Fuzzy inference - cs.bgu.ac.ilsipper/courses/ecal051/lecon2.pdf · Gas Low OR Pressure Temp. & & & High Hot Cold Low OR High. Logic Systems Laboratory − Swiss Federal Institute

Fuzzy inference

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50

Partial

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Cloudy Sunny

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Fuzzyfication Implication

48

MediumLow High

Defuzzyfication

Knowledge base: defines rules and membership functions

Defuzzyfier: translates fuzzy outputs into crisp values

Fuzzyfier: translates crisp inputs into fuzzy values

Inference engine: applies reasoning to compute fuzzy outputs

DefuzzifierInferenceEngine

Fuzzifier

Rule base

Database

Knowledge base

FuzzyOutputFuzzy Output

CrispInputCrisp

Input

..

.. Logic Systems Laboratory − Swiss Federal Institute of Technology

Carlos Andres Pena Reyes

Fuzzy inference systems

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Logic Systems Laboratory − Swiss Federal Institute of Technology Lausanne

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Carlos Andrés Peña−Reyes

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Network−like view of a fuzzy system

Membership function values

Gas

OR

Hot

Cold

Low

High

Pressure

Temp.

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HighOR

. .Logic Systems Laboratory − Swiss Federal Institute of Technology Lausanne


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Operational parameters.

RulesConsequents

Weights

Gas

Pressure

Temp.

Antecedents

Low

OR

Hot

Cold

High

&

&

&

High

LowOR



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Connective parameters

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.

Low

{ Membership functionsRulesRelevant variables

Pressure

Temp.

Number of

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Hot

Cold

High

Low

Gas

High

&

&

OR

&



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Structural parameters

Pressure

Defuzzification methodMembership function typesFuzzy operatorsReasoning mechanism&

&

&

Temp.

Low

High

OR

ORGas

Low

Hot

Cold

High

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. Logic Systems Laboratory − Swiss Federal Institute of Technology Lausanne


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Logical parameters

Parameters of a fuzzy system

Relevant variables

Number of membership functions

Number of rules

Consequents of rules

Defuzzification method

Antecedents of rules

Operational

Connection

Fuzzy operatorsLogic

Class

Membership function types

Membership function values

Rule weights

Structural

Database

Rulebase

Knowledge base

Defuzzifier

Fuzzi- and defuzzifier

Inference engine

ComponentParameters

..

..

Carlos Andrés Peña-Reyes

Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne

Reasoning mechanism

If Tempearture is WARM then Ventilator is Low

If Temperature is HOT then Ventilator is Medium

InterpretabilityPrecision

Linguistic

... Let’s go to the lake!!!

If Temperature is VERY−HOT then Ventilator is High

If Temperature is HELLISH then Ventilator is Off, and...

If Temperature is COOL then Ventilator is Off

Numeric



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Dual external nature

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Universal approximator

Numeric mapping: Crisp inputs / Crisp outputs

Uncertainty management: noise and low quality of data

Nonlinear behavior, but linearity not excluded

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Numeric issues

Number of elements: Compatible with human capabilities

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●

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Distinguishability: Each linguistic label has semantic meaning

Coverage: Any element belongs to at least one fuzzy set

Normalization: At least one element has unitary membership

Complementarity: For each element, the sum of memberships is one

Semantics: the study of meanings

0

1Cold Cool Warm

Temperature

Hot

0

1Cold

Temperature

HotCool Warm

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Interpretability considerations: semantic criteria

●

●

Consistency: rules firing simultaneously must have similar consequents

Rule readability: small number of conditions in rule antecedents

Rule−base simplicity: Set of rules as small as possible

●

Completeness: for any input, at least one rule must fire●

Syntax: the way in which linguistic elements are put together

5R

0RR2

R

R

1R

R A

7R 9R

5

R

4R 6RR

1R 3R2R

RAR4

RB B

5

8

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Interpretability considerations: syntactic criteria.

●

●

Linguistic labels shared by all rules

Normal, orthogonal membership functions

Default rule

●

●

Don’t care conditions

5 R

0R

A

RB

R


Logic Systems Laboratory − Swiss Federal Institute of Technology. .

. .

17 20 26 29

Cold Warm Hot

Strategies to satisfy interpretability criteria

What do you know about the modeled system?

Have you preferences or restrictions to the model?

How do you search?

Search methodi.e. do a well suited and/or well known technique exists?

i.e. what is predefined and what looked for?Search space

Constraintsi.e. do issues like size, speed, or simplicity matter?

