Type-2 fuzzy rough sets based on extended t-norms

A type-2 fuzzy rough set based on extended t-norms is investigated from constructive and axiomatic approaches.An approximation algorithm of type-2 fuzzy rough sets is listed based on α-cut sets.Weak type-2 fuzzy topologies are proposed to investigate the topological properties of type-2 fuzzy rough sets.An interval-valued type-2 fuzzy rough set is proposed and investigated. This paper is devoted to investigate type-2 fuzzy rough sets based on extended t-norms with respect to type-2 fuzzy relations with convex normal fuzzy truth values. Type-2 fuzzy rough sets are proposed on two finite universes of discourse and characterized from both constructive and axiomatic approaches. In the constructive approach, different classes of type-2 fuzzy rough sets are investigated. An approximate algorithm of type-2 fuzzy rough sets based on α-cut sets is given with numerical examples. In the axiomatic approach, the axiomatic characterizations of type-2 fuzzy rough approximation operators are studied. Moreover, the topological properties of type-2 fuzzy rough sets are discussed. In the end, this paper introduces interval-valued type-2 fuzzy rough sets and discusses their properties.

[1]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[2]  Wei-Zhi Wu,et al.  Generalized fuzzy rough sets , 2003, Inf. Sci..

[3]  J. Mendel.  Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions , 2001 .

[4]  Jerry M. Mendel,et al.  Type-2 fuzzy logic systems , 1999, IEEE Trans. Fuzzy Syst..

[5]  Wei-Zhi Wu,et al.  Constructive and axiomatic approaches of fuzzy approximation operators , 2004, Inf. Sci..

[6]  Zhudeng Wang,et al.  Some properties of L-fuzzy approximation spaces based on bounded integral residuated lattices , 2014, Inf. Sci..

[7]  Wei-Zhi Wu,et al.  On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse , 2009, Int. J. Approx. Reason..

[8]  Bao Qing Hu,et al.  Fuzzy rough sets based on generalized residuated lattices , 2013, Inf. Sci..

[9]  Hung T. Nguyen,et al.  A note on the extension principle for fuzzy sets , 1978 .

[10]  Zhiming Zhang,et al.  A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets , 2013 .

[11]  J. Goguen.  L-fuzzy sets , 1967 .

[12]  R. Lowen.  Fuzzy topological spaces and fuzzy compactness , 1976 .

[13]  Jerry M. Mendel,et al.  Applications of Type-2 Fuzzy Logic Systems to Forecasting of Time-series , 1999, Inf. Sci..

[14]  Shyi-Ming Chen,et al.  Fuzzy decision making systems based on interval type-2 fuzzy sets , 2013, Inf. Sci..

[15]  Anna Maria Radzikowska,et al.  A comparative study of fuzzy rough sets , 2002, Fuzzy Sets Syst..

[16]  Zhiming Zhang,et al.  On characterization of generalized interval type-2 fuzzy rough sets , 2013, Inf. Sci..

[17]  Jerry M. Mendel,et al.  Centroid of a type-2 fuzzy set , 2001, Inf. Sci..

[18]  Jianhua Dai,et al.  Uncertainty measurement for interval-valued information systems , 2013, Inf. Sci..

[19]  Zexuan Ji,et al.  Interval-valued possibilistic fuzzy C-means clustering algorithm , 2014, Fuzzy Sets Syst..

[20]  Duoqian Miao,et al.  Positive approximation and converse approximation in interval-valued fuzzy rough sets , 2011, Inf. Sci..

[21]  Oscar Castillo,et al.  Optimal design of type-2 and type-1 fuzzy tracking controllers for autonomous mobile robots under perturbed torques using a new chemical optimization paradigm , 2013, Expert Syst. Appl..

[22]  Jerry M. Mendel,et al.  On a 50% savings in the computation of the centroid of a symmetrical interval type-2 fuzzy set , 2005, Inf. Sci..

[23]  Haoyang Wu,et al.  An Interval Type-2 Fuzzy Rough Set Model for Attribute Reduction , 2009, IEEE Transactions on Fuzzy Systems.

[24]  Mohammad Hossein Fazel Zarandi,et al.  Interval Type-2 Relative Entropy Fuzzy C-Means clustering , 2014, Inf. Sci..

