Universal approximation theorem for uninorm-based fuzzy systems modeling.
type: Item
instance of: scholarly article
title: Universal approximation theorem for uninorm-based fuzzy systems modeling.
zbMATH Open document ID: 1040.93043
DOI: 10.1016/S0165-0114(02)00521-3
author: Ronald R. Yager
author: Vladik Kreinovich
published in: Fuzzy Sets and Systems
publication date: 2003-12-04T00:00:00Z
review text: The fundamental result of this study concerns the universal approximation property of fuzzy systems endowed with arbitrary uninorms. Uninorms are two-argument functions \(U:[0,1]^2\to [0,1]\) satisfying the following properties: (a) identity \(U(g,x)=x,(b)\) commutativity \(U(x,y)=U(y,x)\), (c) associativity \(U(x,U (y,z))= U(U(x, y),z)\), (d) monotonicity \(U(x,y)\leq U(x',y')\) for all \(x\leq x'\) and \(y\leq y'\). For \(g=0\) the uninorm becomes a certain s-norm while for \(g=1\) the definition coincides with the one for any t-norm. Furthermore the authors introduce a relevancy transformation \(h:[0,1]^2\to[0,1]\) such that (a) \(h(1,x)=x\); \(h(0,x)= g\); (b) if \(y<y'\) then \(h(x,y)\leq h(x,y')\), and (c) if \(x<x'\) then \(h(x,y)\leq h(x',y)\) for \(y\leq g\) and \(h(x,y)\geq h(x',y)\) for \(y\geq g\). The property of universal approximation is derived for the Mamdani type of model with the fuzzy relation of the fuzzy system being computed in the following form \[ R(x,y)=U \biggl( h\bigl(A_1(x), B_1(y) \bigr), \dots,h \bigl(A_N(x), B_N(y)\bigr)\biggr). \] Here the model is based on a collection of \(N\)-rules of the form ``if \(A_i\) then \(B_i\)'' involving fuzzy sets \(A_i\) and \(B_i\) defined in the input and output space, respectively.
Mathematics Subject Classification ID: 93C42
Mathematics Subject Classification ID: 93A30
zbMATH DE Number: 2013121
zbMATH Keywords: Fuzzy connectives and aggregation operations
zbMATH Keywords: Fuzzy systems modeling
zbMATH Keywords: Universal approximation
zbMATH Keywords: uninorms
zbMATH Keywords: relevancy transformation
zbMATH Keywords: Mamdani model
reviewed by: Witold Pedrycz
MaRDI profile type: Publication
cites work: Q4879332
cites work: Structure of Uninorms
cites work: Q3838055
cites work: Q4241985
cites work: Fuzzy control as a universal control tool
cites work: Q4211213
cites work: Q4945171
cites work: Uninorms in fuzzy systems modeling
cites work: Uninorm aggregation operators
description: scientific article; zbMATH DE number 2013121