混合不确定性表示及应用研究
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摘要
各种实际信息系统中都存在着不确定性，并且多种不确定性往往同时存在。如何有效合理地表示这些混合不确定性是信息表示和处理中的基础性关键问题。为此，本文研究了由随机性、模糊性和粗糙性等几种基本不确定性组合构成的三类常见混合不确定性的表示处理，并研究了它们在回归分析、模式分类以及证据推理等方面的应用。本文主要内容和创新总结如下：
第一章详细介绍了本文的研究背景，回顾了混合不确定性表示的目前国内外研究现状，并介绍了作者的主要工作。
第二章介绍了已有的几种不确定性表示理论，包括概率论、证据理论、模糊集理论和粗糙集理论。
第三章首先介绍了基于模糊规则的模糊系统建模方法，给出了概率模糊规则基的形式。然后提出了从数据生成概率模糊规则基的方法，实现概率模糊系统建模。当后件变量（输出变量）为定量连续变量时，提出了一种从数据生成Mamdani型概率模糊规则方法；当后件变量为定性离散变量时，提出了一种从数据生成概率模糊分类规则方法。将两类概率模糊系统分别应用到非参数回归分析与时间序列预测以及模式分类中，取得了不错的结果。
第四章提出了一种连续区间上具有模糊值信任函数的模糊证据理论。鉴于模糊刻画的普遍性以及连续区间论域的常见性，连续区间上证据理论的一些模糊扩展被提出。在这些扩展中，或多或少存在一些问题，比如信任函数对焦元的重大改变不敏感，缺少对信任函数可以解释为下概率这一特性的验证，点值信任函数与模糊证据体一般不等价等等。因此，本章定义了连续区间上具有模糊值的信任函数。对于一大类模糊证据体，从理论上证明了其与所定义的模糊值信任函数的等价性。并证明了这样定义的模糊值信任函数具有相应的概率语义解释。
第五章提出了用于模糊粗糙集分析的从模糊属性数据构造模糊T相似性关系的方法。作为粗糙集的模糊推广，模糊粗糙集分析是从连续或模糊数据中发现知识和提取规则的有效方式。模糊关系同时对模糊性和粗糙性实现了表示，是模糊粗糙集分析的前提和基础。一般要求模糊关系满足一定的条件，即是一个模糊T相似性关系。本章给出了从模糊属性刻画数据生成模糊T相似性关系的方法，为后续的模糊粗糙分析处理打下了基础。
第六章给出了本文的总结和对未来工作的展望。
Uncertainty is pervasive in practical information systems, and several differentkinds of uncertainty often exist simultaneously in practical situations. It is a critical andfundamental problem to effectively represent hybrid uncertainty in informationrepresentation and processing. Therefore, the representation and processing of threecommonly encountered classes of hybrid uncertainty consisting of the combinations ofthe elementary uncertainty, randomness, fuzziness and roughness, are studied in thispaper. Their applications in fields such as nonparametric regression analysis, patternclassification and evidential reasoning are also investigated.
The paper is organized as follows.
In the first chapter, the research background is introduced in detail at first. Then thecurrent research status of hybrid uncertainty representation and author’s primary workand main contributions are also presented subsequently.
The second chapter reviews several fundamental theories for uncertaintyrepresentation, including probability theory, evidence theory, fuzzy set theory and roughset theory.
In the third chapter, the rule-based fuzzy system and the form of probabilistic fuzzyrule base are firstly surveyed. Then we present two methods for generating probabilisticfuzzy rule base from data set, based on which two classes of probabilistic fuzzy systemare realized. A Mamdani type probabilistic fuzzy rule base is derived for the case thatthe consequent variable is quantitatively continuous, while a probabilistic fuzzyclassification rule base is induced for the case that the consequent variable isqualitatively discrete. These two classes of probabilistic fuzzy system are applied tononparametric regression analysis, time series prediction and pattern classificationrespectively, with which better application results are obtained as compared with that ofsimilar approaches.
A fuzzy evidence theory with fuzzy-valued belief function in continuous interval isproposed in the fourth chapter. Considering that the fuzzy descriptions are pervasive andthe infinite universes of discourse are commonly encountered in practical problems,some fuzzy generalizations of evidence theory in continuous interval have beenaddressed. More or less some deficiencies occur in these generalizations, such as theinsensitivity of belief functions to the significant changes in focal elements, there is no reasonable interpretation for belief functions as lower probabilities, and resultingpoint-valued belief functions usually are not equivalent to the fuzzy bodies of evidence.Therefore, fuzzy-valued belief function in continuous interval is defined in this paper.The equivalence of such defined belief function with the fuzzy body of evidence isproved for a large class of fuzzy bodies of evidence. Moreover, the induced belieffunction and plausibility function possess corresponding probabilistic interpretations.
The fifth chapter presents a method for constructing fuzzy T-similarity relationfrom fuzzy attribute data for fuzzy rough set analysis. As a fuzzy extension of rough settheory, fuzzy rough set analysis is an effective tool to discover knowledge and extractrules from data set with continuous or fuzzy attribute values. The fuzziness androughness are represented simultaneously by using fuzzy relation which is theprecondition and basis of fuzzy rough set analysis. It is generally required the fuzzyrelation to be a fuzzy T-similarity relation. The presented fuzzy T-similarity relationgeneration method lays foundation for subsequent fuzzy rough analysis.
Finally, some conclusions have been drawn and the directions for future work havealso been listed in the last chapter.
第一章详细介绍了本文的研究背景，回顾了混合不确定性表示的目前国内外研究现状，并介绍了作者的主要工作。
第二章介绍了已有的几种不确定性表示理论，包括概率论、证据理论、模糊集理论和粗糙集理论。
第三章首先介绍了基于模糊规则的模糊系统建模方法，给出了概率模糊规则基的形式。然后提出了从数据生成概率模糊规则基的方法，实现概率模糊系统建模。当后件变量（输出变量）为定量连续变量时，提出了一种从数据生成Mamdani型概率模糊规则方法；当后件变量为定性离散变量时，提出了一种从数据生成概率模糊分类规则方法。将两类概率模糊系统分别应用到非参数回归分析与时间序列预测以及模式分类中，取得了不错的结果。
第四章提出了一种连续区间上具有模糊值信任函数的模糊证据理论。鉴于模糊刻画的普遍性以及连续区间论域的常见性，连续区间上证据理论的一些模糊扩展被提出。在这些扩展中，或多或少存在一些问题，比如信任函数对焦元的重大改变不敏感，缺少对信任函数可以解释为下概率这一特性的验证，点值信任函数与模糊证据体一般不等价等等。因此，本章定义了连续区间上具有模糊值的信任函数。对于一大类模糊证据体，从理论上证明了其与所定义的模糊值信任函数的等价性。并证明了这样定义的模糊值信任函数具有相应的概率语义解释。
第五章提出了用于模糊粗糙集分析的从模糊属性数据构造模糊T相似性关系的方法。作为粗糙集的模糊推广，模糊粗糙集分析是从连续或模糊数据中发现知识和提取规则的有效方式。模糊关系同时对模糊性和粗糙性实现了表示，是模糊粗糙集分析的前提和基础。一般要求模糊关系满足一定的条件，即是一个模糊T相似性关系。本章给出了从模糊属性刻画数据生成模糊T相似性关系的方法，为后续的模糊粗糙分析处理打下了基础。
第六章给出了本文的总结和对未来工作的展望。
Uncertainty is pervasive in practical information systems, and several differentkinds of uncertainty often exist simultaneously in practical situations. It is a critical andfundamental problem to effectively represent hybrid uncertainty in informationrepresentation and processing. Therefore, the representation and processing of threecommonly encountered classes of hybrid uncertainty consisting of the combinations ofthe elementary uncertainty, randomness, fuzziness and roughness, are studied in thispaper. Their applications in fields such as nonparametric regression analysis, patternclassification and evidential reasoning are also investigated.
The paper is organized as follows.
In the first chapter, the research background is introduced in detail at first. Then thecurrent research status of hybrid uncertainty representation and author’s primary workand main contributions are also presented subsequently.
The second chapter reviews several fundamental theories for uncertaintyrepresentation, including probability theory, evidence theory, fuzzy set theory and roughset theory.
In the third chapter, the rule-based fuzzy system and the form of probabilistic fuzzyrule base are firstly surveyed. Then we present two methods for generating probabilisticfuzzy rule base from data set, based on which two classes of probabilistic fuzzy systemare realized. A Mamdani type probabilistic fuzzy rule base is derived for the case thatthe consequent variable is quantitatively continuous, while a probabilistic fuzzyclassification rule base is induced for the case that the consequent variable isqualitatively discrete. These two classes of probabilistic fuzzy system are applied tononparametric regression analysis, time series prediction and pattern classificationrespectively, with which better application results are obtained as compared with that ofsimilar approaches.
