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Hedge. Young, middle aged, old → Primary term ; not → negation ; Very, more or less, quite, extremely → hedges ; And, or, either, neither → connection. Hedge … continued.

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hedge
Hedge

Young, middle aged, old → Primary term ; not → negation ;

Very, more or less, quite, extremely → hedges ;

And, or, either, neither → connection

hedge continued
Hedge…continued
  • Concentration and dilation of linguistic values let A be a linguistic value characterized by a fuzzy set with membership function μA(•), then AK is interpreted as a modified version of the original linguistic value expressed

in particular, the operation of concentration is defined as

CON (A) = A2

while the operation of dilation is defined as

DIL (A) = A0.5

other appropriate values of K are possible

hedge continued2
Hedge…continued
  • Negative and connectives
  • Composite linguistic term

not young and not old

hedge continued3
Hedge…continued

young but not too young

fuzzy if then rules
Fuzzy If-Then Rules
  • Or called fuzzy rules
  • A fuzzy inference system consists of a set of fuzzy if-then rules
  • Have been applied to
    • Control systems, decision-making, pattern recognition, system modeling
  • a fuzzy if-then rule associates
    • A condition described using linguistic variables and fuzzy sets to a conclusion

Ex.If a tomato is red, then it is ripe

fuzzy if then rules1
Fuzzy If-Then Rules
  • Fuzzy rule-based inference can be understood from several points of view
    • (i) a multi-expert decision making metaphor
    • (ii) mathematically, fuzzy rule-based inference can be viewed as an interpolation scheme
    • (iii) from a logic view point, fuzzy rule-based inference is a generalization of a logic reasoning scheme – modes ponens
fuzzy if then rules2
Fuzzy If-Then Rules
  • The elastic condition and the consequent of a fuzzy rule are often described by words (ie. Linguistic labels/ values/terms)

Ex. Linguistic variable:〝age〞

Term set:

T:{young, not young, very young, not very young, ….

middle aged, not middle aged, ……..

old, not old, very old, more or less old, not very old,

not very young and very old, …….}

Use 〝age is young〞to denote the assignment of the linguistic value 〝young〞to the linguistic variable 〝age〞

fuzzy if then rules3
Fuzzy If-Then Rules
  • Fuzzy if-then rules (fuzzy rules)
    • a fuzzy if-then assume the form if x is A, then y is B

A, B:linguistic values defined by fuzzy sets on universes X and Y

x, y: linguistic variables

“xis A”: antecedent / premise

“y is B”:consequence / conclusion

Ex. If pressure is high, then volume is small if the speed is high, the apply the brake a little

fuzzy if then rules4
Fuzzy If-Then Rules
  • Abbreviated as A→B
    • Such a expression describes a relation between two variables x and y

This suggests that a fuzzy if-then rule be defined as a binary fuzzy relation on the product space X * Y

  • Classical implication A→B
fuzzy if then rules5
Fuzzy If-Then Rules
  • Fuzzy implication (A coupled with B)
fuzzy if then rules6
Fuzzy If-Then Rules
  • Ex. If x is medium, then y is small

X={2,3,4,5,6,7,8,9}

Y={1,2,3,4,5,6}

Apply Rm:

Medium ≡ 0.1/2 + 0.3/3 + 0.7/4 + 1/5 + 1/6 + 0.7/7 + 0.5/8 + 0.2/9

small ≡ 1/1 + 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6

fuzzy if then rules7
Fuzzy If-Then Rules

Apply Rp:

Apply Rbp:

(2,1):0 ν(0.1+1-1)=0.1

(2,2):0 ν(0.1+1-1)=0.1

(2,3):0 ν(0.1+0.9-1)=0

(2,4):0 ν(0.1+0.6-1)=0.0

(4,3):0 ν(0.7+0.9-1)=0.6

  • Fuzzy implication (A entails B)
    • 1 Λ (1-a+b)
    • (1-a) ν b
approximate reasoning
Approximate reasoning
  • Modus ponens (離斷律)
    • Basic rule of inference in traditional logic

Premise 1 (fact) : x is A

Premise 2 (rule) : if x is A then y is B

Consequence (conclusion):y is B

  • Ex.

fact : The tomato is red

rule : if the tomato is red then the tomato is ripe

conclusion:the tomato is ripe

approximate reasoning1
Approximate reasoning
  • Generalized modus ponens (GMP)
    • How about〝the tomato is more or less red〞?