..

..

The general modeling problem



Database

Rule base

InferenceEngine Defuzzifier

some parameter pre−definition is thus required.The number of parameters is too high to perform a full search,

and, or, not, ...

f1, ... , fm

P1 P2 P3 P4

R1, ... , Rn

If V1 is Low AND ....

According with the searched parameters we can have:

System design.

Structural parameters:

Behavior learning.Connective parameters:

Operational parameters:

Logical parameters:

Knowledge tuning.

Structure learning.

..



Knowledge base

Fuzzifier

. .

Search space in fuzzy modeling

"Classic" identification methods

Knowledge engineering

Evolutionary fuzzy modeling techniques

Neuro−fuzzy systems

Machine learning approaches

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..

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Search methods: Fuzzy modeling techniques

− Diagnostic: Overall performance, sensitivity, specificity

− Data mining: Completness, complexity.

− Availability: Continuity of explanations (time to provide them)− Interpretability: Allowed complexity.

Who is going to interact with the system?

− Control: Dynamic response, adaptability, robustness, etc.

− Speed: Real−time constraints, computing resources.

− Size: Available memory, computing platform.

− Classification: Classification performance, quadratic error.

What is the fuzzy system expected to do?

How is the system expected to do it?



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Usual constraints in fuzzy modeling

. .

.

all rules share the same MFs

null and unity membership

Rule−specific MFs are not allowed

Orthogonal MFs with well defined



.

. .

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Interpretability−related constraints

Fuzzy modeling: direct approach

This approach is also called knowledge engineering

Domainexpert

Knowledgeengineer

Fuzzymodel

Design loop

Validation loop

Domain expert

Fuzzy modeling: data-driven approaches

This approaches are also denominatedknowledge discovering

Fuzzymodel

Design loop

Validation loop

Domain data Building algorithm

Neuro−fuzzy systems

Fuzzy system

Identification−based

Constructive−learningFuzzy system

Fuzzy system

DataOperational

Connective

Structural

Structural

Connective

Logicdesign

algorithmEstimation

Operational

ANN−like training algorithm

Human

. .

. .


Logic Systems Laboratory − Swiss Federal Institute of Technology

Fuzzy modeling: some data−driven approaches

- Genome encodes values P, Q, and C

- Knowledge is tuned by evolution,

which searches for membership function values

- Rules of type:

if X1=Low and X2=Normal then Output = Ci

- Fixed rule base (completness)X2

X1

R9

R8R5

R4

R6

R7

R3

R1

R2

P3P2P1

Q2

Low Mid High

Q3

Big

Nor

mal

Sm

all

Q1

(3*P + 3*Q + 9*C) * 5 bits = 75 bits

..

..



Evolutionary knowledge tuning (database)

Don’t care conditions and default rule

- Three main approches to evolutionary behavior learning

- Evolution can be used to find a minimal (or fixed size) rule base,

(i.e. fuzzy system)

- Two strategies for reducing this number:

- Number of rules exploses rapidly


RiR3

R2

Ra, Rb, Rc ...

R1

Iterative Rule Learning

Incremental construction ofthe knowledge base

Evolution finds the best rule


R1

Michigan

Individual = One rule

Population = Rule base

.

R2

Pittsburgh

(rule base or knowledge base)Individual = Entire system

Population of systems

.

.

Ri

Evolutionary behavior learning (rule base)

R3

.

- Genome encodes rules: Antecedents and consecuents

- Evolution searches for a subset of N rules (fixed by the designer)

5 functions + 3 don’t care

- Space of 625 rules (1295 including don’t care conditions)

- 4 input variables, 5 membership functions per variable{Tiny, Small, Normal, Big, Huge}

IF V1 is Tiny AND ... AND V4 is Normal then Out = Huge


R2 .... Ri


RnR1 ....

5 * 3 bits

15 bits

N rules * 15 bits

A1 A3 CA4A2

Evolutionary behavior learning: An example

. .

. .

Behavior learning

Evolutionary knowledge base learning

attributeFuzzy systemType of

valuesquantityUsual

type

Operational

Connective

class

Knowledge tuning 10 - 1000 Real-valued Database

Modeling

Symbolic Rule base10 - 1000

Parameter

- Computation requirements

- Parameter representation

- Size of the search space

- Tight interdependency

(Knowledge base = rule base + database)

Critical issues for applying evolution:



. .

. .


