S. Santos, General formulation of sequential automata, Information and control, 12 (1968),##[26] W. Shukla and A. K. Srivastava, A topology for automata: A note, Information and Control,##32 (1976), 163{168.##[27] A. K. Srivastava and W. Shukla, A topology for automata II, International Journal of Math##ematics and Mathematical Sciences, 9 (1986), 425{428.##[28] S. P. Tiwari and S. Sharan, Fuzzy automata based on latticeordered monoid with algebraic##and topological aspects, Fuzzy Information and Engineering, 4 (2012), 155{164.##[29] S. P. Tiwari and A. K. Singh, On minimal realization of fuzzy behavior and associated cate##gories, Journal of Applied Mathematics and Computing, 45 (2014), 223{234.##[30] S. P. Tiwari, A. K. Singh and S. Sharan, Fuzzy subsystems of fuzzy automata based on lattice##ordered monoid, Annals of Fuzzy Mathematics and Informatics, 7 (2013), 437{445.##[31] S. P. Tiwari, A. K. Singh, S. Sharan and V. K. Yadav, Bifuzzy core of fuzzy automata, Iranian##Journal of Fuzzy Systems, 12 (2015), 63{73.##[32] D. Todinca and D. Butoianu, VHDL framework for modeling fuzzy automata, in: Proc. 14th##International Symposium on Symbolic and Numeric Algorithms for Scientic Computing,##IEEE, (2012), 171{178.##]
BATHTUB HAZARD RATE DISTRIBUTIONS AND FUZZY LIFE TIMES
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The development of life time analysis started back in the $20^{textit{th}}$ century and since then comprehensive developments have been made to model life time data efficiently. Recent development in measurements shows that all continuous measurements can not be measured as precise numbers but they are more or less fuzzy. Life time is also a continuous phenomenon, and has already been shown that life time observations are not precise measurements but fuzzy. Therefore, the corresponding analysis techniques employed on the data require to consider fuzziness of the observations to obtain appropriate estimates.In this study generalized estimators for the parameters and hazard rates are proposed for bathtub failure rate distributions to model fuzzy life time data effectively.
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Muhammad
Shafiq
Institute of Statistics and Mathematical Methods in Economics,
Vienna University of Technology, Wien, Austria
Institute of Statistics and Mathematical
Austria
shafiq@kust.edu.pk


