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Extension principle Fuzzy relations – linguistic variable, linguistic value Fuzzy ifthen rules – other aspect of fuzzy relations Compositional rule of inference Fuzzy reasoning A general procedure for extending crisp domains of mathematical expressions to fuzzy domains. Fuzzifying crisp functions is called an extension principle. Application of the extension principle to fuzzy sets with discrete universes Let A = 0.1 2+0.4 1+0.8 0+0.9 1+0.3 2 and f(x) = x2 – 3 Applying the extension principle, we obtain: B = 0.1 1+0.4 2+0.8 3+0.9 2+0.3 1 = 0.8 3+(0.4V0.9) 2+(0.1V0.3) 1 = 0.8 3+0.9 2+0.3 1 where “V” represents the “max” operator Same reasoning for continuous universes
Fuzzy Rules and Fuzzy Reasoning Chap 3: Fuzzy Rules and Fuzzy Reasoning Outline Extension principle Fuzzy relations – linguistic variable, linguistic value Fuzzy if-then rules – other aspect of fuzzy relations Compositional rule of inference Fuzzy reasoning Extension Principle • A general procedure for extending crisp domains of mathematical expressions to fuzzy domains • Fuzzifying crisp functions is called an extension principle Extension Principle A is a fuzzy set on X : A = µ A ( x1 ) / x1 + µ A ( x2 ) / x2 ++ µ A ( x n ) / x n The image of A under f( ) is a fuzzy set B = f(A): B = µ B ( x1 ) / y1 + µ B ( x2 ) / y2 ++ µ B ( x n ) / y n where yi = f(xi), i = to n If f( ) is a many-to-one mapping, then µ B ( y ) = max µ A ( x ) x= f −1 ( y) Example Application of the extension principle to fuzzy sets with discrete universes Let A = 0.1 / -2+0.4 / -1+0.8 / 0+0.9 / 1+0.3 / and f(x) = x2 – Applying the extension principle, we obtain: B = 0.1 / 1+0.4 / -2+0.8 / -3+0.9 / -2+0.3 /1 = 0.8 / -3+(0.4V0.9) / -2+(0.1V0.3) / = 0.8 / -3+0.9 / -2+0.3 / where “V” represents the “max” operator - Same reasoning for continuous universes Fuzzy Relations – fuzzy set with various dimensional membership function A fuzzy relation R is a 2D MF: R = {(( x , y ), µ R ( x , y ))|( x , y ) ∈ X ì Y} Examples: x is close to y (x and y are numbers) • x depends on y (x and y are events) • x and y look alike (x, and y are persons or objects) • If x is large, then y is small (x is an observed reading and Y is a corresponding action) Examples - Let X = Y = IR+ and R(x,y) = “y is much greater than x” The MF of this fuzzy relation can be subjectively defined as: y−x , if y > x µ R ( x, y ) = x + y + , if y ≤ x if X = {3,4,5} & Y = {3,4,5,6,7} Then R can be Written as a matrix: 0 0.111 0.200 0.273 0.333 R = 0 0.091 0.167 0.231 0.077 0.143 0 where R{i,j} = µ[xi, yj] – x is close to y (x and y are numbers) – x depends on y (x and y are events) – x and y look alike (x and y are persons or objects) – If x is large, then y is small (x is an observed reading and Y is a corresponding action) Max-Min Composition The max-min composition of two fuzzy relations R1 (defined on X and Y) and R2 (defined on Y and Z) is Properties: • Associativity: • Distributivity over union: • Week distributivity over intersection: • Monotonicity: µ R1 R2 ( x , z ) = ∨[µ R1 ( x , y ) ∧ µ R2 ( y , z )] y R (S T ) = ( R S ) T R (S T ) = ( R S ) ( R T ) R ( S T ) ⊆ ( R S ) ( R T ) S ⊆ T ⇒ (R S) ⊆ (R T ) Max-Star Composition In general, we have max-* composition: where * is a T-norm operator For Example, µ R R ( x , z ) = ∨[µ R ( x , y ) * µ R ( y , z )] Max-product composition: y µ R R ( x , z ) = ∨[µ R ( x , y )µ R ( y , z )] y • Contrast intensification the operation of contrast intensification on a linguistic value A is defined by 2 A if ≤ µ A ( x) ≤ 0.5 INT( A ) = ơ2( ơA ) if 0.5 A ( x ) ≤ - INT increases the values of µA(x) which are greater than 0.5 & decreases those which are less than or equal to 0.5 - Contrast intensification has effect of reducing the fuzziness of the linguistic value A Orthogonality A term set T = t1,…, tn of a linguistic variable x on the universe X is orthogonal if: n ∑ µ t i (x) = 1, ∀x ∈ X i =1 Where the ti’s are convex & normal fuzzy sets defined on X Fuzzy If-Then Rules General format: If x is A then y is B Examples: • If pressure is high, then volume is small • If the road is slippery, then driving is dangerous • If a tomato is red, then it is ripe • If the speed is high, then apply the brake a little Fuzzy If-Then Rules Two ways to interpret “If x is A then y is B”: y A coupled with B B y A entails B B x A x A Fuzzy If-Then Rules Two ways to interpret “If x is A then y is B”: • A coupled with B: (A and B) R = A→ B = Aì B = ~ A ( x ) ∗ µ B ( y )|( x , y ) • A entails B: (not A or B) - Material implicationA → B = ¬A ∪ B - Propositional calculus A → B = ¬A ∪ ( A ∩ B ) - Extended propositional calculus A → B = ( ¬A ∩ ¬B ) A ∪ B - Generalization of modus ponens sup{c|u(x) * c