A fuzzy evidence theory with fuzzy-valued belief function in continuous interval isproposed in the fourth chapter. Considering that the fuzzy descriptions are pervasive andthe infinite universes of discourse are commonly encountered in practical problems,some fuzzy generalizations of evidence theory in continuous interval have beenaddressed. More or less some deficiencies occur in these generalizations, such as theinsensitivity of belief functions to the significant changes in focal elements, there is no reasonable interpretation for belief functions as lower probabilities, and resultingpoint-valued belief functions usually are not equivalent to the fuzzy bodies of evidence.Therefore, fuzzy-valued belief function in continuous interval is defined in this paper.The equivalence of such defined belief function with the fuzzy body of evidence isproved for a large class of fuzzy bodies of evidence. Moreover, the induced belieffunction and plausibility function possess corresponding probabilistic interpretations.
The fifth chapter presents a method for constructing fuzzy T-similarity relationfrom fuzzy attribute data for fuzzy rough set analysis. As a fuzzy extension of rough settheory, fuzzy rough set analysis is an effective tool to discover knowledge and extractrules from data set with continuous or fuzzy attribute values. The fuzziness androughness are represented simultaneously by using fuzzy relation which is theprecondition and basis of fuzzy rough set analysis. It is generally required the fuzzyrelation to be a fuzzy T-similarity relation. The presented fuzzy T-similarity relationgeneration method lays foundation for subsequent fuzzy rough analysis.
Finally, some conclusions have been drawn and the directions for future work havealso been listed in the last chapter.
引文
[1] Zadeh L A. Fuzzy Sets [J]. Information and Control,1965,(8):338~353.
[2] Zadeh L A. Probability Measures of Fuzzy Events [J]. Journal of MathematicalAnalysis and Applications,1968,(23):421~427.
[3] Chen D G., Yang W X, Li F C. Measures of General Fuzzy Rough Sets on AProbabilistic Space [J]. Information Sciences,2008,(178):3177~3187.
[4] Wu W Z, Leung Y, Mi J S. On Generalized Fuzzy Belief Functions in InfiniteSpaces [J]. IEEE Transactions on Fuzzy Systems,2009,(17):385~397.
[5] Yager R R. Decision Making with Fuzzy Probability Assessments [J]. IEEETransactions on Fuzzy Systems,1999,(7):462~467.
[6] Yager R R, Kreinovich V. Decision Making under Interval Probabilities [J].International Journal of Approximate Reasoning,1999,(22):195~215.
[7] Liu Z, Li H X. A Probabilistic Fuzzy Logic System for Modeling and Control [J].IEEE Transactions on Fuzzy Systems,2005,(13):848~859.
[8] Wang L X, Mendel J M. Generating Fuzzy Rules by Learning from Examples [J].IEEE Transactions on System, Man and Cybernetics,1992,(22):1414~1427.
[9] Meghdadi A H, Akbarzadeh-T M R. Probabilistic Fuzzy Logic and ProbabilisticFuzzy Systems [C]//Proceeding of the10th IEEE International Conference onFuzzy Systems,2001:1127~1130.
[10] van den Berg J, Kaymak U, van den Bergh W M. Financial Markets Analysis byUsing a Probabilistic Fuzzy Modelling Approach [J]. International Journal ofApproximate Reasoning,2004,(35):291~305.
[11] Hong Sungjun, Lee Heesung, Kim Euntai. A New Probabilistic Fuzzy Model:Fuzzification–Maximization (FM) Approach [J]. International Journal ofApproximate Reasoning,2009,(50):1129~1147.
[12] Almeida R J, Kaymak U. Probabilistic Fuzzy Systems in Value-at-Risk Estimation[J]. Intelligent Systems in Accounting, Finance and Management,2009,(16):49~70.
[13] Yager R R, Filev D P. Including Probabilistic Uncertainty in Fuzzy LogicController Modeling Using Dempster-Shafer Theory [J]. IEEE Transactions onSystem, Man and Cybernetics,1995,(25):1221~1230.
[14] Kaymak U, van den Bergh W M, van den Berg J. A Fuzzy Additive ReasoningScheme for Probabilistic Mamdani Fuzzy Systems [C]//Proceeding of the12thIEEE International Conference on Fuzzy Systems,2003:331~336.
[15] Kaymak U, van den Berg J. On constructing probabilistic fuzzy classifiers fromweighted fuzzy clustering[C]//Proceedings of2004IEEE InternationalConference on Fuzzy Systems,2004:395~400.
[16] van den Berg J, Kaymak U, van den Bergh W M. Fuzzy classification usingprobability-based rule weighting[C]//Proceedings of2002IEEE InternationalConference on Fuzzy Systems,2002:991~996.
[17] Lee H-E, Park K-H, Bien Z Z. Iterative fuzzy clustering algorithm withsupervision to construct probabilistic fuzzy rule base from numerical data [J].IEEE Transactions on Fuzzy Systems,2008,(16):263~277.
[18] Cordón O, del Jesus M J, Herrera F. A Proposal on Reasoning Methods in FuzzyRule-Based Classification Systems [J]. International Journal of ApproximateReasoning,1999,(20):21~45.
[19] Liu Z, Li H X. A Probabilistic Fuzzy Logic System for Uncertainty Modeling
[C]//Proceedings of2005IEEE International Conference on System, Man andCybernetics,2005:3853~3858.
[20] Li H X, Liu Z. A Probabilistic Neural-Fuzzy Learning System for StochasticModeling [J]. IEEE Transactions on Fuzzy Systems,2008,(16):898~908.
[21] Liu Z, Li H X. Probabilistic Fuzzy Logic System: a tool to process stochastic andimprecise information[C]//Proceedings of2009IEEE International Conferenceon Fuzzy Systems,2009:20~24.
[22] Dempster A P. Upper and Lower Probabilities Induced by a Multivalued Mapping[J]. The Annals of Mathematical Statistics,1967,(38):325~339.
[23] Shafer G. A Mathematical Theory of Evidence [M]. Princeton, NJ: PrincetonUniversity Press,1976.
[24] Smets P, Kennes R. The Transferable Belief Model [J]. Artificial Intelligence,1994,(66):191~234.
[25] Denoeux T. Constructing belief functions from sample data using multinomialconfidence regions [J]. International Journal of Approximate Reasoning,2006,(42):228~252.
[26] Ngwenyama O K, Bryson N. Generating belief functions from qualitativepreferences: An approach to eliciting expert judgments and deriving probabilityfunctions [J]. Data&Knowledge Engineering,1998,(28):145~159.
[27] Wong S, Lingras P. Representative of Qualitative User Preference by QualitativeBelief Functions [J]. IEEE Transactions on Knowledge and Data Engineering,1994,(6):72~78.
[28] Bryson N, Mobolurin A. A process for generating quantitative belief functions [J].European Journal of Operational Research,1999,(115):624~633.
[29] Bryson N, Mobolurin A. A Qualitative Discriminant Approach for GeneratingQuantitative Belief Functions [J]. IEEE Transactions on Knowledge and DataEngineering,1998,(10):345~348.
[30] Yager R R. On the Dempster–Shafer framework and new combination rules [J].Information Sciences,1987,(41):93~138.
[31] Dubois D, Prade H. Representation and combination of uncertainty with belieffunctions and possibility measures [J]. Computational Intelligence,1998,(4):244~264.
[32] Smets P. The combination of evidence in the transferable belief model [J]. IEEETransactions on Pattern Analysis and Machine Intelligence,1990,(12):447~458.
[33] Murphy C K. Combining belief functions when evidence conflicts [J]. DecisionSupport Systems,2000,(29):1~9.
[34] Lefevre E, Colot O, Vannoorenberghe P. Belief function combination and conflictmanagement [J]. Information Fusion,2002,(3):149~162.
[35] Denoeux T. Conjunctive and disjunctive combination of belief functions inducedby nondistinct bodies of evidence [J]. Artificial Intelligence,2008,(172):234~264.
[36] Liu W R. Analyzing the degree of conflict among belief functions [J]. ArtificialIntelligence,2006,(170):909~924.
[37] Petit-Renaud S, Denoeux T. Nonparametric regression analysis of uncertain andimprecise data using belief functions [J]. International Journal of ApproximateReasoning,2004,(35):1~28.
[38] Denoeux T, Masson M H. EVCLUS: Evidential Clustering of Proximity Data [J].IEEE Transactions on System, Man and Cybernetics—Part B: Cybernetics,2004,(34),95~109.
[39] Denoeux T. A k-nearest neighbor classification rule based on Dempster-Shafertheory [J]. IEEE Transactions on System, Man and Cybernetics,1995,(25):804~903.
[40] Quost B, Denoeux T, Masson M H. Pairwise classifier combination using belieffunctions [J]. Pattern Recognition Letters,2007,(28):644~653.
[41] Zadeh L A. Fuzzy sets and information granularity [C]//. Gupta M M(ed.).Advances in Fuzzy Set Theory and Applications. Amsterdam, the Netherlands:North-Holland Press,1979:3~18.
[42] Ishizuka M, Fu K S, Yao J T P. Inference procedures and uncertainty forproblem-reduction method [J]. Information Sciences,1982,(28):179~206.
[43] Ogawa H, Fu K S. An inexact inference for damage assessment of existingstructures [J]. International Journal of Man-Machine Studies,1985,(22):295~306.
[44] Smets P. The degree of belief in a fuzzy event [J]. Information Sciences,1981,(25):1~19.