Reinhard
Viertl
Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wien, Austria
Institute of Statistics and Mathematical
Austria
r.viertl@tuwien.ac.at
Bathtub failure rate
Fuzzy number
Life time
Nonprecise data
[[1] P. Abbas, P. G. Ali and S. Mansour, Reliability estimation in Rayleigh distribution based on##fuzzy lifetime data, International Journal of System Assurance Engineering and Management,##5(5) (2013), 487{494.##[2] G. Barbato, A. Germak, G. Genta and A. Barbato, Measurements for Decision Making.##Measurements and Basic Statistics, Esculapio, Bologna, 2013. ##[3] M. R. Casals, A. Colubi, N. Corral, M. A. Gil, M. Montenegro, M. A. Lubiano, A. B. Ramos##Guajardo, B. Sinova and others, Random fuzzy sets: a mathematical tool to develop statistical##fuzzy data analysis, Iranian Journal of Fuzzy Systems, 10(2) (2013), 1{28.##[4] Z. Chen, A new twoparameter lifetime distribution with bathtub shape or increasing failure##rate function, Statistics & Probability Letters, 49(2) (2000), 155{161.##[5] V. Couallier, L. GervilleReache, C. HuberCarol, N. Limnios and M. Mesbah, Statistical##Models and Methods for Reliability and Survival Analysis, Wiley, London, 2013.##[6] M. S. Hamada, A. Wilson, C. S. Reese and H. Martz, Bayesian Reliability, Springer, New##York, 2008.##[7] E. Haupt and H. Schabe, A new model for a lifetime distribution with bathtub shaped failure##rate, Microelectronics Reliability, 32(5) (1992), 633{639.##[8] D. W. Hosmer and S. Lemeshow, Applied Survival Analysis: Regression Modeling of Time##to Event Data, Wiley, New York, 1999.##[9] H. Huang, M. J. Zuo and Z. Sun, Bayesian reliability analysis for fuzzy lifetime data, Fuzzy##Sets and Systems, 157(12) (2006), 1674{1686.##[10] N. Hung T and W. Berlin, Fundamentals of Statistics with Fuzzy Data, Springer, New York,##[11] J. G. Ibrahim, M. H. Chen and D. Sinha, Bayesian Survival Analysis, Springer, New York,##[12] D. G. Kleinbaum and M. Klein, Survival Analysis: A selflearning Text, Springer, New York,##[13] G. J. Klir and Y. Bo, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice  Hal,##New Jersey, 1995.##[14] K. H. Lee, First Course on Fuzzy Theory and Applications, Springer, London, 2006.##[15] E. T. Lee and J. W.Wang, Statistical Methods for Survival Data Analysis, Wiley, New Jersey,##[16] W. Q. Meeker and L. A. Escobar, Statistical Methods for Reliability Data, Wiley, New York,##[17] R. G. Miller, Survival Analysis, Wiley, New York, 2011.##[18] W. B. Nelson, Applied Life Data Analysis, Wiley, New Jersey, 2005.##[19] S. G. Tzafestas and A. N. Venetsanopoulos, Fuzzy Reasoning in Information, Decision, and##Control Systems, Kluwer Academic Publishers, Norwell, MA, 1994.##[20] R. Viertl, Beschreibung Und Analyse Unscharfer Information: Statistische Methoden Fur##unscharfe Daten, Springer, Wien, 2006.##[21] R. Viertl, On reliability estimation based on fuzzy lifetime data, Journal of Statistical Planning##& Inference, 139(5) (2009), 1750{1755.##[22] R. Viertl, Statistical Methods for Fuzzy Data, Wiley, Chichester, 2011.##[23] R. Viertl, Univariate statistical analysis with fuzzy data, Computational Statistics & Data##Anaysis, 51(1) (2006), 133{147.##[24] H. C. Wu, Fuzzy bayesian estimation on lifetime data, Computational Statistics, 19(4)##(2004), 613633.##[25] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338{353.##[26] H. J. Zimmermann, Fuzzy Set Theory  and Its Applications, Kluwer Academic Publishers,##Massachusetts, 2001.##]
ADAPTIVE FUZZY OUTPUT FEEDBACK TRACKING CONTROL FOR A CLASS OF NONLINEAR TIMEVARYING DELAY SYSTEMS WITH UNKNOWN BACKLASHLIKE HYSTERESIS
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This paper considers the problem of adaptive output feedback tracking control for a class of nonstrictfeedback nonlinear systems with unknown timevarying delays and unknown backlashlike hysteresis. Fuzzy logic systems are used to estimate the unknown nonlinear functions. Based on the Lyapunov–Krasovskii method, the control scheme is constructed by using the backstepping and adaptive technique. The proposed adaptive controller guarantees that all the closedloop signals are semiglobally uniformly ultimately bounded and the tracking error can converge to a small neighborhood of the origin. Finally, Simulation results further show the effectiveness of the proposed approach.
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64


Mohsen
Hasanpour Naseriyeh
Department of Electrical Engineering, Shahid Bahonar University, Kerman, Iran
Department of Electrical Engineering, Shahid
Iran
mohsen.hasanpour@eng.uk.ac.ir


Adeleh
Arabzadeh Jafari
Department of Electrical Engineering, Shahid Bahonar University, Kerman, Iran
Department of Electrical Engineering, Shahid
Iran
aarabzadeh@eng.uk.ac.ir


