On interval additive generators of interval overlap functions and interval grouping functions

Abstract Overlap and grouping functions have been proposed by Bustince et al. for applications in image processing, classification problems and decision making based on fuzzy preference relations. In this paper, firstly, we generalize the concepts of overlap and grouping functions to the notions of interval overlap functions and interval grouping functions and investigate some vital related properties. In particular, we give the construction methods to obtain interval overlap functions and interval grouping functions, which are the best interval representation of the overlap and grouping functions, respectively. And then, we introduce the concepts of interval additive generators of interval overlap functions and interval grouping functions, which offer convenience for the selection of proper interval overlap functions or interval grouping functions in concrete problems. We also prove that the best interval representation of the additive generator pair of an additively generated overlap function is an interval additive generator pair of the best interval representation of this overlap function and the case for grouping function is analogous.

[1]  Jamel Louati,et al.  Interval computation and constraint propagation for the optimal design of a compression spring for a linear vehicle suspension system , 2015 .

[2]  Humberto Bustince,et al.  Mathematical analysis of interval-valued fuzzy relations: Application to approximate reasoning , 2000, Fuzzy Sets Syst..

[3]  Humberto Bustince,et al.  On additive generators of overlap functions , 2016, Fuzzy Sets Syst..

[4]  Frank Mueller,et al.  Preface , 2009, 2009 IEEE International Symposium on Parallel & Distributed Processing.

[5]  I. Turksen.  Interval valued fuzzy sets based on normal forms , 1986 .

[6]  Benjamín R. C. Bedregal,et al.  On interval fuzzy negations , 2010, Fuzzy Sets Syst..

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

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

[9]  Benjamín R. C. Bedregal,et al.  On the laws of contraposition for residual implications derived from overlap functions , 2015, 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[10]  R. Callejas Bedregal,et al.  Intervals as Domain Constructor , 2001 .

[11]  Humberto Bustince,et al.  Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets , 2000, Int. J. Approx. Reason..

[12]  Humberto Bustince,et al.  Overlap Index, Overlap Functions and Migrativity , 2009, IFSA/EUSFLAT Conf..

[13]  Benjamín R. C. Bedregal,et al.  On (G, N)-implications derived from grouping functions , 2014, Inf. Sci..

[14]  Eyke Hüllermeier,et al.  Grouping, Overlap, and Generalized Bientropic Functions for Fuzzy Modeling of Pairwise Comparisons , 2012, IEEE Transactions on Fuzzy Systems.

[15]  Sadaaki Miyamoto,et al.  Remarks on basics of fuzzy sets and fuzzy multisets , 2005, Fuzzy Sets Syst..

[16]  Humberto Bustince,et al.  n-Dimensional overlap functions , 2016, Fuzzy Sets Syst..

[17]  Benjamín R. C. Bedregal,et al.  Interval Valued Versions of T-Conorms, Fuzzy Negations and Fuzzy Implications , 2006, 2006 IEEE International Conference on Fuzzy Systems.

[18]  Mikel Galar Idoate,et al.  Fuzzy rule-based classification systems for multi-class problems using binary decomposition strategies: on the influence of n-dimensional overlap functions in the fuzzy reasoning method , 2016 .

[19]  Benjamín R. C. Bedregal,et al.  Interval additive generators of interval t-norms and interval t-conorms , 2011, Inf. Sci..

[20]  Benjamín R. C. Bedregal,et al.  Interval t-norms as Interval Representations of t-norms , 2005, The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ '05..

[21]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[22]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[23]  Maciej Wygralak,et al.  An algorithmic study of relative cardinalities for interval-valued fuzzy sets , 2016, Fuzzy Sets Syst..

[24]  R. Tennant.  Algebra , 1941, Nature.

[25]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[26]  L. A. Zadeh,et al.  Fuzzy logic and approximate reasoning , 1975, Synthese.

[27]  Francisco Herrera,et al.  Enhancing Multiclass Classification in FARC-HD Fuzzy Classifier: On the Synergy Between $n$-Dimensional Overlap Functions and Decomposition Strategies , 2015, IEEE Transactions on Fuzzy Systems.

[28]  B. Hu,et al.  A novel approach in uncertain programming part I: new arithmetic and order relation for interval numbers , 2006 .

[29]  Benjamín R. C. Bedregal,et al.  On interval fuzzy S-implications , 2010, Inf. Sci..

[30]  Francisco Herrera,et al.  A Historical Account of Types of Fuzzy Sets and Their Relationships , 2016, IEEE Transactions on Fuzzy Systems.

[31]  Humberto Bustince,et al.  New results on overlap and grouping functions , 2013, Inf. Sci..

