Fuzzy Rules and Fuzzy Reasoning Chapter 3 Fuzzy

Fuzzy Rules and Fuzzy Reasoning Chapter 3: Fuzzy Rules and Fuzzy Reasoning J. -S. Roger Jang (張智星) CS Dept. , Tsing Hua Univ. , Taiwan Modified by Dan Simon Cleveland State University

Fuzzy Rules and Fuzzy Reasoning Outline • Extension principle • Fuzzy relations • Fuzzy if-then rules • Compositional rule of inference • Fuzzy reasoning 2

Fuzzy Rules and Fuzzy Reasoning Extension Principle A is a fuzzy set on X : The image of A under f(. ) is a fuzzy set B: where yi = f(xi), for i = 1 to n. If f(. ) is a many-to-one mapping, then 3

Fuzzy Rules and Fuzzy Reasoning Example: Extension Principle Example 1 y = x 2 (x) 0 1 2 3 -1 0 1 2 3 (y) 0 1 4 (x) Example 2 4 9 0 1 4 9

Fuzzy Rules and Fuzzy Reasoning Fuzzy Relations A fuzzy relation R is a 2 D MF: 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 instrument reading and y is a corresponding control action) 5

Fuzzy Rules and Fuzzy Reasoning Example: x is close to y 6

Fuzzy Rules and Fuzzy Reasoning Example: X is close to Y 7

Fuzzy Rules and Fuzzy Reasoning Max-Min Composition The max-min composition of two fuzzy relations R 1 (defined on X and Y) and R 2 (defined on Y and Z): (max) (min) • Associativity: • Distributivity over union: • Weak distributivity over intersection: • Monotonicity: 8

Fuzzy Rules and Fuzzy Reasoning Max-Star Composition Max-product composition: In general, we have max * compositions: where * is a T-norm operator. 9

Fuzzy Rules and Fuzzy Reasoning Example 3. 4 – Max * Compositions R 1: x is relevant to y y= x=1 0. 1 x=2 0. 4 x=3 0. 6 y= 0. 3 0. 2 0. 8 R 2: y is relevant to z z=a y= 0. 9 y= 0. 2 y= 0. 5 z=b 0. 1 0. 3 0. 6 y= 0. 2 How relevant is x=2 to z=a? 10 0. 7 y= 0. 5 0. 8 0. 3 y= 0. 7 0. 9 0. 2

Fuzzy Rules and Fuzzy Reasoning Example 3. 4 (cont’d. ) x y z 1 0. 9 0. 4 0. 2 0. 5 0. 8 0. 9 0. 7 a 2 3 11 b

Fuzzy Rules and Fuzzy Reasoning Linguistic Variables A numerical variable takes numerical values: Age = 65 A linguistic variables takes linguistic values: Age is old A linguistic value is a fuzzy set. All linguistic values form a term set (set of terms): T(age) = {young, not young, very young, . . . middle aged, not middle aged, . . . old, not old, very old, more or less old, . . . not very young and not very old, . . . } 12

Fuzzy Rules and Fuzzy Reasoning Operations on Linguistic Values (very) Concentration: Dilation: (more or less) Contrast intensification: 13 intensif. m

Fuzzy Rules and Fuzzy Reasoning Linguistic Values (Terms) How are these derived from the above MFs? 14 complv. m

Fuzzy Rules and Fuzzy Reasoning Fuzzy If-Then Rules General format: If x is A then y is B This is interpreted as a fuzzy set Examples: • • 15 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 Rules and Fuzzy Reasoning Fuzzy If-Then Rules Two ways to interpret “If x is A then y is B” A is coupled with B: (x is A) (y is B) A entails B: (x is not A) (y is B) y y B B x A 16 x A

Fuzzy Rules and Fuzzy Reasoning Fuzzy If-Then Rules Example: if (profession is athlete) then (fitness is high) Coupling: Athletes, and only athletes, have high fitness. The “if” statement (antecedent) is a necessary and sufficient condition. Entailing: Athletes have high fitness, and nonathletes may or may not have high fitness. The “if” statement (antecedent) is a sufficient but not necessary condition. 17

Fuzzy Rules and Fuzzy Reasoning Fuzzy If-Then Rules Two ways to interpret “If x is A then y is B”: • A coupled with B: (A and B – T-norm) • A entails B: (not A or B) - Material implication - Propositional calculus - Extended propositional calculus - Generalization of modus ponens 18

Fuzzy Rules and Fuzzy Reasoning Fuzzy If-Then Rules Fuzzy implication A coupled with B (bell-shaped MFs, T-norm operators) Example: only fit athletes satisfy the rule 19 fuzimp. m

Fuzzy Rules and Fuzzy Reasoning Fuzzy If-Then Rules A entails B (bell-shaped MFs) Arithmetic rule: (x is not A) (y is B) (1 – x) + y Example: everyone except non-fit athletes satisfies the rule 20 fuzimp. m

Fuzzy Rules and Fuzzy Reasoning Compositional Rule of Inference Derivation of y = b from x = a and y = f(x): y y b b y = f(x) a 21 x a and b : points a and b : intervals y = f(x) : a curve Crisp : if x = a, then y=b y = f(x) : interval-valued function Fuzzy : if (x is a) then (y is b)

Fuzzy Rules and Fuzzy Reasoning Compositional Rule of Inference A is a fuzzy set of x and y = f(x) is a fuzzy relation: 22 cri. m

Fuzzy Rules and Fuzzy Reasoning Single rule with single antecedent Rule: if x is A then y is B Premise: x is A’, where A’ is close to A Conclusion: y is B’ Use max of intersection between A and A’ to get B’ A’ A B w X A’ x is A’ 23 Y B’ X y is B’ Y

Fuzzy Rules and Fuzzy Reasoning Single rule with multiple antecedents Rule: if x is A and y is B then z is C Premise: x is A’ and y is B’ Conclusion: z is C’ Use min of (A A’) and (B B’) to get C’ A’ A B’ B C w X A’ x is A’ 24 Y B’ X Z y is B’ C’ Y z is C’ Z

Fuzzy Rules and Fuzzy Reasoning Multiple rules with multiple antecedents Rule 1: if x is A 1 and y is B 1 then z is C 1 Rule 2: if x is A 2 and y is B 2 then z is C 2 Premise: x is A’ and y is B’ Conclusion: z is C’ Use previous slide to get C 1’ and C 2’ Use max of C 1’ and C 2’ to get C’ (next slide) 25

Fuzzy Rules and Fuzzy Reasoning Multiple rules with multiple antecedents A’ A 1 B’ B 1 C 1 w 1 X A’ A 2 Z Y B’ B 2 C 2 w 2 X A’ Z Y B’ C’ x is A’ 26 X y is B’ Y z is C’ Z

Fuzzy Rules and Fuzzy Reasoning: MATLAB Demo >> ruleview mam 21 (Matlab Fuzzy Logic Toolbox) 27

Fuzzy Rules and Fuzzy Reasoning Other Variants Some terminology: • Degrees of compatibility (match between input variables and fuzzy input MFs) • Firing strength calculation (we used MIN) • Qualified (induced) MFs (combine firing strength with fuzzy outputs) • Overall output MF (we used MAX) 28