Seyed Mohammad Ali
Mohammadi
Department of Electrical Engineering, Shahid Bahonar University, Kerman, Iran
Department of Electrical Engineering, Shahid
Iran
a_mohammadi@uk.ac.ir
Adaptive fuzzy control
Backstepping design technique
Backlashlike hysteresis
Nonstrictfeedback form
Nonlinear control
[[1] B. Chen, C. Lin, X. Liu and K. Liu, Observerbased adaptive fuzzy control for a class of##nonlinear delayed systems, IEEE Trans. Syst., Man, Cybern. Syst., 46(1) (2016), 27{36.##[2] B. Chen, X. Liu and K. Liu and C. Lin, Direct adaptive fuzzy control of nonlinear strict##feedback systems, Automatica, 45(6) (2009), 1530{1535.##[3] B. Chen, X. P. Liu, S. S. Ge and C. Lin, Adaptive fuzzy control of a class of nonlinear systems##by fuzzy approximation approach, IEEE Trans. Fuzzy Syst, 20(6) (2012), 1012{1021.##[4] S. S. Ge, F. Hong and T. H. Lee, Adaptive neural network control of nonlinear systems with##unknown time delays, IEEE Trans. Autom. Control, 48(11) (2003), 2004{2010.##[5] F. Gmez Estern, A. Barreiro, J. Aracil and F. Gordillo, Robust generation of almostperiodic##oscillations in a class of nonlinear systems, Int. J. Robust and Control, 16(18) (2006),##[6] C. Hua and X. Guan, Output feedback stabilization for timedelay nonlinear interconnected##systems using neural networks, IEEE Trans. Neural Netw., 19(4) (2008), 673{688.##[7] L. R. Hunt and G. Meyer, Stable inversion for nonlinear systems, Automatica, 33(8) (1997),##1549{1554.##[8] M. Krstic, I. Kanellakopoulos and P. V. Kokotovic, Nonlinear and Adaptive Control Design,##New York, NY, USA: Wiley, 1995.##[9] H. Lee and M. Tomizuka, Robust adaptive control using a universal approximation for SISO##nonlinear systems, IEEE Trans. Fuzzy Syst., 8(1) (2001), 95{106.##[10] Y. Li and S. C. Tong, Adaptive fuzzy outputfeedback stabilization control for a class of##switched nonstrictfeedback nonlinear systems, IEEE Trans. Cybern., 99 (2016), 1{10.##[11] Y. Li, S. C. Tong, Y. Liu and T. Li, Adaptive fuzzy robust output feedback control of nonlinear##systems with unknown dead zones based on a smallgain approach, IEEE Trans. Fuzzy Syst,##22(1) (2014), 164176.##[12] Z. Liu, G. Lai, Y. Zhang, X. Chen and C. L. P. Chen, Adaptive neural control for a class of##nonlinear timevarying delay systems with unknown hysteresis, IEEE Trans. Neural Netw.##Learn. Syst, 25(12) (2014), 2129{2140.##[13] A. E. Ougli and B. Tidhaf, Optimal type2 fuzzy adaptive control for a class of uncertain##nonlinear systems using an LMI approach, Int. J. Innov. Comput. I., 11(3) (2015), 851863.##[14] B. Ren, S. S. Ge, C. Y. Su and T. H. Lee, Adaptive neural control for a class of uncertain##nonlinear systems in purefeedback form with hysteresis input, IEEE Trans. Syst., Man,##Cybern. Syst., 39(2) (2009), 431{443.##[15] C. Y. Su, M. Oya and H. Hong, Stable adaptive fuzzy control of nonlinear systems preceded##by unknown backlashlike hysteresis, IEEE Trans. Fuzzy Syst, 11(1) (2003), 1{8.##[16] C. Y. Su, Y. Stepanenko, J. Svoboda and T. P. Leung, Robust adaptive control of a class##of nonlinear systems with unknown backlashlike hysteresis, IEEE Trans. Autom. Control,##45(12) (2000), 2427{2432.