1 chap 4: fuzzy expert systems part 2 asst. prof. dr. sukanya pongsuparb dr. srisupa palakvangsa na...

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Chap 4: Fuzzy Expert SystemsPart 2

Asst. Prof. Dr. Sukanya PongsuparbDr. Srisupa Palakvangsa Na AyudhyaDr. Benjarath Pupacdi

SCCS451 Artificial IntelligenceWeek 10

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Fuzzy Logic

Unclear

The formal systematic study of the principles of valid inference and correct reasoning

a form of multi-valued logic derived from fuzzy set theory to deal with reasoning that is approximate rather than precise. (Wikipedia)

a theoretical system used in mathematics, computing and philosophy to deal with statements which are neither true nor false (dictionary.cambridge.org)

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Classical (Crisp) Set

fA (x) called the characteristic function of APrinciple of Dichotomy: a classic set theory imposes a sharp boundary

fA(x): X {0, 1}, where

Ax

Axxf A if0,

if 1,)(

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Fuzzy Set

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.

A(x) : membership function of set Amembership value (0<= degree <=1): shows the degree of membershipBasic idea: an element belongs to a fuzzy set with a certain degree of membershipfuzzy set is a set with fuzzy boundaries

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Linguistic Variables

IF <antecedent> THEN <consequent>

valueobject valueobject

Examples:• IF wind is strong THEN sailing is good• IF speed is slow THEN stopping_distance is short• IF project_duration is long THEN completion_risk is high

Linguistic variable Linguistic value

* linguistic variable == fuzzy variable

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Hedges

Andy quite likes Thai foodhedges

Mary looks very much like her mother

Jim has been to several attractions in Thailand

Hedges: terms that modify the shape of fuzzy sets e.g. very, somewhat, quite, more or less, and slightly

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Representation of hedges in fuzzy logicHedge Mathematical

ExpressionHedge MathematicalExpression Graphical Representation

Very very

More or less

Indeed

Somewhat

2 [A ( x )]2

A ( x )

A ( x )

if 0 A 0.5

if 0.5 < A 1

1 2 [1 A ( x )]2

[A ( x )]4 Concentration

dilation

dilation

intensification

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Operations of fuzzy sets

The classical set theory developed in the late 19th century by Georg Cantor describes how crisp sets caninteract. These interactions are called operations:

- Complement- Containment- Intersection- Union

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Operations of fuzzy sets

Complement

0x

1

( x )

0x

1

Containment

0x

1

0x

1

A B

Not A

A

Intersection

0x

1

0x

A B

Union0

1

A BA B

0x

1

0x

1

B

A

B

A

( x )

( x )

( x )

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Set Properties

The properties of set used in crisp sets can also be used in fuzzy setsFrequently used properties:-

CommutativityAssociativelyDistributivityIdempotencyIdentityInvolutionTransitivityDe Morgan’s Laws

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How can we combine “fuzziness”to our rules ???

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Fuzzy Rules

Classical IF-THEN ruleRule 1:IF speed is > 100THEN stopping distance is

longRule 2:IF speed is < 70THEN stopping distance is

short

Fuzzy IF-THEN ruleRule 1:IF speed is fastTHEN stopping distance is

longRule 2:IF speed is slowTHEN stopping distance is

shortFuzzy Rules: If the antecedent is true to some degree of membership, then the consequent is also true to that same degree.

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Tall men Heavy men

180

Degree ofMembership1.0

0.0

0.2

0.4

0.6

0.8

Height, cm

190 200 70 80 100160

Weight, kg

120


0.0

0.2

0.4

0.6

0.8

How to reason?

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 tallTHEN weight is heavy

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How to reason? (cont.)

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.

Tall menHeavy men

180


0.0

0.2

0.4

0.6

0.8

Height, cm

190 200 70 80 100160

Weight, kg

120


0.0

0.2

0.4

0.6

0.8

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How about multiple antecedents?

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Multiple Antecedents

ExamplesIF project_duration is longAND project_staffing is largeAND project_funding is inadequateTHEN risk is high

IF service is excellentOR food is deliciousTHEN tip is generous

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Multiple Consequences

ExampleIF temperature is hotTHEN hot_water is reduced;

cold_water is increased

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Fuzzy Inference

Fuzzy inference: a process of mapping from a given input to an output, using the theory of fuzzy setsExamples of inference techniques

Mamdani-Style InferenceSugeno-Style Inference

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Mamdani-Style Inference

Linguistic variables and linguistic values must be identified

Rule 1:IF project_funding is adequateOR project_staffing is smallTHEN risk is lowRule 2:IF project_funding is marginalAND project_staffing is largeTHEN risk is normalRule 3:IF project_funding is inadequateTHEN risk is high

Rule 1:IF x is A3OR y is B1THEN z is C1Rule 2:IF x is A2AND y is B2THEN z is C2Rule 3:IF x is A1THEN z is C3

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Mamdani-Style Inference (cont.)