[45] Yager R R. Generalized probabilities of fuzzy events from fuzzy belief structures[J]. Information Sciences,1982,(28):45~62.
[46] Yen J. Generalizing the Dempster–Shafer theory to fuzzy sets [J]. IEEETransactions on System, Man, and Cybernetics,1990,(20):559~570.
[47] Lucas C, Araabi B N. Generalization of the Dempster-Shafer theory: afuzzy-valued measure [J]. IEEE Transactions on Fuzzy Systems,1999,(7):255~270.
[48] Biacino L. Fuzzy subsethood and belief functions of fuzzy events [J]. Fuzzy Setsand Systems,2007,(158):38~49.
[49] Denoeux T. Modeling Vague Beliefs using Fuzzy-Valued Belief Structures [J].Fuzzy Sets and Systems,2000,(116):167~199.
[50] Pawlak Z. Rough sets [J]. International Journal of Computational InformationScience,1982,(11):341~356.
[51] Chmielewski M R, Grzymala-Busse J W. Global Discretization of ContinuousAttributes as Preprocessing for Machine Learning [J]. International Journal ofApproximate Reasoning,1996,(15):319~331.
[52] Pawlak Z. Rough sets: Theoretical Aspects of Reasoning about Data [M]. Boston:Kluwer Academic Publishers,1991.
[53] Chan C C. A Rough Set Approach to Attribute Generalization in Data Mining [J].Information Sciences,1998,(107):169~176.
[54] Lingras P J, Yao Y Y. Data Mining using Extensions of the Rough Set Model [J].Journal of the American Society for Information Science,1998,(49):415~22.
[55] McSherry D. Knowledge Discovery by Inspection [J]. Decision Support Systems,1997,(21):43~47.
[56] Pawlak Z. Rough Set Approach to Knowledge–Based Decision Support [J].European Journal of Operational Research,1997,(99):48~57.
[57] Pomerol J C. Artificial Intelligence and Human Decision Making [J]. EuropeanJournal of Operational Research,1997,(99):3~25.
[58] Dubois D, Prade H. Rough Fuzzy Sets and Fuzzy Rough Sets [J]. InternationalJournal of General Systems,1990,(17):191~209.
[59] Dubois D, Prade H. Putting Rough Sets and Fuzzy Sets Together [C]//SlowinskiR (ed.). Intelligent Decision Support: Handbook of Applications and Advances ofthe Rough Sets Theory. Dordrecht: Kluwer,1992:203~222.
[60] Morsi N N, Yakout M M. Axiomatics for Fuzzy Rough Sets [J]. Fuzzy Sets andSystems,1998,(100):327~342.
[61] Radzikowska A M, Kerre E E. A Comparative Study of Fuzzy Rough Sets [J].Fuzzy Sets and Systems,2002,(126):137~155.
[62] Mi J S, Zhang W X. An Axiomatic Characterization of A Fuzzy Generalization ofRough Sets [J]. Information Sciences,2004,(160):235~249.
[63] Yeung D S, Chen D G, Tsang E C C, Lee J W T, Wang X Z. On the Generalizationof Fuzzy Rough Sets [J]. IEEE Transactions on Fuzzy Systems,2005,(13):343~361.
[64] Wang Y F. Mining Stock Price using Fuzzy Rough Set System [J]. Expert Systemswith Applications,2003,(24):13~23.
[65] Jensen R, Shen Q. Fuzzy-Rough Data Reduction with Ant Colony Optimization[J]. Fuzzy Sets and Systems,2005,(149):5–20.
[66] Srinivasan P, Ruiz M E, Kraft D H, Chen J H. Vocabulary Mining for InformationRetrieval: Rough Sets and Fuzzy Sets [J]. Information Processing andManagement,2001,(37):15~38.
[67] Pal S K, Mitra P. Case Generation using Rough Sets with Fuzzy Representation[J]. IEEE Transactions on Knowledge Data Engineering,2004,(16):293~300.
[68] Pawlak Z, Wong S K M, Ziarko W. Rough Sets: Probabilistic VersusDeterministic Approach [J]. International Journal of Man-Machine Studies,1988,(29):81~95.
[69] Yao Y Y. A Decision Theoretic Framework for Approximating Concepts [J].International Journal of Man-Machine Studies,1992,(37):793~809.
[70]张文修,吴伟志.基于随机集的粗糙集模型(Ⅰ)[J].西安交通大学学报,2000,（34）:75~79.
[71] Yao Y Y. Probabilistic Approaches to Rough Sets [J]. Expert Systems,2003,(20):287~297.
[72] Yao Y Y, Wong S.K.M. A Decision Theoretic Framework for ApproximatingConcepts [J]. International Journal of Man–machine Studies,1992,(37):793~809.
[73] Ziarko W. Variable Precision Rough Set Model [J]. Journal of Computer andSystem Science,1993,(46):39~59.
[74] Slezak D, Ziarko W. The Investigation of the Bayesian Rough Set Model [J]International Journal of Approximate Reasoning,2005,(40):81~91.
[75] Yao Y Y. Probabilistic Rough Set Approximations [J]. International Journal ofApproximate Reasoning,2008,(49):255~271.
[76] Feller W. An Introduction to Probability Theory and Its Applications [M]. NewYork: Wiley, vol.2,2nd Ed,1971.
[77] Smets P. The Normative Representation of Quantified Beliefs by Belief Functions[J]. Artificial Intelligence,1997(92):229~242.
[78] Smets P. The Transferable Belief Model for Quantified Belief Representation [C]//Gabbay D M, Smets P (eds.). Handbook of Defeasible Reasoning and UncertaintyManagement Systems. Doordrecht, the Netherlands: Kluwer,1998, vol.1:267~301.
[79] Zadeh L A. Fuzzy Sets as A Basis for A Theory of Possibility [J]. Fuzzy Sets andSystem,1978,(1):3~28.
[80] Dubois D, Prade H. Possibility Theory [M]. New York: Plenum Press,1988.
[81] Zhu W. Generalized Rough Sets Based on Relations [J]. Information Sciences,2007,(177):4997~5011.
[82] Zhu W, Wang F Y. Reduction and Axiomization of Covering Generalized RoughSets [J]. Information Sciences,2003,(152):217~230.
[83] Zhu W, Wang F Y. On Three Types of Covering Rough Sets [J]. IEEETransactions on Knowledge and Data Engineering,2007,(19):1131~1144.
[84] Yao Y Y, L ingras P. Interpretations of Belief Functions in The Theory of RoughSets [J]. Information Sciences,1998,(104):81~106.
[85] Nanda S, Majumdar S. Fuzzy Rough Sets [J]. Fuzzy Sets and Systems,1992,(45):157~160.
[86] Sugeno M, Tanaka K. Successive Identification of A Fuzzy Model and ItsApplication to Prediction of Complex Systems [J]. Fuzzy Sets and Systems,1994,(42):315~324.
[87] Tang M, Chen X, Hu W D, Yu W X. Generation of a Probabilistic Fuzzy RuleBase by Learning from Examples [J]. Information Sciences,2012,(217):21~30.
[88] Abe S, Lan M S. Fuzzy Rules Extraction Directly From Numerical Data forFunction Approximation [J]. IEEE Transactions on System, Man, and Cybernetics,1995,(25):119~129.
[89] Alcalá R, Alcalá-Fdez J, Herrera F, Otero J. Genetic Learning of Accurate andCompact Fuzzy Rule Based Systems Based on The2-tuples LinguisticRepresentation [J]. International Journal of Approximate Reasoning,2007,(44):45~64.
[90] Berenji H R, Khedkar P. Learning and Tuning Fuzzy Logic Controllers throughReinforcements [J]. IEEE Transactions on Neural Networks,1992,(3):724~740.
[91] Choi J N, Oh S K, Pedrycz W. Structural and Parametric Design of FuzzyInference Systems using Hierarchical Fair Competition-Based Parallel GeneticAlgorithms and Information Granulation [J]. International Journal of ApproximateReasoning,2008,(49):631~648.
[92] delaOssa L, Gámez J A, Puerta J M. Learning Weighted Linguistic Fuzzy Rulesby using Specifically-Tailored Hybrid Estimation of Distribution Algorithms [J].International Journal of Approximate Reasoning,2009,(50):541~560.
[93] Gobi A F, Pedrycz W. Fuzzy Modelling through Logic Optimization [J].International Journal of Approximate Reasoning,2007,(45):488~510.
[94] González J, Rojas I, Pomares H, Herrera L J, Guillén A, Palomares J M, Rojas F.Improving The Accuracy while Preserving The Interpretability of Fuzzy FunctionApproximators by Means of Multi-Objective Evolutionary Algorithms [J].International Journal of Approximate Reasoning,2007,(44):32~44.
[95] Kim E, Park M, Ji S, Park M. A New Approach to Fuzzy Modeling [J]. IEEETransactions on Fuzzy Systems,1997,(5):328~337.
[96] Kim E, Pedrycz W. Information Granulation as a Basis of Fuzzy Modeling [J].Journal of Intelligent&Fuzzy Systems,2007,(18):123~148.
[97] Kukolj D. Design of Adaptive Takagi–Sugeno–Kang Fuzzy Models [J].Applied Soft Computing,2002,(2):89~103.