##[17] X. Tan and J. S. Baras, Modelling and control of hysteresis in magnetostrictive actua##tors,Automatica, 40(9) (2004), 1469{1480.##[18] G. Tao and P. V. Kokotovic, Adaptive control of plants with unknown hysteresis, IEEE Trans.##Autom. Control, 40(2) (1995), 200{212. ##[19] G. Tao and P. V. Kokotovic, Continuoustime adaptive control of systems with unknown##backlash, IEEE Trans. Autom. Control, 40(6) (2002), 1083{1087.##[20] S. C. Tong and Y. Li, Adaptive fuzzy output feedback tracking backstepping control of strict##feedback nonlinear systems with unkown dead zones, IEEE Trans. Fuzzy Syst, 20(1) (2012),##[21] S. C. Tong, Y. Li and S. Sui, Adaptive Fuzzy Tracking Control Design for SISO Uncertain##NonStrict Feedback Nonlinear Systems, IEEE Trans. Fuzzy Syst., 24(6) (2016), 1441{1454.##[22] S. C. Tong, S. Sui and Y. Li, Adaptive fuzzy decentralized output stabilization for stochas##tic nonlinear largescale systems with unknown control directions, IEEE Trans. Fuzzy Syst,##22(5) (2014), 1365{1372.##[23] S. C. Tong, X. L. He and H. G. Zhang, A combined backstepping and smallgain approach to##robust adaptive fuzzy output feedback contro, IEEE Trans. Fuzzy Syst, 17(5) (2009), 1059{##[24] H. Wang, B. Chen, K. Liu, X. Liu and C. Lin, Adaptive neural tracking control for a class of##nonstrictfeedback stochastic nonlinear systems with unknown backlashlike hysteresis, IEEE##Trans. Neural Netw. Learn. Syst., 25(5) (2014), 947{958.##[25] M. Wang, B. Chen, X. Liu and P. Shi, Adaptive fuzzy tracking control for a class of perturbed##strictfeedback nonlinear timedelay systems, Fuzzy Set. Syst., 159(8) (2008), 949{967.##[26] Y. Yang and C. Zhou, Adaptive fuzzy H stabilization for strictfeedback canonical non##linear systems via backstepping and smallgain approach, IEEE Trans. Fuzzy Syst, 13(1)##(2005),104{114.##[27] J. Yu, P. Shi, H. Yu, B. Chen and C. Lin, Approximationbased discretetime adaptive position##tracking control for interior permanent magnet synchronous motors, IEEE Trans. Cybern,##45(7) (2015), 1363{1371.##[28] X. Zhou, P. Shi and X. Zheng, Fuzzy adaptive control design and discretization for a class##of nonlinear uncertain systems, IEEE Trans. Cybern, 46(6) (2016), 1476{1483.##[29] Q. Zhou, P. Shi, S. Xu and H. Li, Adaptive output feedback control for nonlinear timedelay##systems by fuzzy approximation approach, IEEE Trans. Fuzzy Syst, 21(2) (2013), 301{313.##[30] J. Zhou, C. Wen and Y. Zhang, Adaptive backstepping control of a class of uncertain non##linear systems with unknown backlashlike hysteresis, IEEE Trans.Autom. Control, 49(10)##(2004), 1751{1759.##[31] J. Zhou, C. Wen and T. Li, Adaptive output feedback control of uncertain nonlinear systems##with hysteresis nonlinearity, IEEE Trans. Autom. Control, 57(10) (2012), 2627{2633.##]
KFLAT PROJECTIVE FUZZY QUANTALES
2
2
In this paper, we introduce the notion of {bf K}flat projective fuzzy quantales, and give an elementary characterization in terms of a fuzzy binary relation on the fuzzy quantale. Moreover, we prove that {bf K}flat projective fuzzy quantales are precisely the coalgebras for a certain comonad on the category of fuzzy quantales. Finally, we present two special cases of {bf K} as examples.
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81


