二型模糊粗糙集的理论及应用研究
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摘要
模糊集和粗糙集是处理数据的两种不同的数学方法,在处理不确定性和不精确性问题方面都推广了经典集合理论.它们不是相互对立的,在处理不完备信息方面可以互为补充.D. Dubois和H. Prade提出对对象进行粗糙近似时结合模糊集理论,建立了基于模糊等价关系的经典模糊粗糙集模型.此后,越来越多的学者投入到模糊粗糙集模型的理论及应用研究中,研究了区间值模糊粗糙集、直觉模糊粗糙集等模型.
本文从两个不同的角度分别研究了二型模糊粗糙集模型,深入探讨了它们的性质.
一方面,将区间值模糊粗糙集模型进行扩展,建立了区间二型模糊粗糙集模型,深入研究了其性质；对二型模糊集取二阶截集,将其转换为区间二型模糊集,建立了基于二阶截集的二型模糊粗糙集模型,探讨了退化模型,验证了此模型可以正确退化为经典模糊粗糙集模型.
另一方面,定义了一种新的二型模糊集的交、并运算,使得二型模糊集的这种运算满足包含关系,从而建立了基于包含关系的粗糙二型模糊集模型和二型模糊粗糙集模型,深入探讨了相关的性质.
Fuzzy sets and rough sets, extended from the classical set theory in handling problems with uncertainty and inaccuracy, are two different mathematical methods to deal with data. They are not contradictory, but complementary to each other while handling incomplete information. D. Dubois and H. Prade established the theory of fuzzy rough sets based on fuzzy equivalence relations while objects are roughly approximated in combination with fuzzy sets theory. Henceforth, more and more scholars focus on the theory and applications of fuzzy rough sets, including interval-valued fuzzy rough models and intuitionistic fuzzy rough models.
This dissertation presents the type-2 fuzzy rough models from two distinct views. Their properties are wholly investigated.
On the one hand, we extend the interval-valued fuzzy rough model to interval type-2 fuzzy rough model and explore its properties deeply. We transform type-2 fuzzy sets into interval type-2 fuzzy sets by employing the secondary cut sets. Afterforward, the type-2 fuzzy rough model based on the secondary cut sets is put forward. The degenerated type-2 fuzzy rough sets are discussed and the model can normally degenerate into classic fuzzy rough model is verified.
On the other hand, after new operations of union and intersection of type-2 fuzzy sets are defined, a new rough type-2 fuzzy model and a new type-2 fuzzy rough model based on the inclusion relation of type-2 fuzzy sets are studied. Their properties are wholly investigated.
本文从两个不同的角度分别研究了二型模糊粗糙集模型,深入探讨了它们的性质.
一方面,将区间值模糊粗糙集模型进行扩展,建立了区间二型模糊粗糙集模型,深入研究了其性质；对二型模糊集取二阶截集,将其转换为区间二型模糊集,建立了基于二阶截集的二型模糊粗糙集模型,探讨了退化模型,验证了此模型可以正确退化为经典模糊粗糙集模型.
另一方面,定义了一种新的二型模糊集的交、并运算,使得二型模糊集的这种运算满足包含关系,从而建立了基于包含关系的粗糙二型模糊集模型和二型模糊粗糙集模型,深入探讨了相关的性质.
Fuzzy sets and rough sets, extended from the classical set theory in handling problems with uncertainty and inaccuracy, are two different mathematical methods to deal with data. They are not contradictory, but complementary to each other while handling incomplete information. D. Dubois and H. Prade established the theory of fuzzy rough sets based on fuzzy equivalence relations while objects are roughly approximated in combination with fuzzy sets theory. Henceforth, more and more scholars focus on the theory and applications of fuzzy rough sets, including interval-valued fuzzy rough models and intuitionistic fuzzy rough models.
This dissertation presents the type-2 fuzzy rough models from two distinct views. Their properties are wholly investigated.
