chapter 8 fuzzy inference 模糊推論

Chapter 8

Fuzzy Inference

模糊推論

8. 模糊推論 G.J. Hwang 2

• Fuzzy set and Fuzzy Logic– why “Fuzzy Subset” ?

Ordinary set -- the foundation of present day mathematics.(S)

S : a set

e5 : an element

But in real world , the relation is usually “fuzzy” !John is 170 cm

John : an element

},,,{65325

eeeeSe

}|{ tallisxxor


S = {x|x is tall}

180cm 高的人 S ? Yes179cm 高的人 S ? Yes(179 和 180 只差 1cm)178cm 高的人 S ? Yes(178 和 179 只差 1cm)

•••

170cm 高的人 S ? Yes(170 和 171 只差 1cm)169cm 高的人 S ? Yes No (169 和 170 只差 1cm)

•••

120cm 高的人 S ? Yes(120 和 121 只差 1cm)

Why? 既然 170 是 ,為何 169 不是 ?


S ={x|x is tall}

假如找 100 個人投票 , 互相推選屬於 S 和不屬於 S的人 150cm 160cm 170cm 180cm

00

1

0.5

John is 180cm John S with degree 1.0

John is 165cm John S with degree 0.5

John is 150cm John S with degree 0

JohnS


• ordinary set is a particular case of the theory of fuzzy subset.

let E be a set and A be a subset of E

A E

Characteristic function (x)

x)= 1 if x A (yes)

x)= 0 if x A (no)

e.g. E={x1,x2,x3,x4,x5}

let A = {x2,x3,x5}

x1) = 0, x2 ) = 1, x3)= 1

x4) = 0, x5) = 1


A different representationA = {(x1,0),(x2,1),(x3,1),(x4,0),(x5,1)}

A A= 0

A A = E

IF x A , x A

(x)= 1, A(x)= 0

consider A ={x2,x3,x5}

A(x1) = 1, A(x2) = 0, A(x3) = 0

A(x4) = 1, A(x5) = 0

A = {(x1,1),(x2,0),(x3,0),(x4,1),(x5,0)}


Given two subsets A and B

(x)= 1, if x A

= 0, if x A

(x)= 1, if x B

= 0, if x

AB(x)= 1, if x A B

= 0, if x A B

AB(x)= (x) • (x)

01

0 1

0 00 1

Booleanproduct


Union

AB (x)= 1, if x A B

= 0, if x A B

AB (x)= (x) (x)

Booleansum

e.g. E = {x1,x2,x3,x4,x5}two subsets A and BA={x2,x3,x5}, B={x1,x3,x5}AB = {(x1,0 1),(x2,1 0), (x3,1 1),(x4,0 0),(x5,1 1)}

= {(x1,1),(x2,1),(x3,1),(x4,0),(x5,1)}

01

0 1

0 11 1


• The concept of Fuzzy Subset xi of E 或多或少是 A 的元素

A = {(x1|0.2),(x2|0),(x3|0.3),(x4|1),(x5|0.8)}

Fuzzy Subset x1 屬於 A 的程度 ( 可能由 0~1.0)

A E A is a Fuzzy Subset of E

x2A

A Ex1 , x2 , x3

0.2 0 0.3

membership

通常是主觀的認定 , 但至少表達了 Xis 之間的相對程度


• Zadehs definition of Fuzzy subset

Let E be a set, denumerable or not, let x be an element of E.

Then Fuzzy subset A of E is a set of ordered pairs {(x|(x)},

xE.

Where (x) : grade of membership of x in A

(x) takes its values in a set M (membership set)

x M

IF M={0,1}

fuzzy subset of A will be a nonfuzzy subset

(or ordinary set)

mapping

(x)


E.g.

Let N be the set of natural numbers

N = {0,1,2,3,4,5,6,...}

consider the fuzzy set A of smallnatural numbers

A = {(0/1),(1/0.8),(2/0.6),(3/0.4),(4/0.2),(5/0),(6/0),...}

用傳統的 ordinary set 很難表達

A = {(0,1),(1,1),(2,1),(3,1),(4,1),(5,0),(6,0),...}


S- Function

S (x; ) =

0 for x 2[(x- )/()]2 for x1- 2[(x- )/()]2 for x1 for x

1

0.5

0


Membership Function

A membership Function for the Fuzzy Set TALL

1.00.9

0.5

6.5Height in Feet

6 7

TALL

S (x; ) =

0 for x 2[(x- 5)/(7)]2 = [(x- 5) 2/2] for 5x1- 2[(x- 7)/(7)]2 =1-[(x- 7) 2 /2] for 6x1 for x


close(x; 1

12

( )x

with crossover pointsx =

1.0

0.5

x

Close- Function

close(x;


