基于严凹函数的粗糙集不确定性度量
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摘要
通过语义分析,提出了一种拓展的粗糙集不确定性度量公理化定义;将香农熵函数推广到严凹函数,提出了一类以条件概率为自变量、基于严凹函数的粗糙集不确定性度量公式,它是严凹函数值的加权平均.在此基础上,得到一系列粗糙集不确定性度量方法.从严凹函数视角讨论了基于模糊熵的不确定性度量方法,发现现有多种能够用于度量粗糙集不确定性的模糊熵函数都是所提出方法的特殊情形.比较了粗糙度、改进粗糙度和所提出方法的区别和联系,最后设计了一些算例,比较了各种方法的异同,验证了基于严凹函数的粗糙集不确定性度量与粗糙集不确定性语义是一致的.
Based on the semantic analysis, a general axiomatic definition of uncertainty measure for rough set is proposed. By extending the Shannon entropy function to strictly concave function, a class of uncertainty measures based on strictly concave function are put forward. They are weighted average of strictly concave function, whose variable is a conditional probability. It follows that a series of measuring methods are developed. The measuring methods based on fuzzy entropy are discussed under the view of strictly concave function. It is proved that they are the special cases of the method proposed in this paper. The difference and relationship among roughness measure, modified rough measure and the uncertainty measure based on strictly concave function are discussed. Finally, some examples are designed to compare the methods discussed in this paper. It is found that the proposed uncertainty measures based on strictly concave function are consistent with the semantics of uncertainty of rough set.
Based on the semantic analysis, a general axiomatic definition of uncertainty measure for rough set is proposed. By extending the Shannon entropy function to strictly concave function, a class of uncertainty measures based on strictly concave function are put forward. They are weighted average of strictly concave function, whose variable is a conditional probability. It follows that a series of measuring methods are developed. The measuring methods based on fuzzy entropy are discussed under the view of strictly concave function. It is proved that they are the special cases of the method proposed in this paper. The difference and relationship among roughness measure, modified rough measure and the uncertainty measure based on strictly concave function are discussed. Finally, some examples are designed to compare the methods discussed in this paper. It is found that the proposed uncertainty measures based on strictly concave function are consistent with the semantics of uncertainty of rough set.
引文
[1]Sun L,Xu JC,Tian Y.Feature selection using rough entropy-based uncertainty measures in incomplete decision systems.Knowledge-Based Systems,2012,36(1):206-216.
[2]Liang JY,Qian YH.Information granules and entropy theory in information systems.Science in China Series F:Information Sciences,2008,51(10):1427-1444.
[3]Deng XF,Yao YY.A multifaceted analysis of probabilistic three-way decisions.Fundamenta Informaticae,2014,132(3):291-313.
[4]Pawlak Z.Rough sets.Int’l Journal of Computer&Information Sciences,1982,11(5):341-356.
[5]Beaubouef T,Petry FE,Arora G.Information-Theoretic measures of uncertainty for rough sets and rough relational databases.Information Sciences,1998,109(1-4):185-195.
[6]Liang JY,Wang JH,Qian YH.A new measure of uncertainty based on knowledge granulation for rough sets.Information Sciences,2009,179:458-470.
[7]Xu BW,Zhou YM,Lu HM.An improved accuracy measure for rough sets.Journal of Computer and System Sciences,2005,71:163-173.
[8]Wang XY,Cai N,Yang J,Liu XJ.A new method for measuring uncertainty in rough sets.Journal of Shanghai Jiaotong University,2006,40(7):1130-1134(in Chinese with English abstract).
[9]Teng SH,Lu M,Yang AF,Zhang J,Zhuang ZW.A weighted uncertainty measure of rough sets based on general binary relation.Chinese Journal of Computers,2014,37(3):649-665(in Chinese with English abstract).
[10]Düntsch I,Gediga G.Uncertainty measures of rough set prediction.Artificial intelligence,1998,106(1):109-137.
[11]Wierman MJ.Measuring uncertainty in rough set theory.Int’l Journal of General System,1999,28(4-5):283-297.
[12]Miao DQ,Wang J.On the relationships between information entropy and roughness of knowledge in rough set.Pattern Recognition and Artificial Intelligence,1998,11(3):34-40(in Chinese with English abstract).
[13]Wang GY,Yu H,Yang DH.Decision table reduction based on conditional information entropy.Chinese Journal of Computers,2002,25(7):759-766(in Chinese with English abstract).
[14]Li J,Shi KQ.Uncertainty measurement of rough sets based on conditional entropy.Systems Engineering and Electronics,2008,30(3):473-476(in Chinese with English abstract).