Jing
Lu
College of Mathematics and Information Science, Shaanxi Normal Univer
sity, Xi'an 710119, P.R. China
College of Mathematics and Information Science,
China
1044250817@qq.com


Kaiyun
Wang
College of Mathematics and Information Science, Shaanxi Normal
University, Xi'an 710119, P.R. China
College of Mathematics and Information Science,
China
wangkaiyun@snnu.edu.cn


Bin
Zhao
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, P.R. China
College of Mathematics and Information Science,
China
zhaobin@snnu.edu.cn
Fuzzy quantale
Fuzzy binary relation
{bf K}flat projective fuzzy quantale
Comonad
[[1] J. Adamek and H. Herrlich and G. E. Strecker, Abstract and Concrete Categories: The Joy##of Cats, John Wiley & Sons, New York, (1990), 1507.##[2] B. Banaschewski, Projective frames: a general view, Cahiers Topologie Geom. Dierentielle##Cat., XLVI (2005), 301312. ##[3] R. Belohlavek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Aca##demic/Plenum Publishers, New York, 20 (2002), 1369.##[4] R. P. Dilworth, Noncommutative residuated lattices, Trans. Amer. Math. Soc., 46 (1939),##[5] L. Fan, A new approach to quantitative domain theory, Electron. Notes Theor. Comput. Sci.,##45(1) (2001), 7787.##[6] G. Gierz, et al., Continuous Lattices and Domains, Encyclopedia of Mathematics and its##Applications, vol. 93, Cambridge University Press, Cambridge, 93 (2003), 1591.##[7] H. Herrlich and G. E. Strecker, Category Theory, An introduction, Second edition, Sigma##Series in Pure Mathematics, vol. 1, Heldermann Verlag, Berlin, 1 (1979), 1400.##[8] P. T. Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics, Cambridge##University Press, Cambridge, 3 (1982), 1370.##[9] D. Kruml and J. Paseka, Algebraic and categorical ascepts of quantales, Handb. Algebra, 5##(2008), 323362.##[10] H. L. Lai and D. X. Zhang, Complete and directed complete ##categories, Theor. Comput.##Sci., 388 (2007), 125.##[11] Y. M. Li, M. Zhou and Z. H. Li, Projective and injective objects in the category of quantales,##J. Pure Appl. Algebra, 176 (2002), 249258.##[12] J. Lu and B. Zhao, The projective objects in the category of fuzzy quantales, J. Shandong##Univ. (Nat. Sci.), (in Chinese), 50(2) (2015), 4754 .##[13] C. J. Mulvey, &, Supplemento ai Rendiconti del Circolo Matematico di Palermo, II(12) (19##86), 99104.##[14] C. J. Mulvey and J. W. Pelletier, On the quantisation of points, J. Pure Appl. Algebra, 159##(2001), 231295.##[15] J. Paseka, Projective quantale: A general view, Int. J. Theor. Phys., 47(1) (2008), 291296.##[16] K. I. Rosenthal, Quantales and their Applications, Pitman Research Notes in Mathematics##Series, vol. 234, Longman Scientic & Technical, Essex, 234 (1990), 1165.##[17] S. A. Solovyov, A representation theorem for quantale algebras, Contrib. Gen. Algebra, 18##(2008), 189198.##[18] K. Y. Wang and B. Zhao, Some properties of the category of fuzzy quantales, J. Shaanxi##Norm. Univ. (Nat. Sci. Ed.), (in Chinese), 41(3) (2013), 16 .##[19] K. Y. Wang, Some researches on fuzzy domains and fuzzy quantales, Ph. D. Thesis, College##of Mathematics and Information Science, Shaanxi Normal University, Xi'an, (2012), 1115.##[20] M. Ward, Structure residuation, Ann. Math., 39 (1938), 558568.##[21] R. Wang and B. Zhao, Quantale algebra and its algebraic ideal, Fuzzy Syst. Math., (in##Chinese), 24 (2010), 4449.##[22] M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc., 45 (1939), 335##[23] W. Yao and L. X. Lu, Fuzzy Galois connections on fuzzy posets, Math. Log. Quart., 55(1)##(2009), 105112.##[24] W. Yao, Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete##posets, Fuzzy Sets Syst., 161(7) (2010), 973987.##[25] W. Yao, An approach to fuzzy frames via fuzzy posets, Fuzzy Sets Syst., 166 (2011), 7589.##[26] W. Yao, A survey of fuzzications of frames, the PapertPapertIsbell adjunction and sobri##ety, Fuzzy Sets Syst., 190 (2012), 6381.##[27] L. A. Zadeh, Fuzzy sets, Inf. Control, 8(3) (1965), 338353.##[28] Q. Y. Zhang and L. Fan, Continuity in quantitative domains, Fuzzy Sets Syst., 154(1)##(2005), 118131.##]
LFUZZY CONVEXITY INDUCED BY LCONVEX FUZZY SUBLATTICE DEGREE
2
2
In this paper, the notion of $L$convex fuzzy sublattices is introduced and their characterizations are given. Furthermore, the notion of the degree to which an $L$subset is an $L$convex fuzzy sublattice is proposed and its some characterizations are given. Besides, the $L$convex fuzzy sublattice degrees of the homomorphic image and preimage of an $L$subset are studied. Finally, we obtain an $L$fuzzy convexity, which is induced by the $L$convex fuzzy sublattice degrees, in the sense of Shi and Xiu.
1

83
102


Juan
Li
School of Mathematics, Beijing Institute of Technology, Beijing 100081,
PR China
School of Mathematics, Beijing Institute
China
lijuan@htu.edu.cn


Fu Gui
Shi
School of Mathematics, Beijing Institute of Technology, Beijing 100081,
PR China
School of Mathematics, Beijing Institute
China
$L$convex fuzzy sublattice
Implication operator
$L$convex fuzzy sublattice degree
$L$fuzzy convexity
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GENERAL FUZZY AUTOMATA BASED ON COMPLETE RESIDUATED LATTICEVALUED
2
2
The present paper has been an attempt to investigate the general fuzzy automata on the basis of complete residuated latticevalued ($L$GFAs). The study has been chiefly inspired from the work by Mockor cite{15, 16, 17}. Regarding this, the categorical issue of $L$GFAs has been studied in more details. The main issues addressed in this research include: (1) investigating the relationship between the category of $L$GFAs and the category of nondeterministic automata (NDAs); as well as the relationship between the category of generalized $L$GFAs and the category of NDAs; (2) demonstrating the existence of isomorphism between the category of $L$GFAs and the subcategory of generalized $L$GFAs and between the category of $L$GFAs and the category of sets of NDAs; (3) and further scrutinizing some specific relationship between the output $L$valued subsets of generalized $L$GFAs and the output $L$valued of NDAs.
1

103
121


K.
Abolpour
Department of Mathematics, Kazerun Branch, Islamic Azad University, Kazerun, Iran
Department of Mathematics, Kazerun Branch,
Iran