E = { x|x= 價格合理的牛排 ?}

220NT 120=220NT

=120NT

0.5

1.0

220NT100 340

close(x;


xforxS

xforxSx

),,;(1

),,;(),;(

2

2

function

0.5

1

x


價格合理的牛排

220)420,320,220;(1

220)220,120,20;(

)200,220;(

xforxS

xforxS

x

0.5

1

220NT120 320


Fuzzy Database systems

找一個停車容易 , 且價格合理的餐廳以停車為優先考慮

E = {x|x = 離火車站近的餐館 }

Km

d1

d2

d5

d4

d3


Fuzzy LogicBinary Logic: The logic associated with the Boolean theory of set

Fuzzy Logic : The Logic associated with the same manner with the theory of fuzzy subsets

dialogue

Laws of thought are Fuzzy


A(x) : membership function of the element x in the fuzzy subset A

M = [0,1]

Let A, B be two fuzzy subsets of E and x is an element of Ea = A(x) , b = A(x) a,b,...M = [0,1]

)()(

1

),(

),(

bababa

aa

baMAXba

baMINba


Commutativity

Associativity

~~~~

~~~~

abba

abba

)()(

)()(

~~~~~~

~~~~~~

cbacba

cbacba

~~

~

~~

~~

~

~~~~~~~

~~~~~~~

)(

11

1

)()()(

)()()(

aa

a

aa

aa

a

cabacba

cabacba

Distributivity


DeMorgan s Law

~~~~

~~~~

baba

baba

are true, but not trivial


5 0 0.005 4 0.085 8 0.326 0 0.506 4 0.826 8 0.987 0 1.00

Tall Not Short

5 0 0.00

5 4 0.085 8 0.326 0 0.506 4 0.826 8 0.987 0 1.00

IF tall THEN not short


Tall Not Tall

5 0 0.005 4 0.085 8 0.326 0 0.506 4 0.826 8 0.987 0 1.00

Complementation

5 0 1.005 4 0.925 8 0.686 0 0.506 4 0.186 8 0.027 0 0.00


Not Tall Not Short Middle-Sized

5 0 1.005 4 0.925 8 0.686 0 0.50 AND6 4 0.186 8 0.027 0 0.00

5 0 0.005 4 0.085 8 0.326 0 0.506 4 0.826 8 0.987 0 1.00

5 0 0.005 4 0.085 8 0.326 0 0.506 4 0.186 8 0.027 0 0.00


Linguistic Hedge Operation- Scalar

Ura(x) = rUa(x)

r=0.7

r=0.5

r=0.3


Uar(x) = ((Ua(x))r

r = 0.5

r = 2

r = 4

Linguistic Hedge Operation- Power


UA =supUA(X)

NORM(A)

A

Linguistic Hedge Operation- Normalization


Ucon(A) = UA2(X)

A

CON(A)

Linguistic Hedge Operation- Concentration


UDIL(A)(X) = UA0.5(X)

DILA

Linguistic Hedge Operation- Dilation


UINT(A)(X) = 2(UA(X))2 0 UA(X) 0.51-2(1-UA(X))2 0.5 UA(X) 1.0

INT(A)A

Linguistic Hedge Operation- Intensification


Very A = CON(A)

More Or less A = DIL(A)

Slightly A = NORM(A and not (very A))

Sort of A = NORM(not (CON(A)2and DIL(A))

Pretty A = NORM(INT(A) and not INT(CON(A)))

Rather A = NORM(INT(A))

Usage of Linguistic Hedge Operations


TrueVery trueMore or less trueCompletely trueFalseVery FalseMore or less falseCompletely falseUnknownUndefined

Linguistic truth value


Fuzzy Proposition

“Mr.Wang is young.” is true.

“Mr.Wang is young.” is very true.

“Mr.Wang is young.”is more or less true.