Sample ru;e: IF V1 is Low and V4 is High THEN Diagnostic is Benign

- A basic approach: Single population, single evolution

Example: Breast cancer diagnosis problem (Peña and Sipper 99)

- Genome encodes: Rule antecedents and membership function parameters

- A simple genetic algorithm searches for the knowledge base

- 9 inputs, 1 output, 2 membership functions per variable

Ai A9....A1 ....

9 antecedents * 2 bits


18 bits

P d


3 bits 3 bits

....

6 bits

.... Ri .... RnV1 V2 Vi .... V9

.

9 Variables * 6 bits Nr rules * 18 bits

Low

dP

High

.

R2R1

.

.


1

Symbolic part: Rule baseNumeric part: Database

- A variation: Single population, double evolution

- The rule base is evolved using genetic programming

Example: Evolving fuzzy rule based classifiers with GA–P (García et al. 99)

- Genome encodes: Complete rule base and membership function parameters

- A simple genetic algorithm searches for the database

00 0 1 0 0 0 1 1 0

.. Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne

. .


X1

X2


Q3

Sm

all

Q1

Big

Low Mid High

Q2

Nor

mal R2

A fuzzy self-organizing map searches for P and Q values

Genome encodes rules: Antecedents and consequent

(J.-F. Philagor, student project SPG, 1999)Example: Breast cancer diagnosis- Evolution searches for a fixed-size rule base

- Hybrid learning: Evolved rule base, learned database

P1 P2 P3

R3R1

- Database is tuned using a neuro-fuzzy approach

R1 R2

.

Ri .... Rn....

A1 CA2 A9. . .

9 * 2 bits + 1 bit

19 bits

N rules * 19 bits

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.



..

. .

The

test


The

feat

ures


The

dat

abas

e

The Wisconsin Breast Cancer Database

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... ...

Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne... ...

else (Output is Malignant)

R1: if (V1 is Low) and (V2 is High) and ... and (V9 is Low) then (output is Benign)R2: if (V1 is Low) and (V2 is Low) and ... and (V9 is None) then (output is Benign)

...Malignant

P1+d1P1 P P2 +d +dP P2 2 9 9

.

9

Low

High

Low

LowLow

None

.....

.....

Benign

Benign

VV V1 2 9

Proposed Fuzzy System

.

.

.


Ai = 0 or 3 (Variable not assigned)d = [1;8]

P = [1;8]


Ai = 1 (Benign)

d

Ai = 2 (Malignant)

.

3 bits 3 bits

P Ai A9....A1 ....

.... R1 R2 .... Ri .... RnV1 V2 Vi .... V9

9 antecedents * 2 bits

9 Variables * 6 bits Nr rules * 18 bits

6 bits 18 bits

Total genome length = 54 + 18Nr bits

Genome encoding

cs bgu

cs bgu

cs bgu

Fv : Number of variables

selection pressure to fine tune parameters

measures the interpretability

the most important performance measure

Fe : Quadratic error

Fc : Classification performance,

F = Fc + a* Fv + b*Fe


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. Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne .

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Fitness function

97.36% (3)

97.80% (4.8)

97.80% (4.7)

97.07% (4)

96.65% (7)

96.35% (3)

Learned Boolean rules Evolved fuzzy rules

97.51% (3.4)

1

2

3

5

4

96.19% (1.8)

Setiono Tahawork (99)

97.21% (4)97.14% (4)

95.42% (2)

Results: Classification performance (Number of variables)

RulesPeña

Setiono (96) Liu (96)This

Sipper (98)Ghosh (97)

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.

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IF the clump of cells is not very thick,

ELSE the case is malignant.

THEN the case is benign;

AND nucleoli are not highly abnormal,

AND there are few bare nuclei,

AND the cell’s size is quite uniform,

The best single-rule system



Benign

Malignant

v1 v2

Low

v6

Low

v8

Low

. .

.

Low

.

2 Cooperative coevolutionA building algorithm:

1 Fuzzy systemsA system model:

Fuzzy

CoCo

Database

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. .



Proposed approach: two elements

represented by different types of values,they can be decomposed in distinct components,

and which are very interdependent.

fuzzy modeling an These features render

COOPERATIVEadequate target for

COEVOLUTION

3 − 10Logical

The required solutions can be very complex,

Real−valued10 − 1000Knowledge tuningOperational (labels)

System design

Symbolic10 − 1000Behavior learningConnective (rules)

Integer5 − 20Structure learningStructural (size)

classParameter

typeModeling

numberUsual Type of

values

. .

. .