There are four stepsFuzzificationRule EvaluationAggregation of the Rule OutputsDefuzzification

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1) FuzzificationTake crisp inputsDetermine the degree of inputs

Crisp Inputy1

0.1

0.71

0y1

B1 B2

Y

Crisp Input

0.20.5

1

0

A1 A2 A3

x1

x1 X

(x = A1) = 0.5

(x = A2) = 0.2

(y = B1) = 0.1

(y = B2) = 0.7

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2) Rule Evaluation Take fuzzified inputsApply them to the antecedents of the fuzzy rules

AB(x) = min [A(x), B(x)], where xXAB(x) = max [A(x), B(x)] , where xX

Rule 1:IF x is A3 (0.0)OR y is B1 (0.1)THEN z is C1C1(z) = max [A3(x), B1(y)] = max[0.0, 0.1] = 0.1

Rule 2:IF x is A2 (0.2)AND y is B2 (0.7)THEN z is C2C2(z) = min[A2(x), B2(y)] = min[0.2, 0.7] = 0.2

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The result of the antecedent evaluation can be applied to the membership function of the consequent. – functions can be clipped or scaled


0.0

0.2

Z

Degree ofMembership

Z

C2

1.0

0.0

0.2

C2

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Correlation Minimum (clipping): cut the consequent membership function at the level of the antecedent truthCorrelation Product (scaling): multiplying all membership degrees of the rule consequent by the rule antecedent

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3) Aggregation of the rule outputs Combine all outputs into a single fuzzy set

00.1

1C1

Cz is 1 (0.1)

C2

0

0.2

1

Cz is 2 (0.2)

0

0.5

1

Cz is 3 (0.5)

ZZZ

0.2

Z0

C30.5

0.1

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4) Defuzzification Centroid technique (or centre of gravity – COG): find the point where a vertical line would slice the aggregate set into two equal masses

b

aA

b

aA

dxx

dxxx

COG

( x )

1.0

0.0

0.2

0.4

0.6

0.8

160 170 180 190 200

a b

210

A

150X

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4) Defuzzification (cont.)A reasonable estimate can be obtained by calculating it over a sample of points.

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1.0

0.0

0.2

0.4

0.6

0.8

0 20 30 40 5010 70 80 90 10060

Z

Degree ofMembership

67.4

4.675.05.05.05.02.02.02.02.01.01.01.0

5.0)100908070(2.0)60504030(1.0)20100(

COG

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Fuzzy Inference

Examples of inference techniquesMamdani-Style InferenceSugeno-Style Inference

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Sugeno-Style Inference

uses a single spike (singleton) as the member function of the rule consequentSingleton: a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere elseFormat of rules:-

IF x is AAND y is BTHEN z is f(x,y)

where x, y and z are linguistic variables; A and B fuzzy sets on universe of discourses X and Y respectively; f(x,y) a mathematical function

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Sugeno-Style Inference (cont.)

Four steps of Sugeno1. Fuzzification -> similar to Mamdani Style2. Rule Evaluation

IF x is AAND y is BTHEN z is k

the output is constant which makes all outputs to be spikes3. Aggregation of the Rule Outputs -> similar to

Mamdani Style

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4. Defuzzification: use weighted average (WA)

z is k1 (0.1) z is k2 (0.2) z is k3 (0.5) 0

1

0.1Z 0

0.5

1

Z0

0.2

1

Zk1 k2 k3 0

1

0.1Zk1 k2 k3

0.20.5

655.02.01.0

805.0502.0201.0

)3()2()1(

3)3(2)2(1)1(

kkk

kkkkkkWA

0 Z

Crisp Outputz1

z1

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Mamdani VS Sugeno

MamdaniCapture expert knowledgeExpress knowledge in more human-like mannerNot computationally efficient

SugenoComputationally effectiveWork well with optimisation and adaptive techniquesSuitable for control problems and dynamic nonlinear systems

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Building Fuzzy Expert Systems

1. Specify the problem and define linguistic variables.2. Determine fuzzy sets.3. Elicit and construct fuzzy rules.4. Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzy inference into the expert system.5. Evaluate and tune the system.

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Building Fuzzy Expert Systems (cont.)