TallHeight Degree of

membership50 0.054 0.158 0.360 0.564 0.868 0.970 1.00

VERY TallHeight Degree of

membership5 0 0.05 4 0.015 8 0.096 0 0.256 4 0.646 8 0.817 0 1.00


~ ~A A

1

0.5

0 X

Figure 5-12Fuzzy Complement

A A = min ( A(X), A(X)) 0.5

A A E

~ ~~

~

~

~~

~~

A A

A A = max ( A(X), A (X) 0.5

~

~ ~


Fuzzy Relation

A crisp relation represents the presence or absence of association, interaction, or interconnectedness between the elements of two or more sets.


Binary Relation• Any relation between two sets X and Y is known as a

binary relation. It is usually denoted by R(X,Y).

Tall 0.00 0.00 0.18 0.50 0.98 1.00 1.00

Heavy(100) (140) (160) (200) (240) (280) (300)

(5 0 ) 0.00 .00 .00 .00 .00 .00 .00 .00(5 4 ) 0.08 .00 .00 .08 .08 .08 .08 .08(5 8 ) 0.32 .00 .00 .18 .32 .32 .32 .32(6 0 ) 0.50 .00 .00 .18 .50 .50 .50 .50(6 4 ) 0.82 .00 .00 .18 .50 .82 .82 .82(6 8 ) 0.98 .00 .00 .18 .50 .98 .98 .98(7 0 ) 1.00 .00 .00 .18 .50 .98 1.00 1.00


Representation of binary relations

Membership matrices

X1X2X3X4

Y1 Y2 Y3 Y4

.9 0 .5 .3 .4 .2 .1 .9 0 0 .5 .6 0 .2 0 .4


R1(x) = 0.6/140 + 0.8/150 + 1.0/160

R2:x 120 130 140 150 160

120 1.0 0.7 0.4 0.2 0.0130 0.7 1.0 0.6 0.5 0.2140 0.4 0.6 1.0 0.8 0.5150 0.2 0.5 0.8 1.0 0.8160 0.0 0.2 0.5 0.8 1.0

y

The Relation APPROXIMATELY EQUAL Defined on Weights

R3(y) = R1(x) R2(x,y)

max min(u1(x),u2(x,y))x

Max_Min Composition


R3(y) = [0.0 0.0 0.6 0.8 1.0]

1.0 0.7 0.4 0.2 0.00.7 1.0 0.6 0.5 0.20.4 0.6 1.0 0.8 0.50.2 0.5 0.8 1.0 0.80.0 0.2 0.5 0.8 1.0R1(x) R2(x,y)

0.0/x1 x1 1.0 y1

+ 0.7

0.0/x2 x2 y2 + 0.4

0.6/x3 x3 y3 + 0.2

0.8/x4 x4 y4 + 0.0

1.0/x5 x5 y5

0.4


R3(120) = max min[(.6,.4),(.8,.2)]= max (.4,.2) = 0.4

R3(130) = max min[(.6,.6),(.8,.5),(1,.2)]= max (.6,.5,.2) = 0.6

R3(140) = max min[(.6,.1),(.8,.8),(1,.5)]= max (.6,.8,.5) = 0.8

R3(150) = max min[(.6,.8),(.8,.1),(1,.8)] = max (.6,.8,.8) = 0.8

R3(160) = max min[(.6,.5),(.8,.8),(1,1)] = max (.5,.8,1) = 1


Composition of Two Fuzzy Relations

R1(x,y)0.1 0.2 0 1 0.70.3 0.5 0 0.2 1 0.8 0 1 0.4 0.3

x1x2x3

y1 y2 y3 y4 y5

z1 z2 z3 z4

0.9 0 0.3 0.40.2 1 0.8 0 0.8 0 0.7 1 0.4 0.2 0.3 0 0 1 0 0.8

y1y2y3y4y5

R2(y,z)

R3(x,z) = ?


0.1 0.9

x1 y1 z1 0.4 0.2 0.2 Max

0 y2 0.8

1 y3 0.7 0.4

y4 0

y5

Mix

R(x) = [0.5 0.2 0.6]R(z) = ?R(z) = R(x) R1(x,y) R2(y,z) = R(x) R3(x,z)


Fuzzy Rules

Membership GradeImage Missile Fighter Airliner

1 1.0 0.0 0.0 2 0.9 0.0 0.1 3 0.4 0.3 0.2 4 0.2 0.3 0.5 5 0.1 0.2 0.7 6 0.1 0.6 0.4 7 0.0 0.7 0.2 8 0.0 0.0 1.0 9 0.0 0.8 0.210 0.0 1.0 0.0