Fuzzy modeling: a coevolvable problem

− Better search power− Lesser computational cost− More−flexible setup

Two evolutionary algorithmssearching for:

membership functions

and rules.

Advantages:− Divide−and−conquer strategy

Modification Modification

Selection

EvaluationEvaluation

Selection

Membership functions Rules

.

. .



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Fuzzy CoCo:A cooperative coevolutionary approach to fuzzy modeling

to form fuzzy systems.2. Individuals are combined with cooperators

individual fitness is then calculated.3. These fuzzy systems are evaluated, and

both fitness−dependent andare selected from generation g−1

1. Cooperators for generation g

randomly

Rules

Fitness

Cooperators

Cooperators

Fitness

MFsRules

Gen

erat

ion

g

g−1

Selected cooperators

MFs

.



.

Fitness evaluation in Fuzzy CoCo

●

●

●

●

● Linguistic fitness: when used, increases selective pressure for interpretability

Normal, orthogonal membership functions: constrained representation

Shared membership functions: reinforced by the existence of a separate species

Default rule: guarantees complete coverage of the input space

Don’t care conditions: encourage shorter rules

R5 RA

R

0

B

R

0

1Cold Cool Warm

Temperature

Hot

..



Interpretability strategies in Fuzzy CoCo

5.9

3.0

3.0150 Virginica

Classes(1) setosa

(3) virginica(2) versicolor

(1) SL Sepal length(2) SW Sepal width(3) PL Petal length(4) PW Petal width

Features

2 Setosa

The

var

iabl

es

5.14.9

1 3.5 Setosa1.41.4

5.1

0.20.2

1.8

Case ClassSL SW PL PW

. Logic Systems Laboratory − Swiss Federal Institute of Technology Lausanne


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The

dat

abas

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Fisher’s Iris Data

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Iris: Variable analysis

Iris proposed solution: Controller−type

Class ClassInputSubsystem

Fuzzy Subsystem

FuzzyInput

Fuzzy

CrispOutput

Output

InputCrisp Fuzzifier Defuzzifier

Knowledge base

Rule base

Database

EngineInference

The selection unit approximates it to the nearest class

The fuzzy subsystem estimates a continuous "class" value

estimation

Stair−function


. .

.


.

1 2

21

1 2 3

3

3

... ... ...

R1: if (SL is A11) and (SW is A12) and (PL is A13) and (PW is A14) then (output is Class1)

R2: if (SL is A21) and (SW is A22) and (PL is A23) and (PW is A24) then (output is Class2)

Rn: if (SL is An1) and (SW is An2) and (PL is An3) and (PW is An4) then (output is Classn)

else (Output is Class0)

setosa


...


11 P21 P31 P12 P22 P32 P24P14

.

.

...

P

versicolor

.

virginica

High

Low

LowLow

None

.....

.....

Medium

SL SW PW

.

Iris controller−type: Proposed Fuzzy System

(setosa)

Iris proposed solution: Classifier−type

Threshold SubsystemClassInput

(versicolor)µ

µ

(virginica)µFuzzyInput

Fuzzy

CrispOutput

Output

InputCrisp Fuzzifier Defuzzifier

Knowledge base

Rule base

Database

EngineInference

Fuzzy Subsystem

The selection unit chooses the most active class

value for each classThe fuzzy subsystem estimates a continuous membership

Maximum and

. .



..

Yes

No

No

NoYes Yes

Yes Yes

Yes

YesNo YesNo

NoNoNo Yes

No

31 P24P14P

None

21P11P


.Logic Systems Laboratory − Swiss Federal Institute of Technology

Low

.....

.....

versicolor

PW

setosa

.

virginicaversicolorsetosa

virginica

virginica

.

versicolorsetosa

Low

Medium

SL

... ... ... ... ...

R1: if (SL is A11) and ... and (PW is A14) then (setosa is Yes),(versicolor is No),(virginica is No)

R2: if (SL is A21) and ... and (PW is A24) then (setosa is No),(versicolor is Yes),(virginica is Yes)

Rn: if (SL is An1) and ... and (PW is An4) then (setosa is No),(versicolor is No),(virginica is Yes)

else (setosa is No),(versicolor is Yes),(virginica is No)

.

Iris classifier−type: Proposed Fuzzy System

Membership functions Rules (Controller/Classifier)

P



3 x 5 bits

P1 P3

55 5

P2

.

SWSL SW

.

.

Genome length = 60 bits

.

