Fuzzy Membership Function

This presentation presents Type-1 fuzzy membership function, Type-2 fuzzy membership function (basic level).

1. Fuzzy Membership Function
Course: Computational Intelligence In Engineering (Soft Computing )
Prof. (Dr.) Pravat Kumar Rout
Department of EEE, ITER
Siksha ‘O’ Anusandhan (Deemed to be University),
Bhubaneswar, Odisha, India
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2. Membership Function
Membershipfunctionscharacterizefuzziness(i.e.,alltheinformation
infuzzyset),whethertheelementsinfuzzysetsarediscreteorcontinuous.
Membershipfunctionscanbedefinedasatechniquetosolvepractical
problemsbyexperienceratherthanknowledge.
Membershipfunctionsallowustographicallyrepresentafuzzyset.Thex
axisrepresentstheuniverseofdiscourse,whereastheyaxisrepresentsthe
degreesofmembershipinthe[0,1]interval.Simplefunctionsareusedto
buildmembershipfunctions.
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Membershipfunctions(MFs)arethebuildingblocksoffuzzysettheory,i.e.,
fuzzinessinafuzzysetisdeterminedbyitsMF.Accordingly,theshapesofMFsare
importantforaparticularproblemsincetheyeffectonafuzzyinferencesystem.
Theymayhavedifferentshapesliketriangular,trapezoidal,Gaussian,etc.
Fuzzificationistheprocessofconvertingacrispinputvaluetoafuzzyvaluethatis
performedbytheuseoftheinformationintheknowledgebase.Althoughvarious
typesofcurvescanbeseeninliterature,Gaussian,triangular,andtrapezoidalMFs
arethemostcommonlyusedinthefuzzificationprocess.
Definition.Fuzzificationistheprocessoftransformingacrispsettoafuzzysetora
fuzzysettofuzzierset.Defuzzificationistheprocessofreducingafuzzysetintoa
crispsetorconvertingafuzzymemberintoacrispmember.
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Themembershipfunctionofafuzzysetisageneralizationoftheindicator
functioninclassicalsets.Infuzzylogic,itrepresentsthedegreeoftruthas
anextensionofvaluation.
Degreesoftruthareoftenconfusedwithprobabilities,althoughtheyare
conceptuallydistinct,becausefuzzytruthrepresentsmembershipin
vaguelydefinedsets,notlikelihoodofsomeeventorcondition.
MembershipfunctionswereintroducedbyZadehinthefirstpaperonfuzzy
sets(1965).
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5. Formal Definition of membership function
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8. Basics of Membership Function 8

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Fuzzy Singleton

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Convex and Non-convex Fuzzy Sets
Bandwith

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Afuzzysetisdeterminedbyitsindeterminateboundaries,thereexistsanuncertainty
aboutthesetboundaries.Ontheotherhand,acrispsetisdefined
bycrispboundaries,andcontainthepreciselocationofthesetboundaries.

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Crisplogic(crisp)isthesameasbooleanlogic(either0or1).Eitherastatementistrue(1)
oritisnot(0),meanwhilefuzzylogiccapturesthedegreetowhichsomethingistrue.

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18. Fuzzy Function Block 18

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20. Example-1
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21. Example-2
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22. Example-3 22

23. Example-423

24. Example-5 24

25. Example-6 25

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28. Major Types of Membership Functions 28

29. Triangular Membership Function 29

30. Trapezoidal Membership Function 30

31. Gaussian Membership Function 31

32. Generalized Bell Membership Function 32

33. Different shapes of
Gaussian MFs with
different values of s
and m.
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34. SigmoidalMembership Function 34
✓AsigmoidalMFisinherentlyopenrightorleft&
thus,itisappropriateforrepresentingconcepts
suchas“verylarge”or“verynegative”.
✓SigmoidalMFmostlyusedasactivation
functionofartificialneuralnetworks(NN).ANN
shouldsynthesizeacloseMFinordertosimulate
thebehaviorofafuzzyinferencesystem.

35. 35Left –Right (LR) MF

36. 2-D Membership Function
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38. Composite and non-composite Membership
Function
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39. Composite two-dimensional Membership
Functions based on min & max operations
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Type-2 fuzzy sets and systems generalize standard Type-1 fuzzy sets and
systems so that more uncertainty can be handled.

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Type-2Fuzzysets(T2FS)representalargeamountofuncertaintyand
complexity;however,thesebenefitsarecloselyrelatedtothedefinitionofits
footprintofuncertainty(FOU).

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Footprint of
uncertainty (FOU).

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