[98] Kukolj D, Levi E. Identification of Complex Systems Based on Neural and Takagi–Sugeno Fuzzy Model [J]. IEEE Transactions on Systems, Man, and Cybernetics,Part B,2004,(34):272~282.
[99] Sugeno M, Kang G T. Structure Identification of a Fuzzy Model [J]. Fuzzy Setsand Systems,1988,(28):15~33.
[100] Wang L X. Training of Fuzzy Logic Systems using Nearest NeighborhoodClustering [C]//Proceeding of1993IEEE International Conference on FuzzySystems. San Francisco, CA:1993:13~17.
[101] Mendel J. Uncertain Rule-Based Fuzzy Logic Systems: Introduction and NewDirections [M]. Prentice Hall,2001.
[102] Wang L X, Mendel J M. Fuzzy Basis Functions, Universal Approximation, andOrthogonal Least Squares Learning [J]. IEEE Transactions on Neural Networks,1992,(3):807~814.
[103] Wang L X, Wei C. Approximation Accuracy of Some Neuro-Fuzzy Approaches[J]. IEEE Transactions on Fuzzy Systems,2000,(8):470~478.
[104] Zeng X J, Singh M G. Approximation Accuracy Analysis of Fuzzy Systems asFunction Approximators [J]. IEEE Transactions on Fuzzy Systems,1996,(4):44~63.
[105] Coppi R, D’Urso P, Giordani P, Santoro A. Least Squares Estimation of A LinearRegression Model with LR Fuzzy Response [J]. Computational Statistics&DataAnalysis,2006,(51):267~286.
[106] Friedman J. Multivariate Adaptive Regression Splines (with discussion)[J].Annals of Statistics,1991,(19):1~141.
[107] Hardle W. Applied Nonparametric Regression [M]. Cambridge: CambridgeUniversity Press,1990.
[108] Miller G A. The Magical Number Seven, Plus or Minus Two: Some Limits on OurCapacity for Processing Information [J]. Psychology Review,1956,(63):81~97.
[109] García S, Fernández A, Luengo J, Herrera F. Advanced Nonparametric Tests forMultiple Comparisons in The Design of Experiments in ComputationalIntelligence and Data Mining: Experimental Analysis of Power [J]. InformationSciences,2010,(180):2044-2064.
[110]唐敏,杨宏文,胡卫东,郁文贤. Mamdani型概率模糊系统建模方法[J].系统工程与电子技术,2012(2):323~327.
[111] Song H-J, Miao C-Y, Shen Z-Q, Roel W, D‘Hondt M, Francky C. A ProbabilisticFuzzy Approach to Modeling Nonlinear Systems [J]. Neurocomputing,2011,(74):1008-1025.
[112] Tang M, Chen X, Hu W D, Yu W X. Genetic Algorithm-based Design of FuzzyClassification Systems from Labeled Data [J]. Pattern Recognition Letter,Submitted.
[113] Ishibuchi H, Nozaki K, Tanaka H. Distributed Representation of Fuzzy Rules andIts Application to Pattern Classification [J]. Fuzzy Sets and Systems,1992,(52):21~32.
[114] Mansoori E G, Zolghadri M J, Katebi S D. A Weighting Function for ImprovingFuzzy Classification Systems Performance [J]. Fuzzy Sets and Systems,2007,(158):583~591.
[115] Abe S, Thawonmas R. A Fuzzy Classifier with Ellipsoidal Regions [J]. IEEETransactions on Fuzzy Systems,1997,(5):358~368.
[116] Lee C Y, Lin C J, Chen H J. A Self-Constructing Fuzzy CMAC Model and ItsApplications [J]. Information Sciences,2007,(177):264~280.
[117] Roubos J A, Setnes M, Abonyi J. Learning Fuzzy Classification Rules fromLabeled Data [J]. Information Sciences,2003,(150):77~93.
[118] Chakraborty D, Pal N R. A Neuro-Fuzzy Scheme for Simultaneous FeatureSelection and Fuzzy Rule-Based Classification [J]. IEEE Transactions on NeuralNetworks,2004,(15):110~123.
[119] Lin C T, Lee C S G. Neural Network-Based Fuzzy Logic Control and DecisionSystem [J]. IEEE Transactions on Computers,1991,(40):1320~1336.
[120] Nauck D, Kruse R. A Neuro-Fuzzy Method to Learn Fuzzy Classification Rulesfrom Data [J]. Fuzzy Sets and Systems,1997,(89):277~288.
[121] Gómez-Skarmeta A F, Valdés M, Jiménez F, Marin-Blázquez J G. ApproximativeFuzzy Rules Approaches for Classification with Hybrid-GA Techniques [J].Information Sciences,2001,(136):193~214.
[122] Hu Y C. Finding Useful Fuzzy Concepts for Pattern Classification using GeneticAlgorithm [J]. Information Sciences,2005,(175):1~19.
[123] Ishibuchi H, Nakashima T, Murata T. Three-Objective Genetics-Based MachineLearning for Linguistic Rule Extraction [J]. Information Sciences,2001,(136):109~133.
[124] Sanchez L, Couso I, Corrales J A. Combining GP Operators with SA Search toEvolve Fuzzy Rule Based Classifiers [J]. Information Sciences,2001,(136):175~191.
[125] Zhou E, Khotanzad A. Fuzzy Classifier Design using Genetic Algorithms [J].Pattern Recognition,2007,(40):3401~3414.
[126] Li M Q, Wang Z C. A Hybrid Coevolutionary Algorithm for Designing FuzzyClassifiers [J]. Information Sciences,2009,(179):1970~1983.
[127] Berlanga F J, Rivera A J, del Jesus M J, Herrera F. GP-COACH: GeneticProgramming-Based Learning of COmpact and ACcurate Fuzzy Rule-BasedClassification Systems for High-Dimensional Problems [J]. Information Sciences,2010,(180):1183~1200.
[128] Sanz J A, Fernández A, Bustince H, Herrera F. Improving the Performance ofFuzzy Rule-Based Classification Systems with Interval-Valued Fuzzy Sets andGenetic Amplitude Tuning [J]. Information Sciences,2010,(180):3674~3685.
[129] Fernández A, del Jesus M J, Herrera F. Hierarchical Fuzzy Rule BasedClassification Systems with Genetic Rule Selection for Imbalanced Data-Sets [J].International Journal of Approximate Reasoning,2009,(50):561~577.
[130] Gonzblez A, Perez R. SLAVE: A Genetic Learning System Based on An IterativeApproach [J]. IEEE Transactions on Fuzzy Systems,1999,(7):176~191.
[131] Ishibuchi H, Yamamoto T, Nakashima T. Hybridization of Fuzzy GBMLApproaches for Pattern Classification Problems [J]. IEEE Transactions onSystems, Man, and Cybernetics, Part B,2005,(35):359~365.
[132] Ishibuchi H, Nojima Y. Analysis of Interpretability-Accuracy Tradeoff of FuzzySystems by Multiobjective Fuzzy Genetics-Based Machine Learning [J].International Journal of Approximate Reasoning,2007,(44):4~31.
[133] Mansoori E G, Zolghadri M J, Katebi S D. SGERD: A Steady-State GeneticAlgorithm for Extracting Fuzzy Classification Rules from Data [J]. IEEETransactions on Fuzzy Systems,2008,(16):1061~1071.
[134] del Jesus M J, Hoffmann F, Navascués L J, Sánchez L. Induction ofFuzzy-Rule-Based Classifiers With Evolutionary Boosting Algorithms [J]. IEEETransactions on Fuzzy Systems,2004,(12):296~308.
[135] Hoffmann F, Baesens B, Mues C, Gestel T V, Vanthienen J. Inferring DescriptiveAnd Approximate Fuzzy Rules for Credit Scoring using Evolutionary Algorithms[J]. European Journal of Operational Research,2007,(177):540~555.
[136] Ho S Y, Chen H M, Ho S J, Chen T K. Design of Accurate Classifiers With ACompact Fuzzy-Rule Base using An Evolutionary Scatter Partition of FeatureSpace [J]. IEEE Transactions on Systems, Man, and Cybernetics, Part B,2004,(34):1031~1044.
[137] Murata T, Ishibuchi H, Gen M. Adjusting Fuzzy Partitions by Genetic AlgorithmsAnd Histograms for Pattern Classification Problems [C]//EvolutionaryComputation Proceedings of The1998IEEE World Congress on ComputationalIntelligence,1998:9~14.
[138] Cano J, Herrera F, Lozano M. Evolutionary Stratified Training Set Selection forExtracting Classification Rules with Trade-Off Precision-Interpretability [J]. Dataand Knowledge Engineering,2007,(60):90~108.
[139] Tang M, Chen X, Hu W D, Yu W X. A Fuzzy Rule-based Classification Systemusing Interval Type-2Fuzzy sets [C]//.2011International Conference onIntergrated Uncertainty in Knowledge Modelling and Decision Making, October2011, Hangzhou, China. LNCS7027Springer-Verlag,72~80.
[140] Ishibuchi H, Nakashima T, Nii M. Classification and Modeling with LinguisticInformation Granules: Advanced Approaches to Linguistic Data Mining [M].Springer-Verlag,2004.
