L2 Fuzzy Theories PDF

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L1 Intro 2Fuzzy Logic...

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Lecture 2. Fundamentals of Fuzzy Logic  Fuzzy sets and Membership functions  Fuzzy operations  Fuzzy rules

FUZZY SETS & MEMBERSHIP FUNCTIONS

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What is ‘fuzzy’ logic? (recap) Is it 150.5kg in weight or 149.5kg??? ……

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LOOK OUT!!!

•“IF falling object is heavier than 150kg, THEN give warning.” • “IF falling object is lighter than 150kg, THEN no warning.”

• “IF falling object is HEAVY, THEN give warning.” • “IF falling object is LIGHT, THEN no warning.”

-- Conventional Logic

-- Human Logic 7ENT1013 Artificial Intelligence

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Boolean logic vs. Fuzzy logic (recap) Degree of ‘tall’ Height (cm)

0

1

short

tall 180

Figure 2.1 (a) Two-valued Boolean logic

0

0.8

1

180

200

Degree of ‘tall’ Height (cm) 150

Figure 2.1 (b) Multi-valued fuzzy logic LM

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Crisp and fuzzy sets of “tall men” (recap) Degree of Membership 1.0

•

The x-axis represents the universe of discourse.

•

Elements of a fuzzy set are taken from a universe, the universe depending on the context.

•

The y-axis represents the membership value of the crisp/fuzzy set.

•

Every element is a member of the fuzzy set to some degree – this degree is given by the membership function µ(x).

•

In our case, the fuzzy set of ‘tall men’ maps height values into the corresponding membership values.

Crisp Set

0.8

Tall Men

0.6 0.4 0.2 0.0 150

160

170

180

190

200

210 Height, cm

Degree of Membership 1.0

Fuzzy Set

0.8 0.6 0.4 0.2 0.0 150

160

170

180

190

200

210 Height, cm

Plot of membership function LM

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Membership functions in Matlab (recap) 1.2 trapmf

gbellmf

trimf

gaussmf

gauss2mf

smf

1 0.8 0.6 0.4 0.2 0

1.2 zmf

psigmf

dsigmf

pimf

sigmf

1 0.8 0.6 0.4 0.2 0

Use Matlab help function to learn more. LM

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Steps to define a fuzzy set 1. First, we define the universe of discourse. In our “tall men” example, the universe of discourse is men’s heights: 150-210 cm.

2. Then we decide the possible fuzzy sets in the defined universe of discourse. For example, we obtain fuzzy sets of tall, short and average.

3. Finally, we determine the membership function for each fuzzy set in the defined universe of discourse. LM

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Crisp and fuzzy sets of short, average and tall men Degree of Membership 1.0

Crisp Sets

Short

0.8

Average

Tall

0.6 0.4 0.2 0.0 150

160

170

180

Degree of Membership 1.0

190

200

210 Height, cm

200

210

Fuzzy Sets

0.8 Short

0.6

Average

Tall

0.4 0.2

As you can see, a man who is 182 cm tall is a member of the ‘average men’ set with a degree of membership of 0.2, and at the same time, he is also a member of the ‘tall men’ set with a degree of 0.32.

0.0 150

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160

170

180

190

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Exercise: Define fuzzy sets that describe the forward speed and the turning speed of a car.

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Hedge • Hedges are terms that modify the shape of fuzzy sets. Hedges can be used as – All-purpose modifiers, such as very, quite or extremely. – Truth-values, such as quite true or mostly false. – Probabilities, such as likely or not very likely. – Quantifiers, such as most, several or few. – Possibilities, such as almost impossible or quite possible.

• As operations, hedges can create new fuzzy sets. For example, ‘very’ derives a subset of very tall men from the set of tall men. LM

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Representation of hedges in fuzzy logic Hedge

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Mathematical Expression

A little

[A ( x )]1.3

Slightly

[A ( x )]1.7

Very

[A ( x)

2

Extremely

[A ( x)

3

Graphical Representation

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Representation of hedges in fuzzy logic Hedge

Very very

Mathematical Expression

Graphical Representation

[A ( x)]4

More or less

A ( x )