[25]  Witold Pedrycz,et al.  Granular Computing: Analysis and Design of Intelligent Systems , 2013 .

[26]  Etienne E. Kerre,et al.  On the relationship between some extensions of fuzzy set theory , 2003, Fuzzy Sets Syst..

[27]  Zdenko Takác,et al.  Aggregation of fuzzy truth values , 2014, Inf. Sci..

[28]  Byung-In Choi,et al.  Interval type-2 fuzzy membership function generation methods for pattern recognition , 2009, Inf. Sci..

[29]  Bao Qing Hu,et al.  On fuzzy-valued operations and fuzzy-valued fuzzy sets , 2015, Fuzzy Sets Syst..

[30]  Elbert A. Walker,et al.  Convex normal functions revisited , 2010, Fuzzy Sets Syst..

[31]  Witold Pedrycz,et al.  Granular computing: an introduction , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[32]  Degang Chen,et al.  Fuzzy rough set theory for the interval-valued fuzzy information systems , 2008, Inf. Sci..

[33]  Ali Selamat,et al.  Improved sensitivity based linear learning method for permeability prediction of carbonate reservoir using interval type-2 fuzzy logic system , 2014, Appl. Soft Comput..

[34]  D. Dubois,et al.  ROUGH FUZZY SETS AND FUZZY ROUGH SETS , 1990 .

[35]  Zhiming Zhang,et al.  On interval type-2 rough fuzzy sets , 2012, Knowl. Based Syst..

[36]  Jerry M. Mendel,et al.  Operations on type-2 fuzzy sets , 2001, Fuzzy Sets Syst..

[37]  Andrzej Skowron,et al.  Rudiments of rough sets , 2007, Inf. Sci..

[38]  Elbert A. Walker,et al.  The algebra of fuzzy truth values , 2005, Fuzzy Sets Syst..

[39]  Andrzej Skowron,et al.  Rough sets and Boolean reasoning , 2007, Inf. Sci..

[40]  Anna Maria Radzikowska,et al.  Fuzzy Rough Sets Based on Residuated Lattices , 2004, Trans. Rough Sets.

[41]  Witold Pedrycz,et al.  Regranulation: A granular algorithm enabling communication between granular worlds , 2007, Inf. Sci..

[42]  Francisco Chiclana,et al.  Defuzzification of the discretised generalised type-2 fuzzy set: Experimental evaluation , 2013, Inf. Sci..

[43]  Mohammad Hossein Fazel Zarandi,et al.  A type-2 fuzzy c-regression clustering algorithm for Takagi-Sugeno system identification and its application in the steel industry , 2012, Inf. Sci..

[44]  Nehad N. Morsi,et al.  Axiomatics for fuzzy rough sets , 1998, Fuzzy Sets Syst..

[45]  Bao Qing Hu,et al.  Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets , 2013, Inf. Sci..

[46]  Bao Qing Hu,et al.  On type-2 fuzzy sets and their t-norm operations , 2014, Inf. Sci..

[47]  Zhoujun Li,et al.  A novel variable precision (θ, σ)-fuzzy rough set model based on fuzzy granules , 2014, Fuzzy Sets Syst..

[48]  Wen-Xiu Zhang,et al.  An axiomatic characterization of a fuzzy generalization of rough sets , 2004, Inf. Sci..

[49]  Andrzej Skowron,et al.  Rough sets: Some extensions , 2007, Inf. Sci..

[50]  Jerry M. Mendel,et al.  Advances in type-2 fuzzy sets and systems , 2007, Inf. Sci..

[51]  Witold Pedrycz,et al.  Knowledge-based clustering - from data to information granules , 2007 .

[52]  Didier Dubois,et al.  Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets , 2012, Fuzzy Sets Syst..

[53]  Bao Qing Hu,et al.  On type-2 fuzzy relations and interval-valued type-2 fuzzy sets , 2014, Fuzzy Sets Syst..

[54]  Z. Pawlak.  Rough Sets: Theoretical Aspects of Reasoning about Data , 1991 .

[55]  Jerry M. Mendel,et al.  New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule , 2007, Inf. Sci..

[56]  Degang Chen,et al.  Fuzzy rough set theory for the interval-valued fuzzy information systems , 2008, Inf. Sci..