[15]Wei W,Wei Q,Wang F.Comparative study of uncertainty measure in rough set.Journal of Nanjing University(Natural Sciences),2015,51(4):714-722(in Chinese with English abstract).
[16]Liang JY,Shi ZZ.The information entropy,rough entropy and knowledge granulation in rough set theory.Int’l Journal of Uncertainty,Fuzziness and Knowledge-Based Systems,2004,19(1):37-46.
[17]Liang JY,Shi ZZ,Wierman MJ.Information entropy,rough entropy and knowledge granulation in incomplete information systems.Int’l Journal of General Systems,2006,35(6):641-654.
[18]Bianucci D,Cattaneo G,Ciucci D.Entropies and co-entropies of coverings with application to incomplete information systems.Fundamenta Informatiace,2007,75(1-4):77-105.
[19]Zhu P,Wen QY.Entropy and co-entropy of a covering approximation space.Int’l Journal of Approximate Reasoning,2012,53(4):528-540.
[20]Qian YH,Liang JY.Combination entropy and combination granulation in rough set theory.Int’l Journal of Uncertainty,Fuzziness and Knowledge-Based Systems,2008,16(2):179-193.
[21]Dai JH,Wang WT,Xu Q,et al.Uncertainty measurement for interval-valued decision systems based on extended conditional entropy.Knowledge-Based Systems,2012,27(1):443-450.
[22]Dai JH,Tian HW.Entropy measures and granularity measures for set-valued information systems.Information Sciences,2013,240(1):72-82.
[23]Chakrabary K,Biswas R,Nanda S.Fuzziness in rough sets.Fuzzy Sets and Systems,2000,110(2):247-25.
[24]Liang JY,Dang CY,Chin KS,et al.A new method for measuring uncertainty and fuzziness in rough set theory.Int’l Journal of General Systems,2002,31(4):331-342.
[25]Wang GY,Zhang QH.Uncertainty of rough sets in different knowledge granularities.Chinese Journal of Computers,2008,31(9):1588-1598(in Chinese with English abstract).
[26]De Luca A,Termini S.A definition of a non-probabilistic entropy in the setting of fuzzy sets theory.Information and Control,1972,20(4):301-312.
[27]Wei W,Liang JY,Qian YH,et al.Can fuzzy entropies be effective measures for evaluating the roughness of a rough set?Information Sciences,2013,232:143-166.
[28]Hu J,Wang GY.Uncertainty measure rule sets on rough sets.Pattern Recognition and Artificial Intelligence,2010,23(5):606-615(in Chinese with English abstract).
[29]Huang GS,Zeng FZ,Wen H.Uncertainty measures of rough set based on conditional possibility.Control and Decision,2015,30(6):1099-1105(in Chinese with English abstract).
[30]Huang GS,Zeng FZ,Chen GY,Wen H.Knowledge granularity and relative granularity based on strictly convex function.Pattern Recognition and Artificial Intelligence,2013,26(10):897-908(in Chinese with English abstract).
[31]Yao YY.The superiority of three-way decisions in probabilistic rough set models.Information Sciences,2011,181(6):1080-1096.
[32]Kuang JC.Applied Inequalities.4th ed.,Ji’nan:Shandong Science and Technology Press,2010(in Chinese).
[33]Pal NR,Pal SK.Entropy:A new definition and its applications.IEEE Trans.on Systems,Man and Cybernetics,1991,21(5):1260-1270.
[34]Bhandari D,Pal NR.Some new information measures for fuzzy sets.Information Sciences,1993,67(3):209-228.
[35]Fan J,Xie W.Distance measure and induced fuzzy entropy.Fuzzy Sets and Systems,1999,104(2):305-314.
[36]Liu XC.Entropy,distance measure and similarity measure of fuzzy sets and their relations.Fuzzy Sets and Systems,1992,52(3):305-318.
[37]Fan JL,Ma YL.Some new fuzzy entropy formulas.Fuzzy Sets and Systems,2002,128(2):277-284.
[38]Fan JL,Ma YL,Xie WX.On some properties of distance measures.Fuzzy Sets and Systems,2001,117(3):355-361.
[39]Yao YY.A note on definability and approximations.In:Peters JF,et al.,eds.Proc.of the Transactions on Rough Sets VII.Berlin,Heidelberg:Springer-Verlag,2007.274-282.
[8]王向阳,蔡念,杨杰,刘小军.基于近似精度和条件信息熵的粗糙集不确定性度量方法.上海交通大学学报,2006,40(7):1130-1134.
[9]腾书华,鲁敏,杨阿峰,张军,庄钊文.基于一般二元关系的粗糙集加权不确定性度量.计算机学报,2014,37(3):649-665.