Somewhat

A ( x ) 2 A ( x ) 2

Indeed

if 0  A  0.5

1  2 1  A ( x ) 2 if 0.5 < A  1

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Fuzzy sets with the hedge ‘very’ Degree of Membership 1.0 Short

0.8

Tall Average

0.6 0.4 0.2

Very Short

Very Tall

0.0 150

160

170

180

190

200

210 Height, cm

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FUZZY OPERATIONS

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Typical Operations on Crisp Sets X

Not A

B A

A

Complement

Containment

X A

B

Intersection LM

X

X A B

Union

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Operations of Fuzzy Sets • The name operations are inherited from classical set theory. • Operations describe how sets (either crisp or fuzzy) can interact with each other. • But an operation might function differently on fuzzy sets than on crisp sets.

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 Not A

Complement – “NOT” Crisp Sets: Who does not belong to the set? Fuzzy Sets: How much do elements not belong to the set?

X

A

• The complement of a fuzzy set is an opposite of this set.

Complement of crisp set A

(x) fuzzy set A

1 0

X

x

 (x) 1

fuzzy set Not A

0

X LM

x

• If A is a crisp set, we remove the entire set A from the universe of discourse and we obtain the complement. • If A is a fuzzy set, its complement A can be found as follows: A(x) = 1  A(x) 7ENT1013 Artificial Intelligence

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Containment – Sub/Superset



Crisp Sets: Which sets belong to which other sets? Fuzzy Sets: Which sets belong to which other sets?

X A

B

• A set can contain other sets. The smaller set is called the subset.

crisp set A is a subset of B

(x) B 1

A

0 (x)

X

1

B

x

A

0

X

A(x) ≤ B(x) x

fuzzy set A is a subset of B LM

• In crisp sets, all elements of a subset entirely belong to a larger set and membership values of both sets are 1 constantly. • In fuzzy sets, each element belongs less to the subset than to the larger set. • Note: fuzzy sets A and B here must have the same UoD. 7ENT1013 Artificial Intelligence

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

Intersection – “AND” Crisp Sets: Which element belongs to both sets? Fuzzy Sets: How much of an element is in both sets?

X A B

• An intersection between two crisp sets contains the elements shared by both sets.

Intersection of crisp sets A and B

• In fuzzy sets, an element may partly belong to both sets with different memberships. • A fuzzy intersection is the lower membership in both sets of each element.

(x) 1

A

0 (x)

B X

x

• The fuzzy intersection of two fuzzy sets A and B on universe of discourse X: AB(x) = min [A(x), B(x)] = A(x)  B(x), 0 x X where xX. Intersection of fuzzy sets A and B A B

1

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

Alternative Function for Intersection • Minimum is not the only function that can be used to express the intersection of two fuzzy sets (although it is the most commonly-used). We could also use the product of the two membership functions: AB(x) = A(x) * B(x) where x X. • Different functions may generate different results for the intersection. • But compare how these functions behave when A and B take values of 0 or 1.

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The Fuzzy Empty Set,  X B A

Membership function of  -- (x) = 0 Membership function of X -- X(x) = 1 LM

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

Crisp Sets: Which element belongs to either set? Fuzzy Sets: How much of an element is in either set?

X A B

Union of crisp set A and B

(x) 1

A

B

0 (x)

X

x

AB

1

Union – “OR”

• The union of two crisp sets consists of every element that falls into either set. • In fuzzy sets, the union is the reverse of the intersection. • That is, the union is the largest membership value of the element in either set. • The union of two fuzzy sets A and B on universe X can be given as: AB(x) = max [A(x), B(x)] = A(x)  B(x),

0 X

x

where xX.

Union of fuzzy set A and B LM

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

Alternative Function for Union

Maximum is not the only function that can be used to express the union of two fuzzy sets (although it is the most commonly-used). We could also use the “probabilistic OR”: AB(x) = A(x) + B(x) - A(x) * B(x) where xX. Again, check how these function behave when A and B take values of 0 or 1.