[141] Bastian A. How to Handle the Flexibility of Linguistic Variables withApplications [J]. International Journal of Uncertainty, Fuzziness andKnowledge-Based Systems,1994,(2):463~484.
[142] Liang Q, Mendel J. MPEG VBR Video Traffic Modeling and Classification UsingFuzzy Technique [J]. IEEE Transactions on Fuzzy Systems,2001,(9):183~193.
[143] Wu H, Mendel J. Classification of Battlefield Ground Vehicles Using AcousticFeatures and Fuzzy Logic Rule-Based Classifiers [J]. IEEE Transactions on FuzzySystems,2007,(15):56~72.
[144] Hwang C, Rhee F. Uncertain Fuzzy Clustering: Interval Type-2Fuzzy Approachto C-means [J]. IEEE Transactions on Fuzzy Systems,2007,(15):107~120.
[145] Starczewski J T. Efficient Triangular Type-2Fuzzy Logic Systems [J].International Journal of Approximate Reasoning,2009,(50):799~811.
[146] Tang M, Hu W D, Yu W X. Fuzzy-valued Belief Functions in Infinite Spaces [J].IEEE Transactions on Fuzzy Systems, Under Review.
[147] Yang M S, Chen T C, Wu K L. Generalized Belief Function, Plausibility Function,and Dempster’s Combinational Rule to Fuzzy Sets [J]. International Journal ofIntelligent Systems,2003,(18):925~937.
[148] Hwang C M, Yang M S. Generalization of Belief and Plausibility Functions toFuzzy Sets Based on the Sugeno Integral [J]. International Journal of IntelligentSystems,2007,(22):1215~1228.
[149] Cornelis C, Deschrijver G, Kerre E E. Implication in Intuitionistic Fuzzy AndInterval-Valued Fuzzy Set Theory: Construction, Classification, Application [J].International Journal of Approximate Reasoning,2004,(35):55~95.
[150] Ammar E, Metz J. On Fuzzy Convexity and Parametric Fuzzy Optimization [J].Fuzzy Sets Systems,1992,(49):135~141.
[151] Yen J. Computing Generalized Belief Functions for Continuous Fuzzy Sets [J].International Journal of Approximate Reasoning,1992,(6):1~31.
[152]唐敏,杨宏文,胡卫东,郁文贤.基于模糊属性刻画的T相似性关系的构造[J].模糊系统与数学,2011(5):136~143.
[153] Jagielska I, Matthews C, Whitfort T. An Investigation into the Application ofNeural Networks, Fuzzy logic, Genetic algorithms, and Rough Sets to AutomatedKnowledge Acquisition for Classification Problems [J]. Neurocomputing,1999,(24):37~54.
[154] Kryszkiewicz M. Rough Set Approach to Incomplete Information Systems [J].Information Sciences,1998,(112):39~49.
[155] Tsumoto S. Automated Extraction of Medical Expert System Rules From ClinicalDatabases Based on Rough Set Theory [J]. Information Sciences,1998,(112):67~84.
[156] Nakamura A. Fuzzy rough sets [J]. Note Multiple-Valued Logic Japan,1988,(9):1–8.
[157] Qin K, Pei Z. On The Topological Properties of Fuzzy Rough Sets [J]. Fuzzy Setsand Systems,2005,(151):601–613.
[158] Thiele H. Fuzzy Rough Sets Versus Rough Fuzzy Sets—An Interpretation and AComparative Study using Concepts of Modal Logic [R]. Univ. Dortmund,Dortmund, Germany, Tech. Rep. ISSN1433-3325,1998.
[159] Yao Y Y. Combination of Rough And Fuzzy Sets Based on Alpha-Level Sets [C]//Lin T Y, Cercone N(eds.). Rough Sets and Data Mining: Analysis for ImpreciseData. Norwell, MA: Kluwer,1997:301~321.
[160] Bodjanova S. Approximation of Fuzzy Concepts in Decision Making [J]. FuzzySets and Systems,1997,(85):23~29.
[161] Kuncheva L I. Fuzzy Rough Sets: Application to Feature Extraction [J]. FuzzySets and Systems,1992,(51):147~153.
[162] Fernández Salido J M, Murakami S. Rough Set Analysis of A General Type ofFuzzy Data using Transitive Aggregations of Fuzzy Similarity Relations [J].Fuzzy Sets and Systems,2003,(139):635~660.
[163] Sudkamp T. Similarity, Interpolation, and Fuzzy Rule Construction [J]. Fuzzy Setsand Systems,1993,(58):73~86.
[164] Fonck P, Fodor J, Roubens M. An Application of Aggregation Procedures to TheDefinition of Measures of Similarities Between Fuzzy Sets [J]. Fuzzy Sets andSystems,1998,(97):67~74.
[165] Ovchinnikov S. Aggregating Transitive Fuzzy Binary Relations [J]. InternationalJournal of Uncertainty, Fuzziness Knowledge-based Systems,1992,(3):47~55.
[166] Beliakov G. Definition of General Aggregation Operators through SimilarityRelations [J]. Fuzzy Sets and Systems,2000,(114):437~453.
[167] Fernández Salido J M, Murakami S. On β-precision aggregation [J]. Fuzzy Setsand Systems,2003,139:547-558.
[2] Zadeh L A. Probability Measures of Fuzzy Events [J]. Journal of MathematicalAnalysis and Applications,1968,(23):421~427.
[3] Chen D G., Yang W X, Li F C. Measures of General Fuzzy Rough Sets on AProbabilistic Space [J]. Information Sciences,2008,(178):3177~3187.
[4] Wu W Z, Leung Y, Mi J S. On Generalized Fuzzy Belief Functions in InfiniteSpaces [J]. IEEE Transactions on Fuzzy Systems,2009,(17):385~397.
[5] Yager R R. Decision Making with Fuzzy Probability Assessments [J]. IEEETransactions on Fuzzy Systems,1999,(7):462~467.
[6] Yager R R, Kreinovich V. Decision Making under Interval Probabilities [J].International Journal of Approximate Reasoning,1999,(22):195~215.
[7] Liu Z, Li H X. A Probabilistic Fuzzy Logic System for Modeling and Control [J].IEEE Transactions on Fuzzy Systems,2005,(13):848~859.
[8] Wang L X, Mendel J M. Generating Fuzzy Rules by Learning from Examples [J].IEEE Transactions on System, Man and Cybernetics,1992,(22):1414~1427.
[9] Meghdadi A H, Akbarzadeh-T M R. Probabilistic Fuzzy Logic and ProbabilisticFuzzy Systems [C]//Proceeding of the10th IEEE International Conference onFuzzy Systems,2001:1127~1130.
[10] van den Berg J, Kaymak U, van den Bergh W M. Financial Markets Analysis byUsing a Probabilistic Fuzzy Modelling Approach [J]. International Journal ofApproximate Reasoning,2004,(35):291~305.
[11] Hong Sungjun, Lee Heesung, Kim Euntai. A New Probabilistic Fuzzy Model:Fuzzification–Maximization (FM) Approach [J]. International Journal ofApproximate Reasoning,2009,(50):1129~1147.
[12] Almeida R J, Kaymak U. Probabilistic Fuzzy Systems in Value-at-Risk Estimation[J]. Intelligent Systems in Accounting, Finance and Management,2009,(16):49~70.
[13] Yager R R, Filev D P. Including Probabilistic Uncertainty in Fuzzy LogicController Modeling Using Dempster-Shafer Theory [J]. IEEE Transactions onSystem, Man and Cybernetics,1995,(25):1221~1230.
[14] Kaymak U, van den Bergh W M, van den Berg J. A Fuzzy Additive ReasoningScheme for Probabilistic Mamdani Fuzzy Systems [C]//Proceeding of the12thIEEE International Conference on Fuzzy Systems,2003:331~336.
[15] Kaymak U, van den Berg J. On constructing probabilistic fuzzy classifiers fromweighted fuzzy clustering[C]//Proceedings of2004IEEE InternationalConference on Fuzzy Systems,2004:395~400.
[16] van den Berg J, Kaymak U, van den Bergh W M. Fuzzy classification usingprobability-based rule weighting[C]//Proceedings of2002IEEE InternationalConference on Fuzzy Systems,2002:991~996.
[17] Lee H-E, Park K-H, Bien Z Z. Iterative fuzzy clustering algorithm withsupervision to construct probabilistic fuzzy rule base from numerical data [J].IEEE Transactions on Fuzzy Systems,2008,(16):263~277.
[18] Cordón O, del Jesus M J, Herrera F. A Proposal on Reasoning Methods in FuzzyRule-Based Classification Systems [J]. International Journal of ApproximateReasoning,1999,(20):21~45.
[19] Liu Z, Li H X. A Probabilistic Fuzzy Logic System for Uncertainty Modeling
[C]//Proceedings of2005IEEE International Conference on System, Man andCybernetics,2005:3853~3858.
[20] Li H X, Liu Z. A Probabilistic Neural-Fuzzy Learning System for StochasticModeling [J]. IEEE Transactions on Fuzzy Systems,2008,(16):898~908.