PL R1 ... Ri ... Rn Co

....A1 Ci

Nr rules * 19 bits

A4

4 * 2 bits

2/3 bits

2/3 bits

10/11 bits

4 Variables * 15 bits

Genome length = 10/11*Nr + 2/3 bits

Iris: the genomes

"Fit" cooperatorsRandom cooperators

1. Fitness function

Population size

Fv : Number of variables

1

1

{60, 70}500 + 100*Nr

{0.02, 0.05, 0.1}{0.1, 0.2}

{1, 2}

Maximum generationsCrossover probabilityMutation probabilityElitism rate

measures the interpretability

encourages not−so−bad errorsFm : 1 − mse (mean square error)

the most important

2. Fuzzy CoCo parameters

Fc : Classification performance,

F = Fc * Fm

b(Fc + a*Fv) * Fm{ b

.


Logic Systems Laboratory − Swiss Federal Institute of Technology Lausanne .

. .

Iris: Fuzzy CoCo set−up

ICANNGA’01

8 97.3 % (2)

Iris results: classification (average rule)

Fuzzy CoCoNeurofuzzyConstructive Learning Methods

2 99.33% (2)

3 100 % (1.7)

4 98 % (2.6) 100 % (2.5)

5 100 % (3.3)

Wu (99)

RulesSimple GA

Shi et al (1999)FuGeNeSysRusso (1998)

Fuzzy CoCo

Hong (00) Hung (99) ICANNGA’01

2 98 % (1.5)

99.33% (2.3)3 96.2 % (4)

4 99.33% (2)97.4 % (4)

Rules

Con

trol

ler

Cla

ssifi

er

.

.

.

.Logic Systems Laboratory − Swiss Federal Institute of Technology Lausanne


PWPLSWSL

PWPLSWSL

ClassPL PW

Class

SW

Class

SL

Class

.


Logic Systems Laboratory − Swiss Federal Institute of Technology ..

.

Iris controller−type: A three−rule system

setosa versic.

setosaPWPLSWSL

versic.setosa virgin.PWPLSL

virgin.

SL SW PL PW setosa versic. virgin.

virgin.

SW

versic.



.

.

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Iris classifier−type: A three−rule system

Fuzzy Subsystem

malignant

benign

DiagnosticInput Threshold SubsystemAppraisal

The

dat

abas

e

The

tes

t

..

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Breast cancer diagnosis: the WBCD problem

solu

tion

Pro

pose

d

RulesMembership functions

P d

6 bits

V9.... ....ViV2

Genome length = 54 bits

3 bits

9 Variables * 6 bits

Ai = 0 or 3 (None)

Ai = 1 (Low)

d = [1;8]

3 bits

Ci = 1 (Benign)P = [1;8]

.

. .



V1

Ai = 2 (Malignant)Ai = 2 (High)

.

Genome length = 19*Nr + 1 bits

Nr rules * 19 bits

19 bits 1 bit

1 bit9 * 2 bits

CiA9A1 ....

CoRn...Ri...R1

The genomes

"Fit" cooperatorsElitism rateMutation probabilityCrossover probabilityMaximum generations

Random cooperators

11000 + 100*Nr

[30−90]

{1,2,3,4}1

{0.1−0.6]

Fuzzy CoCo parameters

[0.02−0.3]

Population size

measures the interpretabilityFv : Number of variables

the most important performance measureFc : Classification performance,

F = Fc − a* Fv

Fitness function

. .

. .



Fuzzy CoCo set−up

98.98% (5)

98.83% (5)

98.83% (5)

98.68% (3)

98.54% (4)

98.54% (5)

97.36% (4)

97.73% (3.9)

97.91% (4.4)

98.12% (4.2)

98.18% (4.6)

98.18% (4.3)

97.36% (4.0)

98.25% (4.7)

Evolved fuzzy rulesLearned Boolean rules

Fuzzy CoCo − IEEE TFS 2001AIM 1999

Fuzzy−geneticNeuroRuleSetiono (2000)

Rules

97.36% (4)1

97.36% (3)2

97.80% (6)98.10% (4)3

97.80% (−)4

97.51% (−)98.24% (5)5

98.10% (−)6

97.95% (−)7

Best

97.07% (4)

Average

WBCD results: classification (longest rule)

. .

. .



Classification rate = 98.54%


Logic Systems Laboratory − Swiss Federal Institute of Technology Lausanne .

if (v1 is Low) and (v3 is Low) and (v5 is Low) then (output is Benign)

v

if (v1 is Low) and (v4 is Low) and (v6 is Low) and (v8 is Low) and (v9 is Low) then (output is Benign)

else (output is Malignant)

1 3

v

.

v6 v8

v

v94 v5

.