Membership Grades for Images


1/M1

. 9/M + .1/A2

.4/M + .3/F + .2/A3

.2/M + .3/F + .5/A4

.1/M + .2/F + .7/A5

.1/M + .6/F + .4/A6

.7/M + .2/A7

1/A8

.8/M + .2/A9

1/F10


IF IMAGE4 THEN TARGET4 = 0.2/M + 0.3/F + 0.5/A

IF IMAGE6 THEN TARGET6 = 0.1/M + 0.6/F + 0.4/A

+ : set union

假設現由二個不同觀測點得到 IMAGE4 及 IMAGE6

TARGET = TARGET 4 + TARGET 6

= 0.2/M + 0.3/F + 0.5/A + 0.1/M + 0.6/F + 0.4/A

= 0.2/M + 0.6/F + 0.5/A


Maximum and Moments Methods

R1: IF MIX is too-wet

THEN Add sand and coarse aggregate

R2: IF MIX is Workable

THEN Leave alone

R3: IF MIX is too-stiff

THEN Decrease sand and coarse aggregate

.

cement watersand

removed


Fuzzy Production Rule Antecedents for Concrete Mixture Process

TOO STIFF WORKABLE TOO WET

3 4 5 6 7 8 9Concrete Slump (inches)

1.00.90.80.70.60.50.40.30.20.10.0

MembershipGrade


IF Concrete-slump = 6THEN MIX = 0.0/Too-stiff + 1.0/workable + 0.0/Too-wet

IF Concrete-slump = 7THEN MIX = 0.0/Too-stiff + 0.3/workable + 0.0/Too-wet

IF Concrete-slump = 4.8THEN MIX = 0.05/Too-stiff + 0.2/workable +0.0/Too-wet

.

.

.


R1: IF MIX is too-wet

THEN Add sand and coarse aggregate

R2: IF MIX is Workable

THEN Leave alone

R3: IF MIX is too-stiff

THEN Decrease sand and coarse aggregate


Fuzzy Production Rule Consequence for Concrete Mixture Process Control

MembershipGrade

DECREASE SAND AND COARSE AGGREGATE

-20 -10 0 +10 +20Change in sand and Coarse Aggregate (%)

1.00.90.80.70.60.50.40.30.20.10.0

LEAVE ALONEADD SAND AND COARSE

AGGREGATE


Fuzzy InferenceRule 1 : If the car is in short distance and is at a low speed

Then keep the speedRule 2 : If the car is in short distance and is at a high speed

Then decrease the speedRule 3 : If the car is in long distance and is at a low speed

Then increase the speedRule 4 : If the car is in long distance and is at a high speed

Then keep the speed

10 20 30 KM 30 50 70 KM -10 0 10 %

Short distances

long distances

lowspeed

highspeed

decreasespeed

keepspeed

increasespeed


low speed

10 20 30 30 50 70 -10 0 10

10 20 30 30 50 70 -10 0 10

10 20 30

10 20 30 30 50 70

30 50 70 -10 0 10

-10 0 10

Mass Center Z

-10 0 10

Distance: 15m Speed: 60m/h

short distance low speedkeep the speed

short distance High speeddecrease the speed

long distance

long distance High speed keep the speed

increase the speed

0.8

0.4

0.75

0.3

0.3

(0.4*0+0.75*(-10)+0.3*10+0.3*0)/

(0.4 +0.75+0.3+0.3))=-2.57

0.8

0.8

0.3

0.3

0.75

0.75

0.4

0.4

miles

miles

miles

miles

meter

meter

meter

meter %

%

%

%


Knowledge Acquisition for Fuzzy Expert Systems

Step 1: Elicit all of the elements (concepts to be learned) from the domain

expert.

Li K Fr F Cl I


Step 2: Elicit attributes ( properties or fuzzy variables).

Li K Fr F Cl I

boiling point LOW;MIDDLE;HIGHatom radius NARROW;NORMAL;WIDE

metalloid WEAK;NORMAL;STRONGnegative charge WEAK;MIDDLE;STRONG


Step 3: Fill all of the [concept, attribute] entries of the grid. A 7-scale (-3 to +3) rating and the degree of certainty(“S”,”N”).