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Properties of Fuzzy Sets (1) • Commutativity:

AB=BA AB=BA

Order of conditions in a rule antecedent is not important. • Associativity:

A  (B  C) = (A  B)  C A  (B  C) = (A  B)  C

• Idempotency:

AA=A AA=A

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Properties of Fuzzy Sets (2) • Identity:

A=A AX=X A= AX=A

• Transitivity:

If (A  B) and (B  C) then A  C

• Involution:

 (A) = A

Exercise: Can you prove Transitivity & Involution? LM

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Properties of Fuzzy Sets (3) • De Morgan’s Laws - true for “dual” operations: (A  B) = A  B (A  B) = A  B • Contradiction – NOT true for fuzzy sets: A  A ≠  • Excluded middle – NOT true for fuzzy sets: A  A ≠ X

Exercise: Can you prove and give examples of Contradiction & Excluded middle ? LM

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• Using fuzzy operations and their properties with hedges, we can create a variety of fuzzy sets based on the existing ones. Exercise: Given the MF of fuzzy set ‘tall’, what is the MF of fuzzy set ‘tall but not very tall’ (i.e. ‘average tall’)?

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FUZZY RULES

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Fuzzy Rules In 1973, Professor Lotfi Zadeh published his second most influential paper. This paper outlined a new approach to analysis of complex systems, in which Zadeh suggested capturing human knowledge in fuzzy rules. Professor at University of California, Berkeley

Syntax of a fuzzy rule • A fuzzy rule can be defined as a conditional statement in the form: IF x is A IF speed is fast THEN y is B THEN stopping_distance is long where x and y are linguistic variables of the rule; A and B are linguistic values of the rule and they are determined by fuzzy sets on the universe of discourses X and Y, respectively. • Universe of discourses X and Y are defined for x and y. And fuzzy sets A and B are defined on X and Y in order to interpret x and y. LM

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Difference between classical and fuzzy rules? A classical IF-THEN rule uses binary/Boolean logic, for example, Rule 1: IF speed is > 100 THEN stopping_distance is 100

Rule 2: IF speed is < 40 THEN stopping_distance is 25

The variable speed can have any numerical value between 0 and 220 km/h (the universe of discourse), but each rule applies fully or not at all.

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We can also represent the stopping distance rules in a fuzzy form: Rule 1: IF speed is fast THEN stopping_distance is long

Rule 2: IF speed is slow THEN stopping_distance is short

In fuzzy rules, the linguistic variable speed also has the same universe of discourse (between 0 and 220 km/h), but this range is interpreted by fuzzy sets, such as slow, medium and fast. The universe of discourse of the linguistic variable stopping_distance can be between 0 and 300 m and is interpreted by fuzzy sets such as short, medium and long.

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 

In a fuzzy system, all rules are true to some degree, or in other words they fire partially. If the antecedent is true to some degree, then the consequent is also true to that same degree - Monotonic Selection.

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Monotonic Selection Degree of Membership 1.0

Degree of Membership 1.0

Tall

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0 160

0.0 180

190 200 Height, cm

IF height is tall THEN weight is heavy LM

70

Heavy

80

100 120 Weight, kg

According to this rule, for a given input value of height=183cm, an output weight=77kg should be generated (the red lines). 7ENT1013 Artificial Intelligence

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A fuzzy rule can have multiple parts of antecedents, for example: IF AND AND THEN

project_duration is long project_staffing is large project_funding is inadequate risk is high

IF service is excellent OR food is delicious THEN tip is generous LM

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

The consequent of a fuzzy rule can also include multiple parts, for instance: IF THEN





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temperature is hot hot_water is reduced; cold_water is increased.

Multiple parts in a rule consequent are independent to each other. So a FIS with two outputs is equivalent to two independent FISs with one output each. The only difference is that the rule antecedent is evaluated once here; while with two rules there will be two evaluations of the rule antecedents. 7ENT1013 Artificial Intelligence

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Summary • Fuzzy logic is a set of mathematical principles for knowledge representation based on degrees of membership rather than on the crisp membership of classical binary logic. • Unlike two-valued Boolean logic, fuzzy logic is multi-valued and allows partial truth. • To represent a fuzzy set in a computer, we express it as a function and then map the elements of the set to their degrees of membership. LM

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• Hedges are fuzzy set qualifiers used to modify the shape of fuzzy sets and create new subsets. • Fuzzy rules are the expert knowledge, which contain linguistic uncertainty. • Fuzzy logic is a logic that describes fuzziness. As fuzzy logic attempts to model human’s sense of words, decision making and common sense, it is leading to more human, intelligent machines.

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