Premise 1 (fact) : x is A’

Premise 2 (rule) : if x is A then y is B

Consequence (conclusion):y is B’

Where A’ is dose to A,B’ is close to B. when A, B, A’ and B’ are fuzzy sets, such an inference procedure is called approximate reasoning (fuzzy reasoning)

approximate reasoning2
Approximate reasoning
  • Modus tolens (逆斷律)

Premise 1 (fact) : y is not B

Premise 2 (rule) : if x is A then y is B

Consequence (conclusion):x is not A

  • Ex. (P.12)

Premise 1 (fact) : x is small

Premise 2 (rule) : if x is medium then y is small

conclusion:? (y is B’)

approximate reasoning3
Approximate reasoning

X={2,3,4,5,6,7,8,9} Y={1,2,3,4,5,6}

smallx = 1/1 + 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6 + 0/7 + 0/8 + 0/9

Mediumx = 0.1/2 + 0.3/3 + 0.7/4 + 1/5 + 1/6 + 0.7/7 + 0.5/8 + 0.2/9

(x is small) ∩ (x is medium)

= 0.1/2 + 0.3/3 + 0.6/4 + 0.3/5 + 0.1/6 + 0/7 + 0/8 + 0/9

smally = 1/1 + 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6

B’ = 0.6/1 + 0.6/2 + 0.6/3 + 0.6/4 + 0.3/5 + 0.1/6

approximate reasoning4
Approximate reasoning

2→1:0.1 (0.1 ν1)

3→1:0.3 (0.13ν1)

4→1:0.6 (0.6 ν0.9)

5→1:0.3

6→1:0.1

7→1:0

8→1:0

9→1:0

ν =>0.6

approximate reasoning5
Approximate reasoning
  • Let A, A’ and B be fuzzy sets of X, X and Y, respectively. Assume that the fuzzy implication A→B is expressed as a fuzzy relation R on X × Y. Then the fuzzy set B induced by〝x is A’〞and the fuzzy rule〝if x is A then y is B〞is defined by
approximate reasoning6
Approximate reasoning
  • Note:a fuzzy rule is represented mathematically as fuzzy relations formed by the Cartesian product of the variables referred in the rule’s if-part and then-part.
  • Ex. If x is A, then y is B =>μR(x,y) = μA×B (x,y)

If we use〝min〞operator for the Cartesian product, R becomes

μR(x,y) = min {μA(x), μB(y)}

approximate reasoning7
Approximate reasoning
  • Single rule with multiple antecedents
    • 〝if x is A and y is B then z is C〞

Premise 1 (fact) : x is A’ and y is B’

Premise 2 (rule) : if x is A and y is B then z is C

Consequence (conclusion):z is C’

≡ μA×B→C (x,y,z) = μR(x,y,z)

Firing strength

approximate reasoning8
Approximate reasoning

Or, C’ = (A’ × B’)。(A × B→ C)

W1, W2:degree of compatibility

W1Λ W2:firing strength

approximate reasoning9
Approximate reasoning
  • Multiple rules with multiple antecedents

Premise 1 (fact) : x is A’ and y is B’

Premise 2 (rule 1) : if x is A1 and y is B1then z is C1

Premise 3 (rule 2) : if x is A2 and y is B2then z is C2

Consequence (conclusion):z is C’

Let

R1 = A1 × B1→C1

R2 = A2 × B2→C2

C’ = (A’ × B’)。(R1∪R2)

= [(A’ × B’)。R1]∪[(A’ × B’)。R2]

= C’1 ∪C’2

fuzzy inference systems
Fuzzy inference systems
  • The basic fuzzy inference system can take either fuzzy inputs or crisp inputs, the outputs are almost always fuzzy sets
  • But sometimes it is necessary to have crisp output, especially in a situation where a fuzzy inference system is used as a controller

=> Need a method of defuzzification to extract a crisp value that best represents a fuzzy set

fuzzy inference systems1

Rule 1

(Fuzzy)

W1

yis B1

Rule 2

(Fuzzy)

(Crisp or Fuzzy)

W2

yis B2

(Fuzzy)

(Crisp)

Aggregator

Defuzzifier

y

‧‧‧

‧‧‧

‧‧‧

Rule r

(Fuzzy)

Wr

yis Br

Fuzzy inference systems
slide27
Mandani fuzzy models
    • If x is A1 and y is B1, then z is C1
    • If x is A2 and y is B2, then z is C2

△ max:T-conorn

mim:T-norn

slide28

△ max:T-conorn

product:T-norn

fuzzy inference systems2
Fuzzy inference systems
  • Defuzzification
    • The method a crisp value is extracted from a fuzzy set as a representative value
  • 5 methods
    • (1) COA (Centroid of area)
fuzzy inference systems3
Fuzzy inference systems
  • (2) BOA (Bisector of area)
  • (3) MOM
    • The average of the maximizing z at which the M.F. reach a maximum μx
fuzzy inference systems4
Fuzzy inference systems
  • (5) LOM (largest of maximum)
    • LOM = Zright
  • (4) SOM (smallest of maximum)
    • SOM = Zleft
the mamdani model
The Mamdani model

Ri:if x1is Ai1 and ….xr is Air, then y is ci

xj, j =1,….,r can be crisp or fuzzy