[21] Liu Z, Li H X. Probabilistic Fuzzy Logic System: a tool to process stochastic andimprecise information[C]//Proceedings of2009IEEE International Conferenceon Fuzzy Systems,2009:20~24.
[22] Dempster A P. Upper and Lower Probabilities Induced by a Multivalued Mapping[J]. The Annals of Mathematical Statistics,1967,(38):325~339.
[23] Shafer G. A Mathematical Theory of Evidence [M]. Princeton, NJ: PrincetonUniversity Press,1976.
[24] Smets P, Kennes R. The Transferable Belief Model [J]. Artificial Intelligence,1994,(66):191~234.
[25] Denoeux T. Constructing belief functions from sample data using multinomialconfidence regions [J]. International Journal of Approximate Reasoning,2006,(42):228~252.
[26] Ngwenyama O K, Bryson N. Generating belief functions from qualitativepreferences: An approach to eliciting expert judgments and deriving probabilityfunctions [J]. Data&Knowledge Engineering,1998,(28):145~159.
[27] Wong S, Lingras P. Representative of Qualitative User Preference by QualitativeBelief Functions [J]. IEEE Transactions on Knowledge and Data Engineering,1994,(6):72~78.
[28] Bryson N, Mobolurin A. A process for generating quantitative belief functions [J].European Journal of Operational Research,1999,(115):624~633.
[29] Bryson N, Mobolurin A. A Qualitative Discriminant Approach for GeneratingQuantitative Belief Functions [J]. IEEE Transactions on Knowledge and DataEngineering,1998,(10):345~348.
[30] Yager R R. On the Dempster–Shafer framework and new combination rules [J].Information Sciences,1987,(41):93~138.
[31] Dubois D, Prade H. Representation and combination of uncertainty with belieffunctions and possibility measures [J]. Computational Intelligence,1998,(4):244~264.
[32] Smets P. The combination of evidence in the transferable belief model [J]. IEEETransactions on Pattern Analysis and Machine Intelligence,1990,(12):447~458.
[33] Murphy C K. Combining belief functions when evidence conflicts [J]. DecisionSupport Systems,2000,(29):1~9.
[34] Lefevre E, Colot O, Vannoorenberghe P. Belief function combination and conflictmanagement [J]. Information Fusion,2002,(3):149~162.
[35] Denoeux T. Conjunctive and disjunctive combination of belief functions inducedby nondistinct bodies of evidence [J]. Artificial Intelligence,2008,(172):234~264.
[36] Liu W R. Analyzing the degree of conflict among belief functions [J]. ArtificialIntelligence,2006,(170):909~924.
[37] Petit-Renaud S, Denoeux T. Nonparametric regression analysis of uncertain andimprecise data using belief functions [J]. International Journal of ApproximateReasoning,2004,(35):1~28.
[38] Denoeux T, Masson M H. EVCLUS: Evidential Clustering of Proximity Data [J].IEEE Transactions on System, Man and Cybernetics—Part B: Cybernetics,2004,(34),95~109.
[39] Denoeux T. A k-nearest neighbor classification rule based on Dempster-Shafertheory [J]. IEEE Transactions on System, Man and Cybernetics,1995,(25):804~903.
[40] Quost B, Denoeux T, Masson M H. Pairwise classifier combination using belieffunctions [J]. Pattern Recognition Letters,2007,(28):644~653.
[41] Zadeh L A. Fuzzy sets and information granularity [C]//. Gupta M M(ed.).Advances in Fuzzy Set Theory and Applications. Amsterdam, the Netherlands:North-Holland Press,1979:3~18.
[42] Ishizuka M, Fu K S, Yao J T P. Inference procedures and uncertainty forproblem-reduction method [J]. Information Sciences,1982,(28):179~206.
[43] Ogawa H, Fu K S. An inexact inference for damage assessment of existingstructures [J]. International Journal of Man-Machine Studies,1985,(22):295~306.
[44] Smets P. The degree of belief in a fuzzy event [J]. Information Sciences,1981,(25):1~19.
[45] Yager R R. Generalized probabilities of fuzzy events from fuzzy belief structures[J]. Information Sciences,1982,(28):45~62.
[46] Yen J. Generalizing the Dempster–Shafer theory to fuzzy sets [J]. IEEETransactions on System, Man, and Cybernetics,1990,(20):559~570.
[47] Lucas C, Araabi B N. Generalization of the Dempster-Shafer theory: afuzzy-valued measure [J]. IEEE Transactions on Fuzzy Systems,1999,(7):255~270.
[48] Biacino L. Fuzzy subsethood and belief functions of fuzzy events [J]. Fuzzy Setsand Systems,2007,(158):38~49.
[49] Denoeux T. Modeling Vague Beliefs using Fuzzy-Valued Belief Structures [J].Fuzzy Sets and Systems,2000,(116):167~199.
[50] Pawlak Z. Rough sets [J]. International Journal of Computational InformationScience,1982,(11):341~356.
[51] Chmielewski M R, Grzymala-Busse J W. Global Discretization of ContinuousAttributes as Preprocessing for Machine Learning [J]. International Journal ofApproximate Reasoning,1996,(15):319~331.
[52] Pawlak Z. Rough sets: Theoretical Aspects of Reasoning about Data [M]. Boston:Kluwer Academic Publishers,1991.
[53] Chan C C. A Rough Set Approach to Attribute Generalization in Data Mining [J].Information Sciences,1998,(107):169~176.
[54] Lingras P J, Yao Y Y. Data Mining using Extensions of the Rough Set Model [J].Journal of the American Society for Information Science,1998,(49):415~22.
[55] McSherry D. Knowledge Discovery by Inspection [J]. Decision Support Systems,1997,(21):43~47.
[56] Pawlak Z. Rough Set Approach to Knowledge–Based Decision Support [J].European Journal of Operational Research,1997,(99):48~57.
[57] Pomerol J C. Artificial Intelligence and Human Decision Making [J]. EuropeanJournal of Operational Research,1997,(99):3~25.
[58] Dubois D, Prade H. Rough Fuzzy Sets and Fuzzy Rough Sets [J]. InternationalJournal of General Systems,1990,(17):191~209.
[59] Dubois D, Prade H. Putting Rough Sets and Fuzzy Sets Together [C]//SlowinskiR (ed.). Intelligent Decision Support: Handbook of Applications and Advances ofthe Rough Sets Theory. Dordrecht: Kluwer,1992:203~222.
[60] Morsi N N, Yakout M M. Axiomatics for Fuzzy Rough Sets [J]. Fuzzy Sets andSystems,1998,(100):327~342.
[61] Radzikowska A M, Kerre E E. A Comparative Study of Fuzzy Rough Sets [J].Fuzzy Sets and Systems,2002,(126):137~155.
[62] Mi J S, Zhang W X. An Axiomatic Characterization of A Fuzzy Generalization ofRough Sets [J]. Information Sciences,2004,(160):235~249.
[63] Yeung D S, Chen D G, Tsang E C C, Lee J W T, Wang X Z. On the Generalizationof Fuzzy Rough Sets [J]. IEEE Transactions on Fuzzy Systems,2005,(13):343~361.
[64] Wang Y F. Mining Stock Price using Fuzzy Rough Set System [J]. Expert Systemswith Applications,2003,(24):13~23.
[65] Jensen R, Shen Q. Fuzzy-Rough Data Reduction with Ant Colony Optimization[J]. Fuzzy Sets and Systems,2005,(149):5–20.
[66] Srinivasan P, Ruiz M E, Kraft D H, Chen J H. Vocabulary Mining for InformationRetrieval: Rough Sets and Fuzzy Sets [J]. Information Processing andManagement,2001,(37):15~38.
[67] Pal S K, Mitra P. Case Generation using Rough Sets with Fuzzy Representation[J]. IEEE Transactions on Knowledge Data Engineering,2004,(16):293~300.
[68] Pawlak Z, Wong S K M, Ziarko W. Rough Sets: Probabilistic VersusDeterministic Approach [J]. International Journal of Man-Machine Studies,1988,(29):81~95.
[69] Yao Y Y. A Decision Theoretic Framework for Approximating Concepts [J].International Journal of Man-Machine Studies,1992,(37):793~809.
[70]张文修,吴伟志.基于随机集的粗糙集模型(Ⅰ)[J].西安交通大学学报,2000,（34）:75~79.
[71] Yao Y Y. Probabilistic Approaches to Rough Sets [J]. Expert Systems,2003,(20):287~297.
[72] Yao Y Y, Wong S.K.M. A Decision Theoretic Framework for ApproximatingConcepts [J]. International Journal of Man–machine Studies,1992,(37):793~809.
[73] Ziarko W. Variable Precision Rough Set Model [J]. Journal of Computer andSystem Science,1993,(46):39~59.
[74] Slezak D, Ziarko W. The Investigation of the Bayesian Rough Set Model [J]International Journal of Approximate Reasoning,2005,(40):81~91.
[75] Yao Y Y. Probabilistic Rough Set Approximations [J]. International Journal ofApproximate Reasoning,2008,(49):255~271.
[76] Feller W. An Introduction to Probability Theory and Its Applications [M]. NewYork: Wiley, vol.2,2nd Ed,1971.