[32]  Didier Dubois,et al.  Random sets and fuzzy interval analysis , 1991 .

[33]  Benjamín R. C. Bedregal,et al.  Archimedean overlap functions: The ordinal sum and the cancellation, idempotency and limiting properties , 2014, Fuzzy Sets Syst..

[34]  V. Torra,et al.  A framework for linguistic logic programming , 2010 .

[35]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[36]  Humberto Bustince,et al.  On Additive Generators of Grouping Functions , 2014, IPMU.

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

[38]  Francisco Herrera,et al.  Improving the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets and genetic amplitude tuning , 2010, Inf. Sci..

[39]  Benjamín R. C. Bedregal,et al.  Interval Valued QL-Implications , 2007, WoLLIC.

[40]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

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

[42]  Mustafa Demirci.  Genuine sets , 1999, Fuzzy Sets Syst..

[43]  Humberto Bustince,et al.  Image thresholding using restricted equivalence functions and maximizing the measures of similarity , 2007, Fuzzy Sets Syst..

[44]  Sankar K. Pal,et al.  Fuzzy sets in pattern recognition and machine intelligence , 2005, Fuzzy Sets Syst..

[45]  Didier Dubois,et al.  Interval-valued Fuzzy Sets, Possibility Theory and Imprecise Probability , 2005, EUSFLAT Conf..

[46]  Lotfi A. Zadeh,et al.  Fuzzy Logic , 2009, Encyclopedia of Complexity and Systems Science.

[47]  Humberto Bustince,et al.  Interval-valued fuzzy sets constructed from matrices: Application to edge detection , 2009, Fuzzy Sets Syst..

[48]  Bao Qing Hu,et al.  On multiplicative generators of overlap and grouping functions , 2018, Fuzzy Sets Syst..

[49]  Victor Joseph. Monte Leon.  Automatic error analysis in finite digital computations, using range arithmetic. , 1966 .

[50]  Renata Reiser,et al.  Analyzing Properties of Fuzzy Implications Obtained via the Interval Constructor , 2006 .

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

[52]  M. H. van Emden,et al.  Interval arithmetic: From principles to implementation , 2001, JACM.

[53]  Humberto Bustince,et al.  Some properties of overlap and grouping functions and their application to image thresholding , 2013, Fuzzy Sets Syst..

[54]  Graçaliz Pereira Dimuro,et al.  ICTM: An Interval Tessellation-Based Model for Reliable Topographic Segmentation , 2004, Numerical Algorithms.

[55]  André Bigand,et al.  Membership function construction for interval-valued fuzzy sets with application to Gaussian noise reduction , 2016, Fuzzy Sets Syst..

[56]  Vladik Kreinovich,et al.  From intervals to domains: towards a general description of validated uncertainty, with potential applications to geospatial and meteorological data , 2007 .

[57]  Juan Fernández-Sánchez,et al.  On (α, β)-homogeneous copulas , 2013, Inf. Sci..

[58]  H. Carter.  Fuzzy Sets and Systems — Theory and Applications , 1982 .

[59]  Ramon E. Moore,et al.  Interval analysis and fuzzy set theory , 2003, Fuzzy Sets Syst..

[60]  Mariano Eriz.  Aggregation Functions: A Guide for Practitioners , 2010 .

[61]  G.P. Dimuro,et al.  Towards interval analysis of the load uncertainty in power electric systems , 2004, 2004 International Conference on Probabilistic Methods Applied to Power Systems.

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

[63]  T. Calvo,et al.  Generation of weighting triangles associated with aggregation functions , 2000 .

[64]  Zdenko Takác,et al.  Subsethood measures for interval-valued fuzzy sets based on the aggregation of interval fuzzy implications , 2016, Fuzzy Sets Syst..

[65]  Vincenzo Cutello,et al.  Recursive connective rules , 1999, Int. J. Intell. Syst..

[66]  Graçaliz Pereira Dimuro,et al.  On residual implications derived from overlap functions , 2015, Inf. Sci..

[67]  Benjamín R. C. Bedregal,et al.  Formal Aspects of Correctness and Optimality of Interval Computations , 2006, Formal Aspects of Computing.

[68]  Lotfi A. Zadeh,et al.  Toward a generalized theory of uncertainty (GTU) - an outline , 2005, GrC.

[69]  Renata Reiser,et al.  Analyzing the Relationship between Interval-valued D-Implications and Interval-valued QL-Implications , 2010 .

[70]  Radko Mesiar,et al.  Triangular Norms , 2000, Trends in Logic.

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

[72]  Benjamín R. C. Bedregal,et al.  The best interval representations of t-norms and automorphisms , 2006, Fuzzy Sets Syst..