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### Fuzzy relations

• 1. FUZZY RELATIONS, FUZZY GRAPHS, AND FUZZY ARITHMETIC
• 2. INTRODUCTION 3 Important concepts in fuzzy logic • Fuzzy Relations • Fuzzy Graphs • Extension Principle -- Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall } Form the foundation of fuzzy rules basis of fuzzy Arithmetic - This is what makes a fuzzy system tick!
• 3. Fuzzy Relations • Generalizes classical relation into one that allows partial membership – Describes a relationship that holds between two or more objects • Example: a fuzzy relation “Friend” describe the degree of friendship between two person (in contrast to either being friend or not being friend in classical relation!) Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
• 4. Fuzzy Relations • A fuzzy relation is a mapping from the Cartesian space X x Y to the interval [0,1], where the strength of the mapping is expressed by the membership function of the relation µ (x,y) • The “strength” of the relation between ordered pairs of the two universes is measured with a membership function expressing various “degree” of strength [0,1] Fuzzy Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill ˜R ˜R
• 5. Fuzzy Cartesian Product Let be a fuzzy set on universe X, and be a fuzzy set on universe Y, then Where the fuzzy relation R has membership function Fuzzy Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill ˜A × ˜B = ˜R ⊂ X × Y µ˜R (x, y) = µ ˜Ax˜B (x, y) = min(µ ˜A (x),µ ˜B (y)) ˜A ˜B
• 6. Fuzzy Cartesian Product: Example Let defined on a universe of three discrete temperatures, X = {x1,x2,x3}, and defined on a universe of two discrete pressures, Y = {y1,y2} Fuzzy set represents the “ambient” temperature and Fuzzy set the “near optimum” pressure for a certain heat exchanger, and the Cartesian product might represent the conditions (temperature- pressure pairs) of the exchanger that are associated with “efficient” operations. For example, let Fuzzy Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill ˜A ˜B ˜A ˜B ˜A = 0.2 x1 + 0.5 x2 + 1 x3 and ˜B = 0.3 y1 + 0.9 y2 } ˜A × ˜B = ˜R = x1 x2 x3 0.2 0.2 0.3 0.5 0.3 0.9         y1 y2
• 7. Fuzzy Composition Suppose is a fuzzy relation on the Cartesian space X x Y, is a fuzzy relation on the Cartesian space Y x Z, and is a fuzzy relation on the Cartesian space X x Z; then fuzzy max-min and fuzzy max-product composition are defined as Fuzzy Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill ˜R ˜S ˜T ˜T = ˜Ro ˜S max − min µ˜T (x,z) = ∨ y∈Y (µ ˜R (x,y) ∧ µ˜S (y,z)) max − product µ˜T (x,z) = ∨ y∈Y (µ ˜R (x,y) • µ˜S (y, z))
• 8. Fuzzy Composition: Example (max-min) Fuzzy Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill X = {x1, x2}, µ˜T (x1,z1) = ∨ y∈Y (µ ˜R (x1,y) ∧ µ˜S (y,z1)) = max[min( 0.7,0.9),min(0.5, 0.1)] = 0.7 Y ={y1, y2},and Z = {z1,z2, z3} Consider the following fuzzy relations: ˜R = x1 x2 0.7 0.5 0.8 0.4     y1 y2 and ˜S = y1 y2 0.9 0.6 0.5 0.1 0.7 0.5     z1 z2 z3 Using max-min composition, }˜T = x1 x2 0.7 0.6 0.5 0.8 0.6 0.4     z1 z2 z3
• 9. Fuzzy Composition: Example (max-Prod) Fuzzy Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill X = {x1, x2}, µ˜T (x2, z2 ) = ∨ y∈Y (µ ˜R (x2 , y) • µ˜S (y, z2)) = max[( 0.8,0.6),(0.4, 0.7)] = 0.48 Y ={y1, y2},and Z = {z1,z2, z3} Consider the following fuzzy relations: ˜R = x1 x2 0.7 0.5 0.8 0.4     y1 y2 and ˜S = y1 y2 0.9 0.6 0.5 0.1 0.7 0.5     z1 z2 z3 Using max-product composition, }˜T = x1 x2 .63 .42 .25 .72 .48 .20     z1 z2 z3
• 10. Application: Computer Engineering Fuzzy Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill Problem: In computer engineering, different logic families are often compared on the basis of their power-delay product. Consider the fuzzy set F of logic families, the fuzzy set D of delay times(ns), and the fuzzy set P of power dissipations (mw). If F = {NMOS,CMOS,TTL,ECL,JJ}, D = {0.1,1,10,100}, P = {0.01,0.1,1,10,100} Suppose R1 = D x F and R2 = F x P ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ˜R1 = 0.1 1 10 100 0 0 0 .6 1 0 .1 .5 1 0 .4 1 1 0 0 1 .2 0 0 0           N C T E J and ˜R2 = N C T E J 0 .4 1 .3 0 .2 1 0 0 0 0 0 .7 1 0 0 0 0 1 .5 1 .1 0 0 0             .01 .1 1 10 100
