Fuzzy Expert System

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Fuzzy Expert System
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Fuzzy Expert System

• Fuzzy Sets
• Fuzzy representation in computer
• Linguistic variables and hedges
• Operations of fuzzy sets
• Fuzzy rules
• Reasoning with fuzzy rules
• Fuzzy inference
• Building fuzzy expert system

Basic Notions
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Fuzzy Logic

Fuzzy logic is determined as a set of
mathematical principles for knowledge
representation based on degrees of membership
rather than on crisp membership of classical
binary logic
Fuzzy Logic
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Fuzzy Logic

• Multi-valued
• Deals with degree of membership
• Degrees of truth
• Uses continuum of logical values between 0
  (completely false) and 1(completely true)

Fuzzy Logic
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Fuzzy Sets

• Refer page 89
• The basic idea of fuzzy set theory an element
  belongs to a fuzzy set with certain degree of
  membership.
• Not either true or false, but partly true(false)
  to any degree
• Taken as a real number in the interval
• Refer table 4.1, fig. 4.2.

Fuzzy set
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Fuzzy set theory

• Crisp set
• Let X be the universe of discourse and its
  elements be denoted as x. crisp set A of X is
  defined as function fA(x) of A
• fA(x) X ? 0,1
• Where

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Fuzzy set theory

• Fuzzy set
• Fuzzy set A of universe X is defined by function
  ?A(x) called membership function of set A
• ? A(x) X ? 0,1
• where ? A(x) 1 if x is totally in A
• ? A(x) 0 if x is not in A
• 0 lt ? A(x) lt 1 if x is partly in A

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The representation of fuzzy set

• Determine the membership function
• Method to determine membership function
• Single expert
• Multiple experts
• Self generated by ANN, learn the data derive
  the fuzzy sets.

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The representation of fuzzy set

• Refer tall men example (figure 4.3, 4.4)
• Fuzzy set of tall men in figure 4.3 can be
  represented as fit-vector
• Tall men (0/180, 0.5/185, 1/190) or
• Tall men (0/180, 1/190)
• Fuzzy set of short and average men
• Short men (1/160, 0.5/165, 0/170) or
• Short men (1/160, 0/170)
• average men (0/165, 1/175, 0/185)

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Linguistic variables and hedges

• A fuzzy variable
• E.g. the statement John is tall implies that
  the linguistic variable John takes the linguistic
  value tall
• In fuzzy ES linguistic variables are used in
  fuzzy rules
• IF wind is strong
• THEN sailing is good
• IF project duration is long
• THEN completion_risk is high
• IF the speed is slow
• THEN stopping_distance is short

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Linguistic variables and hedges

• E.g. The linguistic variable speed have range
  between 0 and 220 km/hour may include fuzzy
  subsets as very slow, slow, medium, fast and very
  fast
• Hedges - fuzzy set qualifiers
• Carries by a linguistic variable
• Terms that modifies fuzzy sets
• Includes adverb I.e. very, somewhat, quite, more
  or less and slightly
• Can modify verbs, adjectives, adverbs or thw
  whole sentence (pg 95)

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Linguistic variables and hedges

• Hedges act as operations
• Very perform concentration and creates new subset
• E.g. tall men derive the subset very tall men
• Dilation the of more or less tall men is
  broader than the set of tall men.
• Refer figure 4.5.
• Refer table 4.2

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Fuzzy sets operations

• Complement
• Containment
• Intersection
• Union
• Commutativity
• Associativity
• Distrubutivity
• Indempotency
• Identity
• Involution
• Transitivity
• De Morgans law

operations
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Fuzzy rules

• Capturing human knowledge in fuzzy rules
• Form of fuzzy rules
• IF x is A
• THEN y is B
• Where x and y are linguistic variables A and B
  are linguistic values determined by fuzzy sets

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Fuzzy rules

• Difference with classical rules
• Classical IF-THEN rule uses binary logic e.g.
• Rule 1
• IF speed is gt 100 THEN the stopping_distance is
  long
• Rule 2
• IF speed is lt 40 THEN stopping_distance is short
• The variable speed can have any numerical value
  between 0-220km/h
• The linguistic variables stopping_distance can
  only take either long or short.

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Fuzzy rules

• Difference with classical rules
• Fuzzy IF-THEN rules uses binary logic e.g.
• Rule 1
• IF speed is fast THEN the stopping_distance is
  long
• Rule 2
• IF speed is slow THEN stopping_distance is short
• The variable speed can have any numerical value
  between 0-220km/h but include fuzzy sets range ,
  slow, medium and fast
• The linguistic variables stopping_distance can be
  between 0 and 300m and may take fuzzy sets as
  short, medium or long
• Fuzzy expert systems merge the rules and
  consequently cut the number of rules at least 90

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Reasoning with Fuzzy rules

• Includes 2 distinct part
• Evaluating the rule antecedent (the IF part)
• Implication or applying the result to the
  consequent (the THEN part)
• Mechanism
• In classical rule based system
• If the rule antecedent is true, the consequent is
  also true
• In fuzzy systems,
• All rules fires to some extent,
• Partially fire
• If the antecedent is true to some degree of
  membership, then the consequent is also true to
  that same degree
• Discuss fig. 4.8, 4.9

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Reasoning with Fuzzy rules

• A fuzzy rule can have
• Multiple parts of antecedent
• Multiple parts of consequent (see example pg 105)
• In general fuzzy expert system incorporates not
  one but several rules that describe expert
  knowledge

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Reasoning with Fuzzy rules

• The output of each rule is a fuzzy set but need
  to obtain a single number representing the ES
  output
• The output of the fuzzy sets are combined and
  transformed into a single number by..
• Aggregates all output fuzzy sets into a single
  output fuzzy set
• Then defuzzifies the resulting fuzzy set into a
  single number
• Fuzzy inference

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Fuzzy inference

• A process of mapping from a given input to an
  output using the fuzzy theory.
• Mamdani-style inference
• Sugeno-style inference

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Fuzzy inference

• Mamdani-style inference
• Involve 4 steps
• Fuzzification of the input variables
• Rule evaluation
• Aggregation of the rule output
• Defuzzification
• Refer example in pg 106-112

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Building fuzzy ES

• An iterative process that involves defining fuzzy
  sets and fuzzy rules, evaluating and the tuning
  the system to meet the specified requirement.

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Building fuzzy ES

• Specify the problem and define linguistic
  variables
• Determine fuzzy sets
• Elicit and construct fuzzy rules
• Encode the fuzzy sets, fuzzy rules and procedures
  to perform fuzzy inference into the ES
• Evaluate and tune the system

5 steps