M. M.
Zahedi
Department of Mathematics, Kerman Graduate University of Advanced Technology, Kerman, Iran
Department of Mathematics, Kerman Graduate
Iran
zahedi_mm@ mail.uk.ac.ir
General fuzzy automata
Active state set
Residuatedlattice
Isomorphism of category
Functor
[[1] K. Abolpour and M. M. Zahedi, Isomorphism between two BLgeneral fuzzy automata, Soft##Comput., 16 (2012), 729736.##[2] M. A. Arbib, rom automata theory to brain theory, Int. J. ManMachine Stud., 7 (1975),##[3] M. A. Arbib and E. G. Manes, A categorist's view of automata and systems, in: E. G. Manes##(Ed.), Category Theory Applied to Computation and Control, Proc. First Internat. Symp.##Amherst MA, 1974, Lecture Notes in Computer Science, Springer, Berlin, 25 (1975), 6278.##[4] M. A. Arbib and E. G. Manes, Basic concepts of category theory applicable to computation##and control, in: E. G. Manes (Ed.), Category Theory Applied to Computation and Control,##Proc. First Internat. Symp. Amherst MA, 1974, Lecture Notes in Computer Science, Springer,##Berlin, 25 (1975), 241.##[5] M. A. Arbib and E. G. Manes, Fuzzy machines in a category, Bull. Anstral. Math. Soc., 13##(1975), 169210.##[6] M. A. Arbib and E. G. Manes, Machines in category: an expository introduction, SIAM Rev.,##16 (1974), 163192.##[7] A. W. Burks, Logic, biology and automata some historical re##ections, Int. J. Man Machine##Stud. 7 (1975), 297312.##[8] W. L. Deng and D. W. Qiu, Supervisory control of fuzzy discrete event systems for simulation##equivalence, IEEE Transactions on Fuzzy Systems, 23 (2015), 178192.##[9] M. Doostfatemeh and S. C. Kremer, New directions in fuzzy automata, International Journal##of Approximate Reasoning, 38 (2005), 175214.##[10] J. A. Goguen, Lfuzzy sets, J. Math. Anal. Appl., 18 (1967), 145174.##[11] Y. M. Li, A categorical approach to latticevalued fuzzy automata, Fuzzy Sets and Systems,##156 (2006), 855864.##[12] D. S. Malik and J. N. Mordeson, Fuzzy Discrete Structures, PhysicaVerlag, Heidelberg, New##York, 2000.##[13] M. L. Minsky, Computation: nite and innite machines, PrenticeHall, Englewood Clis,##NJ, Chapter 3, (1967), 3266.##[14] J. Mockor, A category of fuzzy automata, Internat. J. General Systems, 20 (1991), 7382.##[15] J. Mockor, Fuzzy and nondeterministic automata, Soft Comput., 3 (1999), 221226.##[16] J. Mockor, Semigroup homomorphisms and fuzzy automata, Soft comput., 6 (2002), 423427.##[17] J. N. Mordeson and D. S. Malik, Fuzzy Automata and languages: Theory and Applications,##Chapman & Hall, CRC, Boca Raton, London, 2002.##[18] W. Omlin, C. L. Giles and K. K. Thornber, Equivalence in knowledge representation: au##tomata, rnns, and dynamical fuzzy systems, Proc. IEEE, 87 (1999), 16231640.##[19] D. W. Qiu, A note on Trillas CHC models, Artif. Intell., 171 (2007), 239254.##[20] D. W. Qiu, Automata theory based on complete residuated latticedvalued logic (I), Sci. China##(Ser. F), 44 (2001), 419429.##[21] D. W. Qiu, Automata theory based on complete residuated latticedvalued logic (II), Sci.##China (Ser. F), 45 (2002), 442452.