Mathematical foundations of fuzzy systems
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This document discusses the mathematical foundations of fuzzy systems, including: - The curriculum covers theory of fuzzy sets, theory of possibility, crisp vs. fuzzy values, model tasks, and possibilistic optimization tasks over two semesters for a total of 324 hours. - The theory of possibility introduced in 1978 uses axiomatic approach and possibility measures to define possibilistic space and possibilistic (fuzzy) variables characterized by possibility distributions. - Model tasks and possibilistic optimization tasks are presented, where the coefficients can be crisp or possibilistic variables.
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Mathematical foundations of fuzzy systems
- 1. Mathematical foundations of fuzzy systems I.S. Soldatenko Department of Information Technologies Tver State University, Tver, Russia A.V. Yazenin
- 2. Curriculum details “Mathematical foundations of fuzzy systems” is a discipline of first year of master program “Fundamental computer sciences and information technologies”. It has 324 hours (180 – first semester, 144 – second semester).
- 3. Theory of fuzzy sets
- 4. Theory of possibility In 1978 Stephen Nahmias introduced a new theory using axiomatic approach and basing on fuzzy sets – theory of possibility. Let – be a model set, – its elements, P() – power set of . Definition 1. A possibility measure : P() E1 is a set function with the following properties: () = 0, () = 1, (iIAi) = supiI (Ai), AiP(),I. Definition 2. A triplet (,P(),) is called possibilistic space. Definition 3. A possibilistic (fuzzy) variable is a real-valued function possible values of which are characterized by possibility distribution µA(x): µA(x) is the possibility that A can have x as a value. ,:)( 1 A .,)(:)( 1 xxAxA
- 5. Crisp vs fuzzy values 1 10987654321 1514131211 crisp values 1 10987654321 1514131211 fuzzy values
- 6. Model task Here and i=1, …, m, aij, bi – are crisp coefficients. max,)(0 xf . ,,,1,0)( N i Rx mixf n j jj xaxf 1 00 )( i n j jiji bxaxf 1 )(
- 7. Possibilistic optimization task Here and i=1, …, m, aij(), bi() – are possibilistic variables. max,k ,),( 00 kxf . ,,,1,0),( N ii Rx mixf n j jj xaxf 1 00 )(),( )()(),( 1 i n j jiji bxaxf
- 8. Fuzzy systems and soft computing Optimization, classification, data-mining, prognosis…