[12]苗夺谦,王珏.粗糙集理论中知识粗糙性与信息熵关系的讨论.模式识别与人工智能,1998,11(3):34-40.
[13]王国胤,于洪,杨大春.基于条件信息熵的决策表约简.计算机学报,2002,25(7):759-766.
[14]李健,史开泉.基于条件粗糙熵的粗集不确定性度量.系统工程与电子技术,2008,30(3):473-476.
[15]魏巍,魏琪,王锋.粗糙集的不确定性度量比较研究.南京大学学报(自然科学),2015,51(4):714-722.
[25]王国胤,张清华.不同知识粒度下粗糙集的不确定性研究.计算机学报,2008,31(9):1588-1598.
[28]胡军,王国胤.粗糙集的不确定性度量准则.模式识别与人工智能,2010,23(5):606-615.
[29]黄国顺,曾凡智,文翰.基于条件概率的粗糙集不确定性度量.控制与决策,2015,30(6):1099-1105.
[30]黄国顺,曾凡智,陈广义,文翰.基于严凸函数的知识粒度与相对粒度.模式识别与人工智能,2013,26(10):897-908.
[32]匡继昌.常用不等式.第4版,济南:山东科学技术出版社,2010.
[2]Liang JY,Qian YH.Information granules and entropy theory in information systems.Science in China Series F:Information Sciences,2008,51(10):1427-1444.
[3]Deng XF,Yao YY.A multifaceted analysis of probabilistic three-way decisions.Fundamenta Informaticae,2014,132(3):291-313.
[4]Pawlak Z.Rough sets.Int’l Journal of Computer&Information Sciences,1982,11(5):341-356.
[5]Beaubouef T,Petry FE,Arora G.Information-Theoretic measures of uncertainty for rough sets and rough relational databases.Information Sciences,1998,109(1-4):185-195.
[6]Liang JY,Wang JH,Qian YH.A new measure of uncertainty based on knowledge granulation for rough sets.Information Sciences,2009,179:458-470.
[7]Xu BW,Zhou YM,Lu HM.An improved accuracy measure for rough sets.Journal of Computer and System Sciences,2005,71:163-173.
[8]Wang XY,Cai N,Yang J,Liu XJ.A new method for measuring uncertainty in rough sets.Journal of Shanghai Jiaotong University,2006,40(7):1130-1134(in Chinese with English abstract).
[9]Teng SH,Lu M,Yang AF,Zhang J,Zhuang ZW.A weighted uncertainty measure of rough sets based on general binary relation.Chinese Journal of Computers,2014,37(3):649-665(in Chinese with English abstract).
[10]Düntsch I,Gediga G.Uncertainty measures of rough set prediction.Artificial intelligence,1998,106(1):109-137.
[11]Wierman MJ.Measuring uncertainty in rough set theory.Int’l Journal of General System,1999,28(4-5):283-297.
[12]Miao DQ,Wang J.On the relationships between information entropy and roughness of knowledge in rough set.Pattern Recognition and Artificial Intelligence,1998,11(3):34-40(in Chinese with English abstract).
[13]Wang GY,Yu H,Yang DH.Decision table reduction based on conditional information entropy.Chinese Journal of Computers,2002,25(7):759-766(in Chinese with English abstract).
[14]Li J,Shi KQ.Uncertainty measurement of rough sets based on conditional entropy.Systems Engineering and Electronics,2008,30(3):473-476(in Chinese with English abstract).
[15]Wei W,Wei Q,Wang F.Comparative study of uncertainty measure in rough set.Journal of Nanjing University(Natural Sciences),2015,51(4):714-722(in Chinese with English abstract).
[16]Liang JY,Shi ZZ.The information entropy,rough entropy and knowledge granulation in rough set theory.Int’l Journal of Uncertainty,Fuzziness and Knowledge-Based Systems,2004,19(1):37-46.
[17]Liang JY,Shi ZZ,Wierman MJ.Information entropy,rough entropy and knowledge granulation in incomplete information systems.Int’l Journal of General Systems,2006,35(6):641-654.
[18]Bianucci D,Cattaneo G,Ciucci D.Entropies and co-entropies of coverings with application to incomplete information systems.Fundamenta Informatiace,2007,75(1-4):77-105.
[19]Zhu P,Wen QY.Entropy and co-entropy of a covering approximation space.Int’l Journal of Approximate Reasoning,2012,53(4):528-540.
[20]Qian YH,Liang JY.Combination entropy and combination granulation in rough set theory.Int’l Journal of Uncertainty,Fuzziness and Knowledge-Based Systems,2008,16(2):179-193.
