Expert System: Fuzzyness

Question 1

What is a fuzzy set?

Answer 1

an element x belongs to a set with a certain degree of membership

Fuzzy set X= {(x1,0.4), (x2,1), (x3,0.1), (x4,0.6), (x5,0.4)}

It is a set of pairs {(xi, μX(xi)},

where μX(xi) is the membership function of element xi in the set X.

alternative representation

tall men 1⁄4 (0/180, 0.5/185, 1/190) or tall men 1⁄4 (0/180, 1/190)

Question 2

What is a Crisp Set?

Answer 2

an element x either belong to X or does not belong to X

Crisp set X= {x1,x2,x3,x4,x5}

Question 3

Membership function

Answer 3

• the fuzzy theory, fuzzy set A of universe X is defined by function μA(x) called the membership function of set A
• This set allows a continuum of possible choices.
• For any element x of universe X, membership function μA(x) equals the degree to which x is an element of set A.
• This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element x in set A.

Question 4

What is Fuzzy inference?

Answer 4

process of mapping from a given input to an output, using the theory of fuzzy sets.

Question 6

What is a LINGUISTIC VARIABLE?

Answer 6

A LINGUISTIC VARIABLE is a fuzzy variable.

John is tall  The linguistic variable John takes the value tall.

Question 7

What is the universe of discourse?

Answer 7

The range of all possible “real-world” values of a linguistic
variable represents the universe of discourse of that variable.

short, tall, average

Question 8

What are hedges?

Answer 8

Hedges are terms that modify the shape of fuzzy sets:

very,
more, quite, less, slightly, likely, very likely

Question 9

What is Defuzzification?

Answer 9

• Fuzziness helps us to evaluate the rules but the final output of a fuzzy system has to be crisp number
• The output of the defuzzification process is the aggregate output fuzzy
  set and the output is a single number (mean of maximum method; centre of gravity method)

Question 10

Name the two most popular defuzzification methods

Answer 10

• Centroid Defuzzification Method ( center of gravity)
• Sugeno Style inference

Question 11

What is the Centroid Defuzzification Method (COG)?

Answer 11

It finds the point where a vertical line would slice the aggregate set into two equal masses.

Question 12

How does one calcuate the COG?

Answer 12

In theory, the COG is calculated over a continuum of points in the aggregate output membership function,

but in practice, a reason- able estimate can be obtained by calculating it over a sample of points,