Case studyA service centre keeps spare parts and repairs failed ones.A customer brings a failed item and receives a spare of the same typeFailed parts are repaired, placed on the shelf, and thus become spares.The objective here is to advise a manager of the service centre on certain decision policies to keep the customers satisfied

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1) Specify the problem and define linguistic variablesaverage waiting time (mean delay) mrepair utilisation factor of the service centre number of servers sinitial number of spare parts n

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Linguistic Variable: Mean Delay, mLinguistic Value Notation Numerical Range (normalised)

Very ShortShortMedium

VSSM

[0, 0.3][0.1, 0.5][0.4, 0.7]

Linguistic Variable: Number of Servers, sLinguistic Value Notation Numerical Range (normalised)

SmallMediumLarge

SML

[0, 0.35][0.30, 0.70]

[0.60, 1]

Linguistic Variable: Repair Utilisation Factor, Linguistic Value Notation Numerical Range

LowMediumHigh

LMH

[0, 0.6][0.4, 0.8][0.6, 1]

Linguistic Variable: Number of Spares, nLinguistic Value Notation Numerical Range (normalised)

Very SmallSmallRather SmallMediumRather LargeLargeVery Large

VSS

RSMRLL

VL

[0, 0.30][0, 0.40]

[0.25, 0.45][0.30, 0.70][0.55, 0.75]

[0.60, 1][0.70, 1]

Linguistic Variable: Mean Delay, m

Linguistic Variable: Number of Servers, s

Linguistic Variable: Repair Utilisation Factor,

Linguistic Variable: Number of Spares, n

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2) Determine Fuzzy Setsfuzzy sets can have a variety of shapesa triangle or a trapezoid are widely used due to the simplification of the computational processKey point – to maintain sufficient overlap in adjacent fuzzy sets for the fuzzy system to respond smoothly

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0.10

1.0

0.0

0.2

0.4

0.6

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Mean Delay (normalised)

SVS M

Degree of Membership

0.10

1.0

0.0

0.2

0.4

0.6

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

M LS


Number of Servers (normalised)

0.10

1.0

0.0

0.2

0.4

0.6

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Repair Utilisation Factor

M HL


0.10

1.0

0.0

0.2

0.4

0.6

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S RSVS M RL L VL


Number of Spares (normalised)

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3) Elicit and construct fuzzy rulesKnowledge acquisition such as

Ask expertsBooksFlow diagramsObservation

Rule construction

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1. If (utilisation_factor is L) then (number_of_spares is S)2. If (utilisation_factor is M) then (number_of_spares is M)3. If (utilisation_factor is H) then (number_of_spares is L)

4. If (mean_delay is VS) and (number_of_servers is S) then (number_of_spares is VL)5. If (mean_delay is S) and (number_of_servers is S) then (number_of_spares is L)6. If (mean_delay is M) and (number_of_servers is S) then (number_of_spares is M)

7. If (mean_delay is VS) and (number_of_servers is M) then (number_of_spares is RL)8. If (mean_delay is S) and (number_of_servers is M) then (number_of_spares is RS)9. If (mean_delay is M) and (number_of_servers is M) then (number_of_spares is S)

10.If (mean_delay is VS) and (number_of_servers is L) then (number_of_spares is M)11.If (mean_delay is S) and (number_of_servers is L) then (number_of_spares is S)12.If (mean_delay is M) and (number_of_servers is L) then (number_of_spares is VS)

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Rule m s n Rule m s n Rule m s n

1 VS S L VS 10 VS S M S 19 VS S H VL

2 S S L VS 11 S S M VS 20 S S H L

3 M S L VS 12 M S M VS 21 M S H M

4 VS M L VS 13 VS M M RS 22 VS M H M

5 S M L VS 14 S M M S 23 S M H M

6 M M L VS 15 M M M VS 24 M M H S

7 VS L L S 16 VS L M M 25 VS L H RL

8 S L L S 17 S L M RS 26 S L H M

9 M L L VS 18 M L M S 27 M L H RS

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VS VS VSVS VS VSVS VS VS

VL L M

HS

VS VS VSVS VS VSVS VS VSM

VS VS VSVS VS VSS S VSL

s

LVS S M

m

MH

VS VS VS

LVS S M

S

m

VS VS VSM

S S VSL

s

S VS VS

MVS S M

m

VS S M

m

S

RS S VSM

M RS SL

s

S

M M SM

RL M RSL

s

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4) Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzy inference into the expert system.

Possible methodsDIY by using programming languages e.g. C, C++fuzzy logic development tool such as MATLAB Fuzzy Logic Toolbox or Fuzzy Knowledge Builder

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5) Evaluate and tune the systemTo check if the system meets the requirementsUse tools to analyse the system performance

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Tuning systemsReview model input and output variables e.g. redefine their rangesReview the fuzzy sets e.g. define additional sets on the universe of discourseProvide sufficient overlap between neighbouring setsReview the existing rules e.g. modify rules, rewrite hedge rulesAdjust the rule execution weightsRevise shapes of the fuzzy sets.

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Applications of Fuzzy Logic

Air ConditioningDigital Image Processing e.g. edge detectionAutomobile and other vehicle subsystems, such as automatic transmissions, ABS and cruise control (e.g. Tokyo monorail)Home appliances e.g. rice cookers, washing machines, dish washers

1 chap 4: fuzzy expert systems part 2 asst. prof. dr. sukanya pongsuparb dr. srisupa palakvangsa na...

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x x rule

rule antecedent slide

mamdani style

form of fuzzy inference

single fuzzy set slide

long rule

c1 rule

c2 rule