[21]Dai JH,Wang WT,Xu Q,et al.Uncertainty measurement for interval-valued decision systems based on extended conditional entropy.Knowledge-Based Systems,2012,27(1):443-450.
[22]Dai JH,Tian HW.Entropy measures and granularity measures for set-valued information systems.Information Sciences,2013,240(1):72-82.
[23]Chakrabary K,Biswas R,Nanda S.Fuzziness in rough sets.Fuzzy Sets and Systems,2000,110(2):247-25.
[24]Liang JY,Dang CY,Chin KS,et al.A new method for measuring uncertainty and fuzziness in rough set theory.Int’l Journal of General Systems,2002,31(4):331-342.
[25]Wang GY,Zhang QH.Uncertainty of rough sets in different knowledge granularities.Chinese Journal of Computers,2008,31(9):1588-1598(in Chinese with English abstract).
[26]De Luca A,Termini S.A definition of a non-probabilistic entropy in the setting of fuzzy sets theory.Information and Control,1972,20(4):301-312.
[27]Wei W,Liang JY,Qian YH,et al.Can fuzzy entropies be effective measures for evaluating the roughness of a rough set?Information Sciences,2013,232:143-166.
[28]Hu J,Wang GY.Uncertainty measure rule sets on rough sets.Pattern Recognition and Artificial Intelligence,2010,23(5):606-615(in Chinese with English abstract).
[29]Huang GS,Zeng FZ,Wen H.Uncertainty measures of rough set based on conditional possibility.Control and Decision,2015,30(6):1099-1105(in Chinese with English abstract).
[30]Huang GS,Zeng FZ,Chen GY,Wen H.Knowledge granularity and relative granularity based on strictly convex function.Pattern Recognition and Artificial Intelligence,2013,26(10):897-908(in Chinese with English abstract).
[31]Yao YY.The superiority of three-way decisions in probabilistic rough set models.Information Sciences,2011,181(6):1080-1096.
[32]Kuang JC.Applied Inequalities.4th ed.,Ji’nan:Shandong Science and Technology Press,2010(in Chinese).
[33]Pal NR,Pal SK.Entropy:A new definition and its applications.IEEE Trans.on Systems,Man and Cybernetics,1991,21(5):1260-1270.
[34]Bhandari D,Pal NR.Some new information measures for fuzzy sets.Information Sciences,1993,67(3):209-228.
[35]Fan J,Xie W.Distance measure and induced fuzzy entropy.Fuzzy Sets and Systems,1999,104(2):305-314.
[36]Liu XC.Entropy,distance measure and similarity measure of fuzzy sets and their relations.Fuzzy Sets and Systems,1992,52(3):305-318.
[37]Fan JL,Ma YL.Some new fuzzy entropy formulas.Fuzzy Sets and Systems,2002,128(2):277-284.
[38]Fan JL,Ma YL,Xie WX.On some properties of distance measures.Fuzzy Sets and Systems,2001,117(3):355-361.
[39]Yao YY.A note on definability and approximations.In:Peters JF,et al.,eds.Proc.of the Transactions on Rough Sets VII.Berlin,Heidelberg:Springer-Verlag,2007.274-282.
[8]王向阳,蔡念,杨杰,刘小军.基于近似精度和条件信息熵的粗糙集不确定性度量方法.上海交通大学学报,2006,40(7):1130-1134.
[9]腾书华,鲁敏,杨阿峰,张军,庄钊文.基于一般二元关系的粗糙集加权不确定性度量.计算机学报,2014,37(3):649-665.
[12]苗夺谦,王珏.粗糙集理论中知识粗糙性与信息熵关系的讨论.模式识别与人工智能,1998,11(3):34-40.
[13]王国胤,于洪,杨大春.基于条件信息熵的决策表约简.计算机学报,2002,25(7):759-766.
[14]李健,史开泉.基于条件粗糙熵的粗集不确定性度量.系统工程与电子技术,2008,30(3):473-476.
[15]魏巍,魏琪,王锋.粗糙集的不确定性度量比较研究.南京大学学报(自然科学),2015,51(4):714-722.
[25]王国胤,张清华.不同知识粒度下粗糙集的不确定性研究.计算机学报,2008,31(9):1588-1598.
[28]胡军,王国胤.粗糙集的不确定性度量准则.模式识别与人工智能,2010,23(5):606-615.
[29]黄国顺,曾凡智,文翰.基于条件概率的粗糙集不确定性度量.控制与决策,2015,30(6):1099-1105.
[30]黄国顺,曾凡智,陈广义,文翰.基于严凸函数的知识粒度与相对粒度.模式识别与人工智能,2013,26(10):897-908.
[32]匡继昌.常用不等式.第4版,济南:山东科学技术出版社,2010.