Two−rules evolved system

.

Five-rule systems: 900.000 fitness evaluations

Five-rule systems: 480.000 fitness evaluations

Single-rule systems: 352.000 fitness evaluations

32.000 * (1000 + 100Nr) {worst case, Ncr=3}

Single-rule systems: 500.000 fitness evaluations

200 * (2000 + 500Nr)Number of fitness evaluations = Np * Gmax

Fuzzy GA: Single population (Peña & Sipper 99)

Computing requirements

Fuzzy CoCo: Cooperative coevolution (CEC-2000)

Number of fitness evaluations = 2 * Np * Gmax * (Ncf + Ncr)


Logic Systems Laboratory - Swiss Federal Institute of Technology Lausanne. .

. .

computer−assistedCOBRA system:

case interpretationreadingprotocol

mammogram

biopsy

recommendation329 benign (neg)187 malignant (pos)

516 readings{Database



. .

. .

The problem: mammography interpretation

input

Fuzzy system

Proposal

Reading form

appraisal

Diagnostic decision unit

BiopsyReading

Database

Web−based user interface

Malignancy


Threshold unit

Logic Systems Laboratory − Swiss Federal Institute of Technology. .

. .

COBRA system: internal view

BinaryContinuousDiscrete

438

Variable type Number

.

. Logic Systems Laboratory − Swiss Federal Institute of Technology


.

.

Understanding the database

not encoded

Continuous variables (e.g., V1):3 var. x 2 par. x 7 bits = 42 bits

Discrete variables (e.g., V3):8 var. x 2 par. x 4 bits = 64 bits

Binary variables (e.g., V2):

Total genome length = 106 bits


.

. .

NoneLow

.

High


Benign Malignant

V V15V3

P1 P15 P’15P’1 P3 P’3

Ri: if (v1 is Ai1) and (v2 is Ai2) and (v3 is Ai3) and ... and (v15 is Ai15) then (output is Ci)

DB

V1 Vi ........ V15V2

.....

Pi P’i

1

Genome encoding for linguistic labels

1 1

20

22

1

Clinical

120

11222222

if Sr = 0

if Sr = 1

+ Radiological

Total genome length = 20 x Nr +1

...Ri...R1

A3A2A1

RB

A15

Co

A11 A12

A9A8A7A4 A5 A6

Rn


Ar2 ... Ar6 Sr CAc1 Ac2 Ac3 Ar1


A10 A14

. .

. .

A13

Genome encoding for rules

Specificity TN

Basic fitness (Fbase)

Accuracy reinforcement

(note: done only if Accuracy > 0.7)

Performance measures and fitness function

αSensitivity + Specificity1 + α

1 + βFbase + Accuracyβ

SensitivityTP

TP + FN

TN + FP

AccuracyTP + TN

TP+TN+FP+FN

TPTP + FN

PPV

..



..

0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.920

5

10

15

20

25

2.502.22Vr

171712

Reff

0.9154

90.89780.8910Fitness

0.9109201510Nr

Best individual

2.4125

2.622.52Vr

15.7814.1512.039.17Reff

2.760.89340.87860.8754Fitness

25201510Nr

Average per class

2.59

2.70

0.8947

5

14

13

2

4

14

22



.

..

.

Fuzzy CoCo results on 65 runs

395/516211/329184/187

Ratio

186/289412/516226/329186/187

Figure

76.55%60.93%

98.40%

Figure

64.36%79.84%68.69%99.47%

Measure

PPV

64.13%SpecificitySensitivity

184/302

Ratio

Accuracy

9−rule17−rule

. .

. .



Performance of two selected systems

184/302395/516

Ratio

63.22%

Threshold = 2 Threshold = 3

SensitivitySpecificityAccuracyPPV

Measure

98.40%64.13%76.55%60.93%

Figure

184/187211/329

187/187208/329

76.55%60.71%

Figure

100.0%

395/516187/308

Ratio


. .

.


.

The 9−rule system with two different thresholds

.

. .


Logic Systems Laboratory − Swiss Federal Institute of Technology.

COBRA system: reading form

Documents

Fuzzy inference - cs.bgu.ac.ilsipper/courses/ecal051/lecon2.pdf · Gas Low OR Pressure Temp. & & & High Hot Cold Low OR High. Logic Systems Laboratory − Swiss Federal Institute