##[22] D. W. Qiu, Pumping lemma in automata theory based on complete residuated latticevalued##logic: a note, Fuzzy Sets and Systems, 157 (2006), 21282138.##[23] D. W. Qiu, Supervisory control of fuzzy discrete event systems: a formal approach, IEEE##Transactions on Systems, Man and CyberneticsPart B, 35 (2005), 7288.##[24] D. W. Qiu and F. C. Liu, Fuzzy discrete event systems under fuzzy observability and a##testalgorithm, IEEE Transactions on Fuzzy Systems., 17(2009), 578589.##[25] J. Tang, M. Luo and J. Tang, Results on the use of category theory for the study of lattice##valued nite state machines, Information Sciences, 288 (2014), 279289.##[26] S. P. Tiwari and A. K. Singh, On minimal realization of fuzzy behaviour and associated##categories, Journal of Applied Mathematics and Computing, 45 (2014), 223234. ##[27] S. P. Tiwari, K. Y. Vijay and A. K. Singh, Construction of a minimal realization and monoid##for a fuzzy language: a categorical approach, Journal of Applied Mathematics and Comput##ing, 47 (2015), 401416.##[28] V. Trnkova, Automata and categories, in: lecture notes computer science, Springer, Berlin,##32 (1975), 160166.##[29] V. Trnkova, Lfuzzy functional automata, in: lecture notes computer science, Springer,##Berlin, 74 (1979), 463473.##[30] V. Trnkova, Relational automata in a category and their languages, in: lecture notes com##puter science, Springer, Berlin, 56 (1977), 340355.##[31] W. G. Wee, On generalization of adaptive algorithm and application of the fuzzy sets concept##to pattern classication, Ph.D. Thesis, Purdue University, Lafayette, IN, 1967.##[32] L. H. Wu and D. Qiu, Automata theory based on complete residuated latticevalued logic:##Reduction and minimization, Fuzzy Sets and Systems, 161 (2010), 16351656.##[33] L. H. Wu, D. Qiu and H. Xing, Automata theory based on complete residuated latticevalued##logic: Turing machines, Fuzzy Sets and Systems, 208 (2012), 4366.##[34] H. Xing and D. Qiu, Automata theory based on complete residuated latticevalued logic: A##categorical approach, Fuzzy Sets and Systems, 160 (2009), 24162428.##[35] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338353.##[36] M. M. Zahedi, M. Horry and K. Abolpour, Bifuzzy (general) topology on maxmin general##fuzzy automata, Advances in Fuzzy Mathematics, 3(1) (2008), 5168.##]
SOME COUPLED FIXED POINT RESULTS ON MODIFIED INTUITIONISTIC FUZZY METRIC SPACES AND APPLICATION TO INTEGRAL TYPE CONTRACTION
2
2
In this paper, we introduce fruitful concepts of common limit range and joint common limit range for coupled mappings on modified intuitionistic fuzzy metric spaces. An illustrations are also given to justify the notion of common limit range and joint common limit range property for coupled maps. The purpose of this paper is to prove fixed point results for coupled mappings on modified intuitionistic fuzzy metric spaces. Moreover, we extend the notion of common limit range property and E.A property for coupled maps on modified intuitionistic fuzzy metric spaces. As an application, we extend our main result to integral type contraction condition and also for finite number of mappings on modified intuitionistic fuzzy metric spaces.
1