[77] Smets P. The Normative Representation of Quantified Beliefs by Belief Functions[J]. Artificial Intelligence,1997(92):229~242.
[78] Smets P. The Transferable Belief Model for Quantified Belief Representation [C]//Gabbay D M, Smets P (eds.). Handbook of Defeasible Reasoning and UncertaintyManagement Systems. Doordrecht, the Netherlands: Kluwer,1998, vol.1:267~301.
[79] Zadeh L A. Fuzzy Sets as A Basis for A Theory of Possibility [J]. Fuzzy Sets andSystem,1978,(1):3~28.
[80] Dubois D, Prade H. Possibility Theory [M]. New York: Plenum Press,1988.
[81] Zhu W. Generalized Rough Sets Based on Relations [J]. Information Sciences,2007,(177):4997~5011.
[82] Zhu W, Wang F Y. Reduction and Axiomization of Covering Generalized RoughSets [J]. Information Sciences,2003,(152):217~230.
[83] Zhu W, Wang F Y. On Three Types of Covering Rough Sets [J]. IEEETransactions on Knowledge and Data Engineering,2007,(19):1131~1144.
[84] Yao Y Y, L ingras P. Interpretations of Belief Functions in The Theory of RoughSets [J]. Information Sciences,1998,(104):81~106.
[85] Nanda S, Majumdar S. Fuzzy Rough Sets [J]. Fuzzy Sets and Systems,1992,(45):157~160.
[86] Sugeno M, Tanaka K. Successive Identification of A Fuzzy Model and ItsApplication to Prediction of Complex Systems [J]. Fuzzy Sets and Systems,1994,(42):315~324.
[87] Tang M, Chen X, Hu W D, Yu W X. Generation of a Probabilistic Fuzzy RuleBase by Learning from Examples [J]. Information Sciences,2012,(217):21~30.
[88] Abe S, Lan M S. Fuzzy Rules Extraction Directly From Numerical Data forFunction Approximation [J]. IEEE Transactions on System, Man, and Cybernetics,1995,(25):119~129.
[89] Alcalá R, Alcalá-Fdez J, Herrera F, Otero J. Genetic Learning of Accurate andCompact Fuzzy Rule Based Systems Based on The2-tuples LinguisticRepresentation [J]. International Journal of Approximate Reasoning,2007,(44):45~64.
[90] Berenji H R, Khedkar P. Learning and Tuning Fuzzy Logic Controllers throughReinforcements [J]. IEEE Transactions on Neural Networks,1992,(3):724~740.
[91] Choi J N, Oh S K, Pedrycz W. Structural and Parametric Design of FuzzyInference Systems using Hierarchical Fair Competition-Based Parallel GeneticAlgorithms and Information Granulation [J]. International Journal of ApproximateReasoning,2008,(49):631~648.
[92] delaOssa L, Gámez J A, Puerta J M. Learning Weighted Linguistic Fuzzy Rulesby using Specifically-Tailored Hybrid Estimation of Distribution Algorithms [J].International Journal of Approximate Reasoning,2009,(50):541~560.
[93] Gobi A F, Pedrycz W. Fuzzy Modelling through Logic Optimization [J].International Journal of Approximate Reasoning,2007,(45):488~510.
[94] González J, Rojas I, Pomares H, Herrera L J, Guillén A, Palomares J M, Rojas F.Improving The Accuracy while Preserving The Interpretability of Fuzzy FunctionApproximators by Means of Multi-Objective Evolutionary Algorithms [J].International Journal of Approximate Reasoning,2007,(44):32~44.
[95] Kim E, Park M, Ji S, Park M. A New Approach to Fuzzy Modeling [J]. IEEETransactions on Fuzzy Systems,1997,(5):328~337.
[96] Kim E, Pedrycz W. Information Granulation as a Basis of Fuzzy Modeling [J].Journal of Intelligent&Fuzzy Systems,2007,(18):123~148.
[97] Kukolj D. Design of Adaptive Takagi–Sugeno–Kang Fuzzy Models [J].Applied Soft Computing,2002,(2):89~103.
[98] Kukolj D, Levi E. Identification of Complex Systems Based on Neural and Takagi–Sugeno Fuzzy Model [J]. IEEE Transactions on Systems, Man, and Cybernetics,Part B,2004,(34):272~282.
[99] Sugeno M, Kang G T. Structure Identification of a Fuzzy Model [J]. Fuzzy Setsand Systems,1988,(28):15~33.
[100] Wang L X. Training of Fuzzy Logic Systems using Nearest NeighborhoodClustering [C]//Proceeding of1993IEEE International Conference on FuzzySystems. San Francisco, CA:1993:13~17.
[101] Mendel J. Uncertain Rule-Based Fuzzy Logic Systems: Introduction and NewDirections [M]. Prentice Hall,2001.
[102] Wang L X, Mendel J M. Fuzzy Basis Functions, Universal Approximation, andOrthogonal Least Squares Learning [J]. IEEE Transactions on Neural Networks,1992,(3):807~814.
[103] Wang L X, Wei C. Approximation Accuracy of Some Neuro-Fuzzy Approaches[J]. IEEE Transactions on Fuzzy Systems,2000,(8):470~478.
[104] Zeng X J, Singh M G. Approximation Accuracy Analysis of Fuzzy Systems asFunction Approximators [J]. IEEE Transactions on Fuzzy Systems,1996,(4):44~63.
[105] Coppi R, D’Urso P, Giordani P, Santoro A. Least Squares Estimation of A LinearRegression Model with LR Fuzzy Response [J]. Computational Statistics&DataAnalysis,2006,(51):267~286.
[106] Friedman J. Multivariate Adaptive Regression Splines (with discussion)[J].Annals of Statistics,1991,(19):1~141.
[107] Hardle W. Applied Nonparametric Regression [M]. Cambridge: CambridgeUniversity Press,1990.
[108] Miller G A. The Magical Number Seven, Plus or Minus Two: Some Limits on OurCapacity for Processing Information [J]. Psychology Review,1956,(63):81~97.
[109] García S, Fernández A, Luengo J, Herrera F. Advanced Nonparametric Tests forMultiple Comparisons in The Design of Experiments in ComputationalIntelligence and Data Mining: Experimental Analysis of Power [J]. InformationSciences,2010,(180):2044-2064.
[110]唐敏,杨宏文,胡卫东,郁文贤. Mamdani型概率模糊系统建模方法[J].系统工程与电子技术,2012(2):323~327.
[111] Song H-J, Miao C-Y, Shen Z-Q, Roel W, D‘Hondt M, Francky C. A ProbabilisticFuzzy Approach to Modeling Nonlinear Systems [J]. Neurocomputing,2011,(74):1008-1025.
[112] Tang M, Chen X, Hu W D, Yu W X. Genetic Algorithm-based Design of FuzzyClassification Systems from Labeled Data [J]. Pattern Recognition Letter,Submitted.
[113] Ishibuchi H, Nozaki K, Tanaka H. Distributed Representation of Fuzzy Rules andIts Application to Pattern Classification [J]. Fuzzy Sets and Systems,1992,(52):21~32.
[114] Mansoori E G, Zolghadri M J, Katebi S D. A Weighting Function for ImprovingFuzzy Classification Systems Performance [J]. Fuzzy Sets and Systems,2007,(158):583~591.
[115] Abe S, Thawonmas R. A Fuzzy Classifier with Ellipsoidal Regions [J]. IEEETransactions on Fuzzy Systems,1997,(5):358~368.
[116] Lee C Y, Lin C J, Chen H J. A Self-Constructing Fuzzy CMAC Model and ItsApplications [J]. Information Sciences,2007,(177):264~280.
[117] Roubos J A, Setnes M, Abonyi J. Learning Fuzzy Classification Rules fromLabeled Data [J]. Information Sciences,2003,(150):77~93.
[118] Chakraborty D, Pal N R. A Neuro-Fuzzy Scheme for Simultaneous FeatureSelection and Fuzzy Rule-Based Classification [J]. IEEE Transactions on NeuralNetworks,2004,(15):110~123.
[119] Lin C T, Lee C S G. Neural Network-Based Fuzzy Logic Control and DecisionSystem [J]. IEEE Transactions on Computers,1991,(40):1320~1336.
[120] Nauck D, Kruse R. A Neuro-Fuzzy Method to Learn Fuzzy Classification Rulesfrom Data [J]. Fuzzy Sets and Systems,1997,(89):277~288.
[121] Gómez-Skarmeta A F, Valdés M, Jiménez F, Marin-Blázquez J G. ApproximativeFuzzy Rules Approaches for Classification with Hybrid-GA Techniques [J].Information Sciences,2001,(136):193~214.
[122] Hu Y C. Finding Useful Fuzzy Concepts for Pattern Classification using GeneticAlgorithm [J]. Information Sciences,2005,(175):1~19.
[123] Ishibuchi H, Nakashima T, Murata T. Three-Objective Genetics-Based MachineLearning for Linguistic Rule Extraction [J]. Information Sciences,2001,(136):109~133.
[124] Sanchez L, Couso I, Corrales J A. Combining GP Operators with SA Search toEvolve Fuzzy Rule Based Classifiers [J]. Information Sciences,2001,(136):175~191.
