Thick gradual intervals: An alternative interpretation of type-2 fuzzy intervals and its potential use in type-2 fuzzy computations

Abstract This paper proposes a new interpretation of type-2 fuzzy intervals (T2FIs) through the joint use of gradual intervals (GIs) and thick intervals (TIs). In this framework, a T2FI is viewed as a thick gradual interval (TGI). This new representation gives an original concept for the manipulation of T2FIs according to the thick gradual representation. Furthermore, this vision allows an extension of the interval arithmetic arsenal and reasoning to the framework of T2FIs. The proposed approach can be regarded as more computationally viable, which will make T2FIs computations more useful in applied scenarios. As an illustration, the proposed concept is used to implement the elementary arithmetic operations on T2FIs. The potentialities of the TGI approach have been validated in the frameworks of T2FI aggregation operators and T2FI regression.

[1]  L. Jaulin,et al.  A Priori Error Analysis with Intervals , 2007 .

[2]  Pei-Yi Hao,et al.  Dual possibilistic regression analysis using support vector networks , 2020, Fuzzy Sets Syst..

[3]  Jerry M. Mendel,et al.  Corrections to “Aggregation Using the Linguistic Weighted Average and Interval Type-2 Fuzzy Sets” , 2008, IEEE Transactions on Fuzzy Systems.

[4]  Luciano Stefanini,et al.  A generalization of Hukuhara difference and division for interval and fuzzy arithmetic , 2010, Fuzzy Sets Syst..

[5]  Didier Dubois,et al.  Gradual Numbers and Their Application to Fuzzy Interval Analysis , 2008, IEEE Transactions on Fuzzy Systems.

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

[7]  Luc Jaulin,et al.  A Priori Error Analysis and Spring Arithmetic , 2009, SIAM J. Sci. Comput..

[8]  Slavka Bodjanova,et al.  Alpha-bounds of fuzzy numbers , 2003, Inf. Sci..

[9]  Luc Jaulin,et al.  Thick set inversion , 2017, Artif. Intell..

[10]  Oscar Castillo,et al.  Edge-Detection Method for Image Processing Based on Generalized Type-2 Fuzzy Logic , 2014, IEEE Transactions on Fuzzy Systems.

[11]  Sylvie Galichet,et al.  Linear Fuzzy Regression Using Trapezoidal Fuzzy Intervals , 2008, IPMU 2008.

[12]  E. Kaucher.  Interval Analysis in the Extended Interval Space IR , 1980 .

[13]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[14]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

[15]  Zenon Kulpa.  A diagrammatic approach to investigate interval relations , 2006, J. Vis. Lang. Comput..

[16]  Susana Cubillo,et al.  Aggregation operators on type-2 fuzzy sets , 2017, Fuzzy Sets Syst..

[17]  Robert Ivor John,et al.  Parallel computation of type-2 fuzzy sets using alpha-cuts , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[18]  Oscar Castillo,et al.  A generalized type-2 fuzzy granular approach with applications to aerospace , 2016, Inf. Sci..

[19]  Wei Chen,et al.  Curve reconstruction based on an interval B-spline curve , 2005, The Visual Computer.

[20]  A. Kaufmann,et al.  Introduction to fuzzy arithmetic : theory and applications , 1986 .

[21]  Mohammad Hossein Fazel Zarandi,et al.  Multi-central general type-2 fuzzy clustering approach for pattern recognitions , 2016, Inf. Sci..

[22]  Jerry M. Mendel,et al.  Aggregation Using the Linguistic Weighted Average and Interval Type-2 Fuzzy Sets , 2007, IEEE Transactions on Fuzzy Systems.

[23]  Pranab K. Muhuri,et al.  Interval type-2 fuzzy sets for enhanced learning in deep belief networks , 2017, 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[24]  Patrick Meyer,et al.  On the use of the Choquet integral with fuzzy numbers in multiple criteria decision support , 2006, Fuzzy Sets Syst..

[25]  Sylvie Galichet,et al.  Min and Max Operators for Gradual Intervals , 2018, IEEE Transactions on Fuzzy Systems.

[26]  C. Harmening,et al.  Choosing the Optimal Number of B-spline Control Points (Part 1: Methodology and Approximation of Curves) , 2016 .

[27]  T. Sunaga.  Theory of an interval algebra and its application to numerical analysis , 2009 .

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

[29]  Ali Vahidian Kamyad,et al.  A Piecewise Type-2 Fuzzy Regression Model , 2017, Int. J. Comput. Intell. Syst..

[30]  Pierpaolo D'Urso,et al.  Least squares estimation of a linear regression model with LR fuzzy response , 2006, Comput. Stat. Data Anal..

[31]  Milan Hladík,et al.  Possibilistic linear regression with fuzzy data: Tolerance approach with prior information , 2018, Fuzzy Sets Syst..

[32]  Feilong Liu,et al.  An efficient centroid type-reduction strategy for general type-2 fuzzy logic system , 2008, Inf. Sci..

[33]  Robert Ivor John,et al.  A novel alpha-cut representation for type-2 fuzzy sets , 2010, International Conference on Fuzzy Systems.

[34]  Robert Ivor John,et al.  Type-2 Fuzzy Alpha-Cuts , 2017, IEEE Transactions on Fuzzy Systems.

[35]  Oscar Castillo,et al.  Generalized Type-2 Fuzzy Systems for controlling a mobile robot and a performance comparison with Interval Type-2 and Type-1 Fuzzy Systems , 2015, Expert Syst. Appl..