• 11. Application: Computer Engineering (Cont) Fuzzy Logic with Engineering Applications: Timothy J. Ross, McGraw-Hill We can use max-min composition to obtain a relation between delay times and power dissipation: i.e., we can compute or˜R3 = ˜R1 o ˜R2 µ˜R3 = ∨(µ ˜R1 ∧ µ ˜R2 ) ˜R3 = 0.1 1 10 100 1 .1 0 .6 .5 .1 .1 .5 1 .5 .2 1 .7 1 0 .2 .4 1 .3 0             .01 .1 1 10 100
• 12. Application: Fuzzy Relation Petite Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall Fuzzy Relation Petite defines the degree by which a person with a specific height and weight is considered petite. Suppose the range of the height and the weight of interest to us are {5’, 5’1”, 5’2”, 5’3”, 5’4”,5’5”,5’6”}, and {90, 95,100, 105, 110, 115, 120, 125} (in lb). We can express the fuzzy relation in a matrix form as shown below: ˜P = 5' 5'1" 5' 2" 5' 3" 5' 4" 5' 5" 5' 6" 1 1 1 1 1 1 .5 .2 1 1 1 1 1 .9 .3 .1 1 1 1 1 1 .7 .1 0 1 1 1 1 .5 .3 0 0 .8 .6 .4 .2 0 0 0 0 .6 .4 .2 0 0 0 0 0 0 0 0 0 0 0 0 0                 90 95 100 105 110 115 120 125
• 13. Application: Fuzzy Relation Petite Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall ˜P = 5' 5'1" 5' 2" 5' 3" 5' 4" 5' 5" 5' 6" 1 1 1 1 1 1 .5 .2 1 1 1 1 1 .9 .3 .1 1 1 1 1 1 .7 .1 0 1 1 1 1 .5 .3 0 0 .8 .6 .4 .2 0 0 0 0 .6 .4 .2 0 0 0 0 0 0 0 0 0 0 0 0 0                 90 95 100 105 110 115 120 125 Once we define the petite fuzzy relation, we can answer two kinds of questions: • What is the degree that a female with a specific height and a specific weight is considered to be petite? • What is the possibility that a petite person has a specific pair of height and weight measures? (fuzzy relation becomes a possibility distribution)
• 14. Application: Fuzzy Relation Petite Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall Given a two-dimensional fuzzy relation and the possible values of one variable, infer the possible values of the other variable using similar fuzzy composition as described earlier. Definition: Let X and Y be the universes of discourse for variables x and y, respectively, and xi and yj be elements of X and Y. Let R be a fuzzy relation that maps X x Y to [0,1] and the possibility distribution of X is known to be Πx(xi). The compositional rule of inference infers the possibility distribution of Y as follows: max-min composition: max-product composition: ΠY (yj ) = max xi (min(ΠX (xi),ΠR (xi, yj ))) ΠY (yj ) = max xi (ΠX (xi) × ΠR(xi, yj ))
• 15. Application: Fuzzy Relation Petite Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall Problem: We may wish to know the possible weight of a petite female who is about 5’4”. Assume About 5’4” is defined as About-5’4” = {0/5’, 0/5’1”, 0.4/5’2”, 0.8/5’3”, 1/5’4”, 0.8/5’5”, 0.4/5’6”} Using max-min compositional, we can find the weight possibility distribution of a petite person about 5’4” tall: Πweight (90) = (0 ∧1) ∨ (0 ∧1) ∨ (.4 ∧ 1) ∨ (.8 ∧1) ∨ (1 ∧ .8) ∨ (.8 ∧ .6) ∨ (.4 ∧ 0) = 0.8 ˜P = 5' 5'1" 5'2" 5'3" 5'4" 5'5" 5'6" 1 1 1 1 1 1 .5 .2 1 1 1 1 1 .9 .3 .1 1 1 1 1 1 .7 .1 0 1 1 1 1 .5 .3 0 0 .8 .6 .4 .2 0 0 0 0 .6 .4 .2 0 0 0 0 0 0 0 0 0 0 0 0 0                 90 95 100 105 110 115 120 125 Similarly, we can compute the possibility degree for other weights. The final result is Πweight = {0.8 /90,0.8 /95,0.8 /100,0.8/105,0.5 /110,0.4 /115, 0.1/120,0 /125}
• 16. Fuzzy Graphs Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall • A fuzzy relation may not have a meaningful linguistic label. • Most fuzzy relations used in real-world applications do not represent a concept, rather they represent a functional mapping from a set of input variables to one or more output variables. • Fuzzy rules can be used to describe a fuzzy relation from the observed state variables to a control decision (using fuzzy graphs) • A fuzzy graph describes a functional mapping between a set of input linguistic variables and an output linguistic variable.