[125] Zhou E, Khotanzad A. Fuzzy Classifier Design using Genetic Algorithms [J].Pattern Recognition,2007,(40):3401~3414.
[126] Li M Q, Wang Z C. A Hybrid Coevolutionary Algorithm for Designing FuzzyClassifiers [J]. Information Sciences,2009,(179):1970~1983.
[127] Berlanga F J, Rivera A J, del Jesus M J, Herrera F. GP-COACH: GeneticProgramming-Based Learning of COmpact and ACcurate Fuzzy Rule-BasedClassification Systems for High-Dimensional Problems [J]. Information Sciences,2010,(180):1183~1200.
[128] Sanz J A, Fernández A, Bustince H, Herrera F. Improving the Performance ofFuzzy Rule-Based Classification Systems with Interval-Valued Fuzzy Sets andGenetic Amplitude Tuning [J]. Information Sciences,2010,(180):3674~3685.
[129] Fernández A, del Jesus M J, Herrera F. Hierarchical Fuzzy Rule BasedClassification Systems with Genetic Rule Selection for Imbalanced Data-Sets [J].International Journal of Approximate Reasoning,2009,(50):561~577.
[130] Gonzblez A, Perez R. SLAVE: A Genetic Learning System Based on An IterativeApproach [J]. IEEE Transactions on Fuzzy Systems,1999,(7):176~191.
[131] Ishibuchi H, Yamamoto T, Nakashima T. Hybridization of Fuzzy GBMLApproaches for Pattern Classification Problems [J]. IEEE Transactions onSystems, Man, and Cybernetics, Part B,2005,(35):359~365.
[132] Ishibuchi H, Nojima Y. Analysis of Interpretability-Accuracy Tradeoff of FuzzySystems by Multiobjective Fuzzy Genetics-Based Machine Learning [J].International Journal of Approximate Reasoning,2007,(44):4~31.
[133] Mansoori E G, Zolghadri M J, Katebi S D. SGERD: A Steady-State GeneticAlgorithm for Extracting Fuzzy Classification Rules from Data [J]. IEEETransactions on Fuzzy Systems,2008,(16):1061~1071.
[134] del Jesus M J, Hoffmann F, Navascués L J, Sánchez L. Induction ofFuzzy-Rule-Based Classifiers With Evolutionary Boosting Algorithms [J]. IEEETransactions on Fuzzy Systems,2004,(12):296~308.
[135] Hoffmann F, Baesens B, Mues C, Gestel T V, Vanthienen J. Inferring DescriptiveAnd Approximate Fuzzy Rules for Credit Scoring using Evolutionary Algorithms[J]. European Journal of Operational Research,2007,(177):540~555.
[136] Ho S Y, Chen H M, Ho S J, Chen T K. Design of Accurate Classifiers With ACompact Fuzzy-Rule Base using An Evolutionary Scatter Partition of FeatureSpace [J]. IEEE Transactions on Systems, Man, and Cybernetics, Part B,2004,(34):1031~1044.
[137] Murata T, Ishibuchi H, Gen M. Adjusting Fuzzy Partitions by Genetic AlgorithmsAnd Histograms for Pattern Classification Problems [C]//EvolutionaryComputation Proceedings of The1998IEEE World Congress on ComputationalIntelligence,1998:9~14.
[138] Cano J, Herrera F, Lozano M. Evolutionary Stratified Training Set Selection forExtracting Classification Rules with Trade-Off Precision-Interpretability [J]. Dataand Knowledge Engineering,2007,(60):90~108.
[139] Tang M, Chen X, Hu W D, Yu W X. A Fuzzy Rule-based Classification Systemusing Interval Type-2Fuzzy sets [C]//.2011International Conference onIntergrated Uncertainty in Knowledge Modelling and Decision Making, October2011, Hangzhou, China. LNCS7027Springer-Verlag,72~80.
[140] Ishibuchi H, Nakashima T, Nii M. Classification and Modeling with LinguisticInformation Granules: Advanced Approaches to Linguistic Data Mining [M].Springer-Verlag,2004.
[141] Bastian A. How to Handle the Flexibility of Linguistic Variables withApplications [J]. International Journal of Uncertainty, Fuzziness andKnowledge-Based Systems,1994,(2):463~484.
[142] Liang Q, Mendel J. MPEG VBR Video Traffic Modeling and Classification UsingFuzzy Technique [J]. IEEE Transactions on Fuzzy Systems,2001,(9):183~193.
[143] Wu H, Mendel J. Classification of Battlefield Ground Vehicles Using AcousticFeatures and Fuzzy Logic Rule-Based Classifiers [J]. IEEE Transactions on FuzzySystems,2007,(15):56~72.
[144] Hwang C, Rhee F. Uncertain Fuzzy Clustering: Interval Type-2Fuzzy Approachto C-means [J]. IEEE Transactions on Fuzzy Systems,2007,(15):107~120.
[145] Starczewski J T. Efficient Triangular Type-2Fuzzy Logic Systems [J].International Journal of Approximate Reasoning,2009,(50):799~811.
[146] Tang M, Hu W D, Yu W X. Fuzzy-valued Belief Functions in Infinite Spaces [J].IEEE Transactions on Fuzzy Systems, Under Review.
[147] Yang M S, Chen T C, Wu K L. Generalized Belief Function, Plausibility Function,and Dempster’s Combinational Rule to Fuzzy Sets [J]. International Journal ofIntelligent Systems,2003,(18):925~937.
[148] Hwang C M, Yang M S. Generalization of Belief and Plausibility Functions toFuzzy Sets Based on the Sugeno Integral [J]. International Journal of IntelligentSystems,2007,(22):1215~1228.
[149] Cornelis C, Deschrijver G, Kerre E E. Implication in Intuitionistic Fuzzy AndInterval-Valued Fuzzy Set Theory: Construction, Classification, Application [J].International Journal of Approximate Reasoning,2004,(35):55~95.
[150] Ammar E, Metz J. On Fuzzy Convexity and Parametric Fuzzy Optimization [J].Fuzzy Sets Systems,1992,(49):135~141.
[151] Yen J. Computing Generalized Belief Functions for Continuous Fuzzy Sets [J].International Journal of Approximate Reasoning,1992,(6):1~31.
[152]唐敏,杨宏文,胡卫东,郁文贤.基于模糊属性刻画的T相似性关系的构造[J].模糊系统与数学,2011(5):136~143.
[153] Jagielska I, Matthews C, Whitfort T. An Investigation into the Application ofNeural Networks, Fuzzy logic, Genetic algorithms, and Rough Sets to AutomatedKnowledge Acquisition for Classification Problems [J]. Neurocomputing,1999,(24):37~54.
[154] Kryszkiewicz M. Rough Set Approach to Incomplete Information Systems [J].Information Sciences,1998,(112):39~49.
[155] Tsumoto S. Automated Extraction of Medical Expert System Rules From ClinicalDatabases Based on Rough Set Theory [J]. Information Sciences,1998,(112):67~84.
[156] Nakamura A. Fuzzy rough sets [J]. Note Multiple-Valued Logic Japan,1988,(9):1–8.
[157] Qin K, Pei Z. On The Topological Properties of Fuzzy Rough Sets [J]. Fuzzy Setsand Systems,2005,(151):601–613.
[158] Thiele H. Fuzzy Rough Sets Versus Rough Fuzzy Sets—An Interpretation and AComparative Study using Concepts of Modal Logic [R]. Univ. Dortmund,Dortmund, Germany, Tech. Rep. ISSN1433-3325,1998.
[159] Yao Y Y. Combination of Rough And Fuzzy Sets Based on Alpha-Level Sets [C]//Lin T Y, Cercone N(eds.). Rough Sets and Data Mining: Analysis for ImpreciseData. Norwell, MA: Kluwer,1997:301~321.
[160] Bodjanova S. Approximation of Fuzzy Concepts in Decision Making [J]. FuzzySets and Systems,1997,(85):23~29.
[161] Kuncheva L I. Fuzzy Rough Sets: Application to Feature Extraction [J]. FuzzySets and Systems,1992,(51):147~153.
[162] Fernández Salido J M, Murakami S. Rough Set Analysis of A General Type ofFuzzy Data using Transitive Aggregations of Fuzzy Similarity Relations [J].Fuzzy Sets and Systems,2003,(139):635~660.
[163] Sudkamp T. Similarity, Interpolation, and Fuzzy Rule Construction [J]. Fuzzy Setsand Systems,1993,(58):73~86.
[164] Fonck P, Fodor J, Roubens M. An Application of Aggregation Procedures to TheDefinition of Measures of Similarities Between Fuzzy Sets [J]. Fuzzy Sets andSystems,1998,(97):67~74.
[165] Ovchinnikov S. Aggregating Transitive Fuzzy Binary Relations [J]. InternationalJournal of Uncertainty, Fuzziness Knowledge-based Systems,1992,(3):47~55.
[166] Beliakov G. Definition of General Aggregation Operators through SimilarityRelations [J]. Fuzzy Sets and Systems,2000,(114):437~453.
[167] Fernández Salido J M, Murakami S. On β-precision aggregation [J]. Fuzzy Setsand Systems,2003,139:547-558.