On the one hand, we extend the interval-valued fuzzy rough model to interval type-2 fuzzy rough model and explore its properties deeply. We transform type-2 fuzzy sets into interval type-2 fuzzy sets by employing the secondary cut sets. Afterforward, the type-2 fuzzy rough model based on the secondary cut sets is put forward. The degenerated type-2 fuzzy rough sets are discussed and the model can normally degenerate into classic fuzzy rough model is verified.
On the other hand, after new operations of union and intersection of type-2 fuzzy sets are defined, a new rough type-2 fuzzy model and a new type-2 fuzzy rough model based on the inclusion relation of type-2 fuzzy sets are studied. Their properties are wholly investigated.
引文
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[6]F.L. Liu, J.M. Mendel. Encoding words into interval type-2 fuzzy sets using an interval approach. IEEE Transactions on Fuzzy Systems,2008,16(6): 1503-1521.
[7]C. Hwang, F.C.H. Rhee. Uncertain fuzzy clustering:Interval Type-2 fuzzy approach to C-means. IEEE Transactions on Fuzzy Systems,2007, 15(1):107-120.
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[9]N. Zhong, J.Z. Dong, S. Ohsuga. Using rough sets with heuristics for feature selection. Journal of Intelligent Information Systems,2001,16(3):199-214.
[10]L.J. Ke, Z. Feng, Z.G. Ren. An efficient ant colony optimization approach to attribute reduction in rough set theory. Pattern Recognition Letters,2008, 29(9):1351-1357.
[11]T.R. Li, D. Ruan, W. Geert, J. Song, Y. Xu. A rough sets based characteristic relation approach for dynamic attribute generalization in data mining. Knowledge-Based Systems,2007,20(5):485-494.
[12]J.Y. Shyng, F.K. Wang, G.H. Tzeng, K.S. Wu. Rough set theory in analyzing the attributes of combination values for the insurance market. Expert Systems with Applications,2007,32(1):56-64.
[13]张恒,冯子亮.一种基于粗糙集合理论的彩色图像分割.计算机技术与发展,2009,19(2)：39-41,44.
[14]高明,王继成.双论域下粗糙集数据约简方法研究.计算机工程与应用,2009,45(2)：144-146.
[15]D. Dubois, H. Prade. Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems,1990,17:191-209.
[16]S.K. Samanta, T.K. Mondal. Intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets. Journal of Fuzzy Mathematics,2001,9:561-582.
[17]L. Zhou, W.Z. Wu. On generalized intuitionistic fuzzy rough approximation operators. Information Sciences,2008,178(11):2448-2465.
[18]B.Z. Sun, Z.T. Gong, D.G. Chen. Fuzzy rough set theory for the interval-valued fuzzy information systems. Information Sciences,2008, 178(13):2794-2815.
[19]N.N. Karnik, J.M. Mendel. An introduction to type-2 fuzzy logic systems. Los Angeles:University of Southern California,1998.
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[21]N.N. Karnik, J.M. Mendel. Operations on type-2 fuzzy sets. Fuzzy Sets and Systems,2001,122(2):327-348.
[22]N.N. Karnik, J.M. Mendel. Centroid of a type-2 fuzzy set. Information Sciences,2001,132(1-4):195-200.
[23]Q.L. Liang, J.M. Mendel. Interval type-2 fuzzy logic systems:theory and design.2000,8(5):535-550.
[24]J.M. Mendel, R.I. John. Type-2 fuzzy sets made simple. IEEE Transactions on Fuzzy Systems,2002,10(2):117-127.
[25]J.M. Mendel. Advances in type-2 fuzzy sets and systems. Information Sciences,2007,177(1):84-110.
[26]H.B. Mitchell. Pattern recognition using type-II fuzzy sets. Information Sciences,2005,170(2-4):409-418.
[27]M.S. Yang, D.C. Lin. On similarity and inclusion measures between type-2 fuzzy sets with anapplication to clustering. Computers and Mathematics with Applications,2009,57(6):896-907.
[28]H.R. Tizhoosh. Image thresholding using type II fuzzy sets. Pattern Recognition,2005,38(12):2363-2372.
[29]陈阳,王涛,刘玉航.二型模糊集下的推理模型及Mamdani推理算法.模糊系统与数学,2008,22(3)：41-48.