• 17. Extension Principle Neuro-Fuzzy and Soft Computing, J. Jang, C. Sun, and E. Mitzutani, Prentice Hall • Provides a general procedure for extending crisp domains of mathematical expressions to fuzzy domains. • Generalizes a common point-to-point mapping of a function f(.) to a mapping between fuzzy sets. Suppose that f is a function from X to Y and A is a fuzzy set on X defined as A = µA(x1) /(x1) + µA(x2 )/(x2 ) + ..... + µA(xn )/(xn ) Then the extension principle states that the image of fuzzy set A under the mapping f(.) can be expressed as a fuzzy set B, B = f(A) = µA(x1) /(y1) + µA(x2 ) /(y2 ) + ..... + µA(xn )/(yn ) Where yi =f(xi), i=1,…,n. If f(.) is a many-to-one mapping then µB(y) = max x= f −1 (y) µA (x)
• 18. Extension Principle: Example Neuro-Fuzzy and Soft Computing, J. Jang, C. Sun, and E. Mitzutani, Prentice Hall Let A=0.1/-2+0.4/-1+0.8/0+0.9/1+0.3/2 and f(x) = x2 -3 Upon applying the extension principle, we have B = 0.1/1+0.4/-2+0.8/-3+0.9/-2+0.3/1 = 0.8/-3+max(0.4, 0.9)/-2+max(0.1, 0.3)/1 = 0.8/-3+0.9/-2+0.3/1
• 19. Extension Principle: Example Neuro-Fuzzy and Soft Computing, J. Jang, C. Sun, and E. Mitzutani, Prentice Hall Let µA(x) = bell(x;1.5,2,0.5) and f(x) = { (x-1)2 -1, if x >=0 x, if x <=0
• 20. Extension Principle: Example Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall Around-4 = 0.3/2 + 0.6/3 + 1/4 + 0.6/5 + 0.3/6 and Y = f(x) = x2 -6x +11
• 21. Arithmetic Operations on Fuzzy Numbers Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall Applying the extension principle to arithmetic operations, we have Fuzzy Addition: Fuzzy Subtraction: Fuzzy Multiplication: Fuzzy Division: µA+ B(z) = ⊕ x ,y x+ y= z µA(x) ⊗ µB (y) µA− B(z) = ⊕ x ,y x− y= z µA(x) ⊗ µB (y) µA× B(z) = ⊕ x ,y x× y=z µA(x) ⊗ µB (y) µA / B(z) = ⊕ x ,y x / y= z µA(x) ⊗ µB (y)
• 22. Arithmetic Operations on Fuzzy Numbers Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall Let A and B be two fuzzy integers defined as A = 0.3/1 + 0.6/2 + 1/3 + 0.7/4 + 0.2/5 B = 0.5/10 + 1/11 + 0.5/12 Then F(A+B) = 0.3/11+ 0.5/12 + 0.5/13 + 0.5/14 +0.2/15 + 0.3/12 + 0.6/13 + 1/14 + 0.7/15 + 0.2/16 + 0.3/13 + 0.5/14 + 0.5/15 + 0.5/16 +0.2/17 Get max of the duplicates, F(A+B) =0.3/11 + 0.5/12 + 0.6/13 + 1/14 + 0.7/15 +0.5/16 + 0.2/17
• 23. Summary Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall • A fuzzy relation is a multidimensional fuzzy set • A composition of two fuzzy relations is an important technique • A fuzzy graph is a fuzzy relation formed by pairs of Cartesian products of fuzzy sets • A fuzzy graph is the foundation of fuzzy mapping rules • The extension principle allows a fuzzy set to be mapped through a function • Addition, subtraction, multiplication, and division of fuzzy numbers are all defined based on the extension principle
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