Fuzzy Expert System

Basic Notions 1.Fuzzy Sets 2.Fuzzy representation in computer 3.Linguistic variables and hedges 4.Operations of fuzzy sets 5.Fuzzy rules 6.Reasoning with fuzzy rules 7.Fuzzy inference 8.Building fuzzy expert system

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

Fuzzy Sets Fuzzy set 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.

Range of logical values in Boolean and fuzzy logic

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 f A (x) of A f A (x): X  0,1 Where

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 <  A (x) < 1 if x is partly in A

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.

n The classical example in fuzzy sets is tall men. The elements of the fuzzy set “tall men” are all men, but their degrees of membership depend on their height.

Crisp and fuzzy sets of “tall men”

The representation of fuzzy set Fuzzy set of tall men 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)

Crisp and fuzzy sets of short, average and tall men

Representation of crisp and fuzzy subsets Typical functions that can be used to represent a fuzzy set are sigmoid, gaussian and pi. However, these functions increase the time of computation. Therefore, in practice, most applications use linear fit functions.

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

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 the whole sentence (pg 95)

How to represent a fuzzy set in a computer? n First, we determine the membership functions. In our “tall men” example, we can obtain fuzzy sets of tall, short and average men. n The universe of discourse  the men’s heights  n consists of three sets: short, average and tall men. n As you will see, a man who is 184 cm tall is a member of the average men set with a degree of membership of 0.1, n and at the same time, he is also a member of the tall men set with a degree of 0.4.

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

Fuzzy sets with the hedge very

Representation of hedges in fuzzy logic

Representation of hedges in fuzzy logic (continued)

Fuzzy sets operations operations 1.Complement 2.Containment 3.Intersection 4.Union 5.Commutativity 6.Associativity 7.Distrubutivity 8.Indempotency 9.Identity 10.Involution 11.Transitivity 12.De Morgan’s law

Cantor’s sets

Operations of fuzzy sets

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

Fuzzy rules Difference with classical rules –Classical IF-THEN rule uses binary logic e.g. Rule 1: IF speed is > 100 THEN the stopping_distance is long Rule 2: IF speed is < 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.

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 km/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%

Fuzzy sets of tall and heavy men These fuzzy sets provide the basis for a weight estimation model. The model is based on a relationship between a man’s height and his weight: IF height is tall THEN weight is heavy

The value of the output or a truth membership grade of the rule consequent can be estimated directly from a corresponding truth membership grade in the antecedent. This form of fuzzy inference uses a method called monotonic selection.

A fuzzy rule can have multiple antecedents, for example: IF project_duration is long AND project_staffing is large AND project_funding is inadequate THEN risk is high IF service is excellent OR food is delicious THEN tip is generous

The consequent of a fuzzy rule can also include multiple parts, for instance: IF temperature is hot THEN hot_water is reduced; cold_water is increased

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

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

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

Fuzzy inference c8_fuzzyES_2.ppt