Degree-of-fit (a fuzzy set and a crisp set)A:Comfortabletemperature(Afuzzyset)B:Just25degree(Acrispset)μ(x)μg(x)xx15C20°℃25°C15°C20°℃25°℃Thedegree-of-fit that"Just 25 degree"is"Comfortabletemperature"max (min(μA(x), μg(x) ))= max μans(x)Thisisthevalueofthemembershipfunctionatx=25.25°℃15°℃20°C16
16 Degree-of-fit (a fuzzy set and a crisp set) 0 1 15 C 20 C 25 C 0 1 15 C 20 C 25 C B: Just 25 degree (A crisp set) The degree-of-fit that “Just 25 degree” is “Comfortable temperature” 0 1 15 C 20 C 25 C max{min{ (x), (x) }} max (x) A B x X A B x X µ µ µ ∈ ∈ = This is the value of the membership function at x = 25. (x) µ A (x) µ B x x x A: Comfortable temperature (A fuzzy set)
Degree-of-fit (Closeness) (Possibility)WithyourabilityyoucangetYouwill passthissubjectB:about50points(Afuzzyset)ifyoutakeover60pointsinthetest.μA(x)A: A crisp setμg(x)(60 points or more)xx605060Calculationofthepossibilityofyoursuccessmax (min(μa(x), μg(x) ))= max μans(x)= 0.2xThepossibilityofyoursuccessis2opercent.506017
17 Degree-of-fit (Closeness) (Possibility) 0 1 60 Calculation of the possibility of your success 0 1 max{min{ (x), (x) }} max (x) A B x X A B x X µ µ µ ∈ ∈ = You will pass this subject if you take over 60 points in the test. 50 60 0 1 50 60 With your ability you can get B: about 50 points (A fuzzy set). = 0.2 The possibility of your success is 20 percent. (x) µ A (x) A µ B : A crisp set (60 points or more) x x x
Fuzzyreasoning,whichisanMax-min operation in reasoningimportantuseoffuzzysettheory,isintroducedherethoughit.Themax-minoperationisalsousedinreasoningdeviatesfromtheflowof lecture..Thepossibilityis defined as themaximum valueamong the minimum guaranteed values.ARBFuzzy relationFuzzy inputFuzzyoutputB= AoR <μg(y)=max(min(μa(x),μr(x,y)))Thepossibilityofoutputisgivenbytheclosenessbetweenthefuzzyinputandthefuzzyrelation.(Forreference:)Adoctor'sdiagnosis istofindadiseasename(A)byseeingsymptoms(B).Thefuzzyrelation(R)isknowledgeofthedoctor.Thisisadifficultproblemcalledaninverseproblem18
18 Max-min operation in reasoning • The max-min operation is also used in reasoning. • The possibility is defined as the maximum value among the minimum guaranteed values. Fuzzy input Fuzzy relation Fuzzy output A R B ( y) max{ min{ A (x), R (x, y) }} x X µ B µ µ ∈ = B = A R ⇔ (For reference:) A doctor’s diagnosis is to find a disease name (A) by seeing symptoms (B). The fuzzy relation (R) is knowledge of the doctor. This is a difficult problem called an inverse problem. The possibility of output is given by the closeness between the fuzzy input and the fuzzy relation. Fuzzy reasoning, which is an important use of fuzzy set theory, is introduced here though it deviates from the flow of lecture
You can infer even if conditions are slightly differentClassical logicP implies Q,，(P=Q)(if PthenQ)and P is true,then Q is true.EvenifPisalmosttrue(say90%),youcannotconcludethatQistruewith9o%.FuzzyreasoningRule:If x is A thenyis BEven if the fact A' is different from A slightly,Fact: x is A'youcanobtainaslightlydifferentconclusion.Conclusion:y is B'19
19 You can infer even if conditions are slightly different. then Q is true. and P is true, P implies Q, ( P ⇒ Q ) ( if P then Q ) Classical logic Fuzzy reasoning y B x A x A y B ′ ′ Conclusion : is Fact : is Rule : If is then is Even if P is almost true (say 90 %), you cannot conclude that Q is true with 90%. Even if the fact is different from A slightly, you can obtain a slightly different conclusion. A′
Fuzzy reasoningμ (x)μg(y)Rule:IfxisAthenyisB1x3h4h5h708090(Ex.)If youstudyforabout4hours,youwill beabletogetabout8opoints intheexam.(A, A= B)→ BB'= A'o(A= B)Rule:Ifxis Athenyis BFact :x is A'μg(y)=max (min(μA(x),μA=B(x,y) ))Conclusion :y is B'XEXHow to define: μA=B(x,y)20
Rule： If x is A then y is B (A′ , A ⇒ B) → B′ B′ = A′ (A ⇒ B) ( y) max{min{ A (x), A B (x, y) }} x X B ′ ⇒ ∈ µ ′ = µ µ 20 Fuzzy reasoning 0 1 (x) µ A (y) µ B x y y B x A x A y B ′ ′ Conclusion is Fact is Rule If is then is ： ： ： (Ex.) If you study for about 4 hours, you will be able to get about 80 points in the exam. 3h 4h 5h 70 80 90 (x, y) How to define: µ A⇒B