Consider the ratings of fuzzy variable ‘boiling point’: 3 means VERY HIGH, 2 means HIGH, 1 means MORE OR LESS HIGH, 0 means MIDDLE,-1 means MORE OR LESS LOW, -2 means LOW,-3 means VERY LOW‘S’ means ‘VERY SURE’, ‘N’ means ‘NOT VERY SURE’

Li K Fr F Cl I

boiling point -1/N 0/N 1/N 1/S 2/S 3/S LOW;MIDDLE;HIGHatom radius -2/S -1/S 1/N 1/S 2/S 3/S NARROW;NORMAL;WIDE

metalloid 1/S 2/S 3/S -3/S -3/S -3/S WEAK;NORMAL;STRONGnegative charge -3/S -3/S -3/S 3/S 2/S 1/S WEAK;MIDDLE;STRONG


Step 4: the first column of the above fuzzy table is translated to the following rule:

IF boiling point is MORE OR LESS LOW, andIF boiling point is MORE OR LESS LOW, and atom radius is NARROW, andatom radius is NARROW, and

metalloid metalloid is MORE OR LESS STRONG, andis MORE OR LESS STRONG, and

negative chargenegative charge is VERY WEAKis VERY WEAK

THEN the element could be Li THEN the element could be Li TRUTH = 0.8TRUTH = 0.8

TRUTHTRUTH = = # " "

(# " " # " "). .

of S

of S of N 08 0 2


Some default functions(LS(x),RS(x),MS(x))

1 f or x

f or x

f or x

f or x

1 2

2

0

2

2

( )

( )

x

x

0 f or x

f or x

f or x

f or x

2

1 2

1

2

2

( )

( )

x

x

0 f or x

f or x+2

f or +2

x

f or x2

f or 2

x

f or x

2

1 2

1 2

2

0

2

2

2

2

( )( )

( )( )

( )( )

( )( )

x

x

x

x


• Numerical value 250 for fuzzy value LOW(LS), 300 for MIDDLE(MS), and 350 for HIGH(RS), membership functions given as below:

F(X) NOT = 1 – X F(X) MORE-OR-LESS = X0.5

• The fuzzy inference process map the inputs to the corresponding membership functions and obtains linguistic variables.

1 - -0.8- -0.6- -0.4- -0.2- -0.0-

250 300 350

350

Low Middle High

High


Knowledge Integration

While elicit expertise from multiple experts, someproblems may arise:

same element or attribute

different vocabularies

Unified vocabularies

Different ratings

Conflict happen

Insist their ratings

?

??

???


CLIPS

InteractiveKnowledgeElicitation

Unit

KnowledgeAnalysis

Unit

Tutoring

Negotiation

JAVA-based Communication Unit Fuzzy Reasoning Interface

KnowledgeBase

GeneratorKnowledge

Base

Computer Networks

CLIPS

Strategy

Unit

Knowledge Integration for a Fuzzy Tutoring System

Educator 1 Educator 2 Educator n

63G.J. Hwang8. 模糊推論

Interactive knowledge elicitation unit

Interactiveuser

interface

Fuzzy tableeditor

Membership

function

builder

Knowledge

base


• NEGOTIATOR:Please give a set of elements(concepts to be learned or decisions to be made).• EDUCATOR: Li, K, Fr, F, Cl, I. . . .• NEGOTIATOR: Select a set of fuzzy values for fuzzy variable “boiling point”: 1. LOW/MIDDLE/HIGH 2. SHORT/MIDDLE/TALL 3. LIGHT/NORMAL/HEAVY 4. SMALL/MIDDLE/BIG 0. Other (user-defined)• EDUCATOR: 1• NEGOTIATOR: Select a set of fuzzy values for fuzzy variable “atom radius”: 1. LOW/MIDDLE/HIGH 2. SHORT/MIDDLE/TALL 3. LIGHT/NORMAL/HEAVY 4. SMALL/MIDDLE/BIG 0. Other (user-defined)• EDUCATOR: 0• NEGOTIATOR: Indicate the lower bound of the fuzzy values.• EDUCATOR: NARROW• NEGOTIATOR: Indicate the middle of the fuzzy values.• EDUCATOR: NORMAL• NEGOTIATOR: Indicates the upper bound of the fuzzy values.• EDUCATOR: WIDE

Interactive user interface


Fuzzy table editor


Membership function builder


Fuzzy reasoning interface

Users

FuzzyInterface

outputs CLIPS

FuzzyInput Data

Membership Function

s

Control Rules

Domain Rules with fuzzy Expressions

input

data Defuzzification

Fuzzification

Fuzzy Inference


Knowledge analysis unit

Li K Fr F Cl I

boiling point -1/N 0/N 1/N 1/S 2/S 3/S LOW;MIDDLE;HIGHatom radius -2/S -1/S 1/N 1/S 2/S 3/S NARROW;NORMAL;WIDE

metalloid 1/S 2/S 3/S -3/S -3/S -3/S WEAK;NORMAL;STRONGnegative charge -3/S -3/S -3/S 3/S 2/S 1/S WEAK;MIDDLE;STRONG

