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The Concept of Fuzzy Theory

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Title: The Concept of Fuzzy Theory


1
The Concept of Fuzzy Theory

2
????
  • ???????????,?????????????????????????????????????
    ??
  • ?????????????????????????????????,??????????
  • ??????????????????????
  • ??????????????????????????????????????????????????
    ???????????????????

3
?????
  • ?????(Fuzzy Set Theory)?1965???????????????(L.A.Za
    deh)?????
  • ???????????????????(uncertainty)????(fuzziness)??,
    ???????????????
  • ??????FA(X)???0?1??,??????X????A???????????

4
???
  • ??????????,?????? (x)U?0,1,??????
  • ???????
  • ???????
  • ?????x0,?? (x0)1?
  • ?????????????,???????,????????????(Triangular
    Fuzzy Number,TFN),??????(Trapezoidal Fuzzy
    Number,TrFN)?

5
?????
µ(x)  
               
 

1
x
0
a
b
c
d
?????
6
FEATURES OF THE MEMBERSHIP FUNCTION(1)
  • All information contained in a fuzzy set is
    described by its membership function.
  • Core
  • Support
  • Boundaries
  • Normal

7
FEATURES OF THE MEMBERSHIP FUNCTION(2)
  • Convex
  • Intersection
  • Crossover points
  • Define fuzzy set A has value equal to 0.5
  • Height
  • Fuzzy number
  • Most common forms of MF are those that are normal
    and convex.

8
FUZZIFICATION(1)
  • Fuzzification is the process of making a crisp
    quantity fuzzy.
  • Digital voltmeter(???)
  • Range of errors

9
MEMBERSHIP VALUE ASSIGNMENTS
  • Intuition
  • Inference
  • Rank ordering
  • Angular fuzzy sets
  • Neural networks
  • Genetic algorithms
  • Inductive reasoning
  • Soft partitioning (ch11)
  • Meta rules (not addressed in this book)
  • Fuzzy statistics (not addressed in this book)

10
Intuition
  • It is simply derived from the capacity of humans
    to develop membership functions through their own
    innate intelligence and understanding.

11
Inference
  • In the inference method we use knowledge to
    perform deductive reasoning.
  • Triangle
  • 4.2 identification of a triangle
  • 4.3 approximate isosceles triangle (?????)
  • 4.4 right triangle(?????)
  • 4.5 intersection of the isosceles and right
    triangle
  • 4.6 equilateral triangle (????)
  • 4.7 all triangular shapes other than I,R and E
  • Example 4.1.

12
Rank Ordering(1)
  • Assessing preference
  • Example4.2
  • 1000 people
  • Pairwise preference among five colors.
  • Xred,organge,yellow,green,blue

13
Rank Ordering(2)
  • Ex4.2 Membership function for best color

14
For example, the membership function of
linguistic variable S
15
????????
  • ???????
  • ????? (a,b,c)
  • ????? (a,b,c,d)
  • ?????
  • ???????

16
???????(1/2)
  1. ???????
  2. ???????
  3. ????????

17
???????(2/2)
  1. ??????? (OR Operation)
  2. ??????? (AND Operation)

18
?????????
  • ??????? (a1,a2,a3)? (b1,b2,b3)???????
  • ???? ( )
  • (a1,a2,a3)()(b1,b2,b3)(a1 b1 , a2 b2 , a3
    b3)
  • ???? ( - )
  • (a1,a2,a3)( - ) (b1,b2,b3)(a1 - b1 , a2 - b2 ,
    a3 - b3)
  • ???? ( )
  • (a1,a2,a3)( )(b1,b2,b3)(a1 b1 , a2 b2 , a3
    b3)
  • ???? ( ) ?b1,b2,b3? 0
  • (a1,a2,a3)( )(b1,b2,b3)(a1b3, a2 b2 , a3b1)

19
?????a-cut
  • ?????????a-cut?? ,???? ,???X??,
  • ?????a-cut,???X????????
  • ?????????a??????????
  • a??????(confidence level),???????,a-cut???????,???
    ??????,???????(crisp set) ?


20
????(Defuzzification)
  • ???????,???????????????,????????????????,?????????
    ???,???????????,??????????????
  • ???(Center of Gravity Method)
  • ????????(Mean of Maximum Method)
  • ???
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