Inference in Fuzzy Systems PDF

Inference in fuzzy systems Artificial intelligence Kristóf Karacs PPKE-ITK 1 Fuzzy relation nGiven two universes and , a fuzzy relation ℛ is ℛ⊂ × where ⊂ denotes a fuzzy subset...

Inference in fuzzy systems Artificial intelligence Kristóf Karacs PPKE-ITK 1 Fuzzy relation nGiven two universes and , a fuzzy relation ℛ is ℛ⊂ × where ⊂ denotes a fuzzy subset n ℛ is defined by ℛ , 2 1  Composition n Given two fuzzy relations ℛ: × → [0,1] : × → [0,1] their composition is defined by = ℛ ∘ : × → [0,1] ℛ∘ , = max min ℛ , , , ∈ n Called an inner or-and product 3 Composition example é 0.4 0.3 0.3ù ê 0 0.4 0 úú é 0.8 1 0.1 0.7 ù R ( X ,Y ) = ê S (Y , Z ) = ê ë 0 0.8 0 0 úû ê 0.3 0.5 0.8ú ê ú ë 0.6 0.7 0.5û ( x , z ) = max ( min ( ( x , y ) , ( y , z ) ) ) i j yk i k k j é 0.6 0.7 0.5ù R oS (X ,Z ) = ê ú ë 0 0.4 0 û 4 2  Form of reasoning n Fuzzy version of generalized modus ponens n Antecedent (premise): x is A’ ¨ Implication: if x is A then y is B ¨ Consequence: y is B’ A 'o RA® B = B ' 5 Implication n Let and be two fuzzy sets in and , respectively n Implication is a relation defined by → ≜ ⊗ , ¨ where ⊗ is the tensor (outer) product of the vectors using the logical operator (∧) n Implication functions ¨ I(x,y) = min(x, y) Mamdani ¨ I(x,y) = max(1-x, y) Dilne, Zadeh ¨ I(x,y) = xy Larsen 6 3  Implication example n Rule ¨ “If temperature is high, then humidity is fairly high.” n Fuzzy variables ¨ tÎUt = {20,30,40} hÎUh = {20,50,70,90} n Fuzzy sets ¨ HT Í Ut mHT(t) = [0.1, 0.5, 0.9]T ¨ FHH Í Uh mFHH(h) = [0.2, 0.6, 0.7 1]T n Fuzzy rule ¨ R(t,h): if t is HT then h is FHH 7 Implication example n mHT(t) = [0.1, 0.5, 0.9]T n mFHH(h) = [0.2, 0.6, 0.7 1]T n → = HT ⊗ FHH 0.1 0.1 0.1 0.1 n → = 0.2 0.5 0.5 0.5 0.2 0.6 0.7 0.9 8 4  Implication example n According to the rule what is the humidity if temperature is fairly high? ¨t = FHT, FHT Í Ut n mFHT(t) = mHT2(t) = [0.01, 0.25, 0.81]T 9 Implication example n ℎ = ∘ → = ∘ → 0.1 0.1 0.1 0.1 = 0.01 0.25 0.81 ∘ 0.2 0.5 0.5 0.5 0.2 0.6 0.7 0.9 = 0.2 0.6 0.7 0.81 10 5  Single rule General fuzzification A®B A' ¾¾¾ Singleton fuzzification B' 11 Superposition of rules A1 ® B1 A'1 A2 ® B2 A'2 ¾¾¾¾¾ B'1, B'2 ¾¾¾¾¾ B'' 12 6  Multiple antecedents A1 Ù A2 ® B A'1, A'2 ¾¾¾¾¾¾ B' 13 Multiple Rules – Fuzzy-Fuzzy 1 A1 A¢ 1 B¢ B1 1 C1 a1 0 0 0 u v w 1 1 A¢ 1 B¢ 1 0 A2 B2 C2 a2 0 0 0 u v min w 14 7  Multiple Rules – Crisp-Fuzzy 1 1 1 a1 C1¢ 0 0 0 u v w 1 C1¢ C2¢ 1 1 1 0 a2 C2¢ 0 0 0 u v min w u0 v0 15 Defuzzification n Converting fuzzy set to crisp data ¨ Mean of maxima (MOM) n : set of points with maximum membership value ∑ = m(y) ¨ Center of area (COA) 1 yBOA ∑ yLOM = ySOM ∑ ¨ Bisectorof Area (BOA) 0 ¨ Smallest of Maximum (SOM) yMOM yCOA y ¨ Largest of Maximum (LOM) 16 8  Model of a fuzzy system Rule 1 (Fuzzy) x is A1 w1 y is B1 Rule 2 X Fuzzy (Fuzzy) x is A2 w2 y is B2 Defuzzifier (Fuzzy) Composition Rule r x is Ar wr y is Br y (Fuzzy) (Crisp) 17 Ingredients of a fuzzy system n Normalization of universes n Fuzzification of crisp input data n Fuzzy inference n Defuzzification n Denormalization 18 9