[36]  Hideo Tanaka,et al.  Interval regression analysis by quadratic programming approach , 1998, IEEE Trans. Fuzzy Syst..

[37]  Oscar Castillo,et al.  Comparative analysis of noise robustness of type 2 fuzzy logic controllers , 2018, Kybernetika.

[38]  Hani Hagras,et al.  zSlices — towards bridging the gap between interval and general type-2 fuzzy logic , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[39]  Simon Coupland,et al.  Type-2 Fuzzy Arithmetic using Alpha-planes , 2009, IFSA/EUSFLAT Conf..

[40]  Luciano Stefanini,et al.  Type-2 fuzzy numbers and operations by F-transform , 2013, 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS).

[41]  Pranab K. Muhuri,et al.  Type-2 fuzzy reliability–redundancy allocation problem and its solution using particle-swarm optimization algorithm , 2018, Granular Computing.

[42]  Etienne E. Kerre,et al.  Aggregation Operators in Interval-valued Fuzzy and Atanassov's Intuitionistic Fuzzy Set Theory , 2008, Fuzzy Sets and Their Extensions: Representation, Aggregation and Models.

[43]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

[44]  Peide Liu,et al.  A weighted aggregation operators multi-attribute group decision-making method based on interval-valued trapezoidal fuzzy numbers , 2011, Expert Syst. Appl..

[45]  Sylvie Galichet,et al.  Model Inversion Using Extended Gradual Interval Arithmetic , 2012, IEEE Transactions on Fuzzy Systems.

[46]  Phil Diamond,et al.  Fuzzy least squares , 1988, Inf. Sci..

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

[48]  Elham Hosseinzadeh,et al.  A weighted goal programming approach to fuzzy linear regression with crisp inputs and type-2 fuzzy outputs , 2015, Soft Comput..

[49]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[50]  Juan R. Castro,et al.  A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems , 2016, Inf. Sci..

[51]  Ramon E. Moore.  Interval arithmetic and automatic error analysis in digital computing , 1963 .

[52]  Pranab K. Muhuri,et al.  Big-data clustering with interval type-2 fuzzy uncertainty modeling in gene expression datasets , 2019, Eng. Appl. Artif. Intell..

[53]  Peide Liu,et al.  A multi-attribute group decision-making method based on weighted geometric aggregation operators of interval-valued trapezoidal fuzzy numbers , 2012 .

[54]  D. Dubois,et al.  Operations on fuzzy numbers , 1978 .

[55]  Sylvie Galichet,et al.  Extended gradual interval (EGI) arithmetic and its application to gradual weighted averages , 2014, Fuzzy Sets Syst..

[56]  Sylvie Galichet,et al.  MIN and MAX Operators for Fuzzy Intervals and Their Potential Use in Aggregation Operators , 2007, IEEE Transactions on Fuzzy Systems.

[57]  Olga Poleshchuk,et al.  A Fuzzy Nonlinear Regression Model for Interval Type-2 Fuzzy Sets , 2014 .

[59]  Sylvie Galichet,et al.  From fuzzy regression to gradual regression: Interval-based analysis and extensions , 2018, Inf. Sci..

[60]  Andrzej Piegat,et al.  Is an interval the right result of arithmetic operations on intervals? , 2017, Int. J. Appl. Math. Comput. Sci..

[61]  Ting-Yu Chen,et al.  An ELECTRE-based outranking method for multiple criteria group decision making using interval type-2 fuzzy sets , 2014, Inf. Sci..

[62]  Luciano Stefanini,et al.  Approximate fuzzy arithmetic operations using monotonic interpolations , 2005, Fuzzy Sets Syst..

[63]  Didier Dubois,et al.  Gradual elements in a fuzzy set , 2008, Soft Comput..

[64]  Oscar Castillo,et al.  Type-2 fuzzy logic aggregation of multiple fuzzy controllers for airplane flight control , 2015, Inf. Sci..

[65]  Jerry M. Mendel,et al.  Interval Type-2 Fuzzy Logic Systems Made Simple , 2006, IEEE Transactions on Fuzzy Systems.

[66]  Pierpaolo D'Urso,et al.  A least-squares approach to fuzzy linear regression analysis , 2000 .

[67]  Jerry M. Mendel,et al.  Analytical solution methods for the fuzzy weighted average , 2012, Inf. Sci..

[68]  Sylvie Galichet,et al.  Model Inversion using Extended Gradual Intervals Arithmetic , 2011, EUSFLAT Conf..

[69]  Benoît Desrochers.  Simultaneous localization and mapping in unstructured environments : a set-membership approach , 2018 .

[70]  R. John,et al.  On aggregating uncertain information by type-2 OWA operators for soft decision making , 2010 .

[71]  Zenon Kulpa,et al.  Diagrammatic representation for interval arithmetic , 2001 .

[72]  Luciano Stefanini,et al.  Approximation of Fuzzy Numbers by F-Transform , 2012, IPMU.

[73]  Jerry M. Mendel,et al.  $\alpha$-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications , 2009, IEEE Transactions on Fuzzy Systems.

[74]  Dug Hun Hong,et al.  Interval regression analysis using quadratic loss support vector machine , 2005, IEEE Transactions on Fuzzy Systems.

[75]  Junzo Watada,et al.  Building a type II fuzzy qualitative regression model , 2012 .