[30]王钰,基于Ⅱ型模糊集的车辆图像二值化算法.北京交通大学学报,2008,32(5)：73-76.
[31]孙曦浩,李医民,贡崇颖.基于Type-2型模糊集的动态生态位模型.模糊系统与数学,2007,21(1)：137-144.
[32]J.M. Mendel. Computing derivatives in interval type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems,2004,12(1):84-98.
[33]C.H. Wang, C.S. Cheng, T.T. Lee. Dynamical optimal training for interval type-2 fuzzy neural network. IEEE Transactions on Systems, Man, and Cybernetics,2004,34(3):1462-1477.
[34]J.M. Mendel, H.W. Wu. Type-2 fuzzistics for nonsymmetric interval type-2 fuzzy sets:forward problems. IEEE Transactions on Fuzzy Systems,2007, 15(5):916-930.
[35]J.M. Mendel, H.W. Wu. New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule. Information Sciences, 2007,177(12):360-377.
[36]D.R. Wu, J.M. Mendel. A vector similarity measure for linguistic approximation:Interval type-2 and type-1 fuzzy sets. Information Sciences, 2008,178(2):381-402.
[37]A.M. Radzikowska, E.E. Kerre. A comparative study of fuzzy rough sets: Fuzzy Sets and Systems,2002,126(2):137-155.
[38]T.Q. Deng, Y.M. Chen, W.L. Xu, Q.H. Dai. A novel approach to fuzzy rough sets based on a fuzzy covering. Information Sciences,2007,177(11): 2308-2326.
[39]T.J. Li, Y. Leung, W.X. Zhang. Generalized fuzzy rough approximation operators based on fuzzy coverings. International Journal of Approximate Reasoning,2008,48(3):836-856.
[40]J.S. Mi, Y.Leung, H.Y. Zhao, T. Feng. Generalized fuzzy rough sets determined by a triangular norm. Information Sciences,2008, 178(16):3203-3213.
[41]W.Z. Wu, W.X. Zhang. Constructive and axiomatic approaches of fuzzy approximation operators. Information Sciences,2004,159(3-4):233-254.
[42]Daniel. S. Yeung, D.G. Chen, E.C.C. Tsang, John W.T. Lee, X.Z. Wang. On the generalization of fuzzy rough sets. IEEE Transactions on Fuzzy Systems, 2005,13(3):343-361.
[43]G.L. Liu. Axiomatic systems for rough sets and fuzzy rough sets. International Journal of Approximate Reasoning,2008,48(3):857-867.
[44]何亚群,李继军,胡寿松.基于模糊相容关系下的相容模糊粗糙集.系统工程学报,2006,21(5)：553-556.
[415]吴正江,秦克云,乔全喜.双论域L模糊粗糙集.计算机工程与应用,2007,43(5)：10,11,23.
[46]R. Jensen, Q. Shen. Fuzzy-rough attribute reduction with application to web categorization. Fuzzy Sets and Systems,2004,141(3):469-485.
[47]E.C.C. Tsang, D.G. Chen, D.S. Yeung, X.Z. Wang, John W.T. Lee. Attribute reduction using fuzzy rough sets. IEEE Transactions on Fuzzy Systems,2008, 16(5):1130-1141.
[48]S.Y. Zhao, Eric C.C. Tsang. On fuzzy approximation operators in attribute reduction with fuzzy rough sets. Information Sciences,2008, 178(16):3163-3176.
[49]F.F. Xu, D.Q. Miao, L. Wei. Fuzzy-rough attribute via mutual information with an application to cancer classification. Computers and Mathematics with applications,2009,57(6):1010-1017.
[50]J.Y. Zhao, Z.L. Zhang. Fuzzy-rough data reduction based on information entropy. IEEE Proceedings of the Sixth International Conference on Machine Learning and Cybernetics, Hong kong,2007:3708-3711.
[51]王丽,冯山.基于模糊粗糙集的两种属性约简算法.计算机应用,2006,26(3)：635-637,672.
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