123
137


Vishal
Gupta
Department of Mathematics, Maharishi Markandeshwar University,
Mullana133207, Ambala, Haryana, India
Department of Mathematics, Maharishi Markandeshwar
India
vishal.gmn@gmail.com


Rajesh
Kumar Saini
Department of Mathematics, Statistics and Computer Applications, Bundelkhand University, Jhansi, U.P., India
Department of Mathematics, Statistics and
India


Ashima
Kanwar
Department of Mathematics, Maharishi Markandeshwar University,
Mullana133207, Ambala, Haryana, India
Department of Mathematics, Maharishi Markandeshwar
India
kanwar.ashima87@gmail.com
Modified intuitionistic fuzzy metric space (MIFMspace)
Coupled maps
Common limit range property
Joint common limit range property
E.A property
Weakly compatible mappings
[[1] M. Aamri and D. EI Moutawakil, Some new common xed point theorems under strict con##tractive conditions, J. Math. Anal. Appl., 270 (2002), 181{188.##[2] M. Abbas, M. Ali Khan and S. Radenovic, Common coupled xed point theorems in cone##metric spaces for wcompatible mappings, Appl. Math. Comput., 217(1) (2010), 195{202.##[3] C. Alaca, D. Turkoglu and C. Yildiz, Fixed points in intuitionistic fuzzy metric spaces, Chaos,##Solitons and Fractals, 29 (2006), 1073{1078.##[4] K. T. Atanassov, Intuitionistic fuzzy set, Fuzzy Sets and Systems, 20(1) (1986), 87{96.##[5] T. G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric##spaces and applications, Nonlinear Anal. TMA., 65 (2006), 1379{1393.##[6] A. Branciari, A xed point theorem for mappings satisfying a general contractive condition##of integral type, Int. J. Math. Sci., 29(9) (2002), 531{536.##[7] G. Deschrijver, C. Cornelis and E. E. Kerre, On the representation of intuitionistic fuzzy##tnorms and tconorms , IEEE Trans Fuzzy System, 12(3) (2004), 45{61.##[8] G. Deschrijver and E. E. Kerre, On the relationship between some extensions of fuzzy set##theory, Fuzzy Sets System, 133(2) (2003), 227{235. ##[9] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems,##64(3) (1994), 395{399.##[10] V. Gregori, S. Romaguera and P. Veereamani, A note on intuitionistic fuzzy metric spaces ,##Chaos, Solitons and Fractals, 28(4) (2006), 902{905.##[11] V. Gupta, A. Kanwar and N. Gulati, Common coupled xed point result in fuzzy metric##spaces using JCLR property, Smart Innovation, Systems and Technologies, Springer, 43(1)##(2016), 201{208.##[12] V. Gupta and A. Kanwar, Fixed point theorem in fuzzy metric spaces satisfying E.A Property##, Indian Journal of Science and Technology, 5(12) (2012), 3767{3769.##[13] S. Jain, S. Jain and L. B. Jain, Compatibility of type (P) in modied intuitionistic fuzzy##metric space , Journal of Nonlinear Science and its Applications, 3(2) (2010), 96{109.##[14] S. M. Kang , V. Gupta, B. Singh and S. Kumar, Some common xed point theorems using##implicit relations in fuzzy metric spaces, International Journal of Pure and Applied Mathe##matics, 87(2) (2013), 333{347.##[15] I. Kramosil and J. Michalek, Fuzzy metric and Statistical metric spaces, Kybernetica, 11##(1975), 326{334.##[16] V. Lakshmikantham and Lj. B. Ciric, Coupled xed point theorems for nonlinear contractions##in partially ordered metric space , Nonlinear Anal. TMA., 70 (2009), 4341{4329.##[17] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22(5) (2004),##1039{1046.##[18] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and##Fractals, 27(2) (2006), 331{344.##[19] R. Saadati, S. Sedghi and N. Shobe, Modied intuitionistic fuzzy spaces and some xed point##theorems, Chaos, Solitons and Fractals, 38(1) (2008), 36{47.##[20] R. K. Saini, V. Gupta and S. B. Singh, Fuzzy version of some xed points theorems on##expansion type maps in fuzzy metric space, Thai Journal of Mathematics, 5(2) (2007), 245{##[21] S. Sedghi, N. Shobe and A. Aliouche, Common xed point theorems in intuitionistic fuzzy##metric spaces through conditions of integral type, Applied Mathematics and Information##Sciences, 2(1) (2008), 61{82.##[22] M. Tanveer , M. Imdad, D. Gopal and D. K. Patel, Common xed point theorems in modied##intuitionistic fuzzy metric spaces with common property (E.A.), Fixed Point Theory and##Applications, doi :10.1186/16871812201236, article 36 (2012), 1{12.##[23] D. Turkoglu, C. Alaca, Y. J. Cho and C. Yildiz, Common xed points in intuitionistic fuzzy##metric spaces, Journal of Applied Mathematics and Computing, 22(12) (2006), 411{424.##[24] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.##]
INTERVAL ANALYSISBASED HYPERBOX GRANULAR COMPUTING CLASSIFICATION ALGORITHMS
2
2
Representation of a granule, relation and operation between two granules are mainly researched in granular computing. Hyperbox granular computing classification algorithms (HBGrC) are proposed based on interval analysis. Firstly, a granule is represented as the hyperbox which is the Cartesian product of $N$ intervals for classification in the $N$dimensional space. Secondly, the relation between two hyperbox granules is measured by the novel positive valuation function induced by the two endpoints of an interval, where the operations between two hyperbox granules are designed so as to include granules with different granularity. Thirdly, hyperbox granular computing classification algorithms are designed on the basis of the operations between two hyperbox granules, the fuzzy inclusion relation between two hyperbox granules, and the granularity threshold. We demonstrate the superior performance of the proposed algorithms compared with the traditional classification algorithms, such as, Random Forest (RF), Support Vector Machines (SVMs), and Multilayer Perceptron (MLP).
1

139
156


Hongbing
Liu
Center of Computing, Xinyang Normal University, Xinyang 464000,
P. R. China
Center of Computing, Xinyang Normal University,
China
liuhbing@126.com


Jin
Li
Center of Computing, Xinyang Normal University, Xinyang 464000, P. R. China
Center of Computing, Xinyang Normal University,
China
lijin@xynu.edu.cn


Huaping
Guo
School of Computer and Information Technology, Xinyang Normal
University, Xinyang 464000, P. R. China
School of Computer and Information Technology,
China
hpguo_cm@163.com


Chunhua
Liu
School of Computer and Information Technology, Xinyang Normal
University, Xinyang 464000, P. R. China
School of Computer and Information Technology,
China
zzdxliuch@163.com
Fuzzy lattice
Granular computing
Hyperbox granule
Fuzzy inclusion relation
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