• Check if conflict occurs and integrate tutoring strategies.• The contents of a fuzzy table is represented as

Fuzzy_value(Educator_ID, Object_name, Fuzzy_variable) and Certainty_Degree (Educator_ID, Object_name, Fuzzy_variable)

for examples, the fuzzy table below can represented as Fuzzy_value(Educator1, Li, boiling point) = -1 Certainty_Degree(Educator1, Li, boiling point) = “N” ...


Knowledge analysis rulesKnowledge analysis rulesRule_analysis_02Rule_analysis_02:

IF (1) IF (1) Current_Phase is Knowledge_Analysis Current_Phase is Knowledge_Analysis andand

(2) Fuzzy_value(Expi, Gk, Vs)(2) Fuzzy_value(Expi, Gk, Vs)Fuzzy_value(Expj, Gk, Vs)Fuzzy_value(Expj, Gk, Vs) < 0 and < 0 and

(3) Certainty_Degree (Expi, Gk, Vs) (3) Certainty_Degree (Expi, Gk, Vs) is is "S""S" and and (4) Certainty_Degree(Expj, Gk, Vs) (4) Certainty_Degree(Expj, Gk, Vs) is is ”N” ”N” andand

THEN (a) Set THEN (a) Set Suggested_Fuzzy_ValueSuggested_Fuzzy_Value be be Fuzzy_value(Expi, Gk, Vs) Fuzzy_value(Expi, Gk, Vs) andand

(b) Set (b) Set Suggested_Certainty_Degree be ”N" Suggested_Certainty_Degree be ”N" andand

(c)(c) Set Set Current_Phase Current_Phase bebe Knowledge_Negotiation Knowledge_Negotiation


Rule_analysis_04:


(2) Fuzzy_value(Expi, Gk, Vs)(2) Fuzzy_value(Expi, Gk, Vs)Fuzzy_value(Expj, Gk, Vs)Fuzzy_value(Expj, Gk, Vs) 0 and 0 and

(3) Certainty_Degree (Expi, Gk, Vs) (3) Certainty_Degree (Expi, Gk, Vs) is is "S""S" and and (4) Certainty_Degree(Expj, Gk, Vs) (4) Certainty_Degree(Expj, Gk, Vs) is is "S” "S” andand

(5) Fuzzy_value(Expi, Gk, Vs) (5) Fuzzy_value(Expi, Gk, Vs) Fuzzy_value(Expj, Gk, Vs) Fuzzy_value(Expj, Gk, Vs) 00

THEN (a) Set THEN (a) Set Suggested_Fuzzy_ValueSuggested_Fuzzy_Value be be Fuzzy_value(Expi, Gk, Vs) Fuzzy_value(Expi, Gk, Vs) andand

(b) Set (b) Set Suggested_Certainty_Degree be "S" Suggested_Certainty_Degree be "S" andand

(c)(c) Set Set Current_Phase Current_Phase bebe Knowledge_Negotiation Knowledge_Negotiation


Rule_analysis_03:


(2) Fuzzy_value(Expi, Gk, Vs)(2) Fuzzy_value(Expi, Gk, Vs)Fuzzy_value(Expj, Gk, Vs)Fuzzy_value(Expj, Gk, Vs) < 0 and < 0 and

(3) Certainty_Degree (Expi, Gk, Vs) (3) Certainty_Degree (Expi, Gk, Vs) is is "S""S" and and

(4) Certainty_Degree(Expj, Gk, Vs) (4) Certainty_Degree(Expj, Gk, Vs) is is "S” "S” andand

THEN (a) Set THEN (a) Set Suggested_Fuzzy_ValueSuggested_Fuzzy_Value be be “Conflict” “Conflict” andand

(b)(b) Set Set Current_Phase Current_Phase bebe Knowledge_Negotiation Knowledge_Negotiation


Tutoring Strategy Negotiation unit

• Present suggestions by knowledge analysis unitPresent suggestions by knowledge analysis unit• When a When a conflictconflict occurs, experts are asked to give suggestions. occurs, experts are asked to give suggestions.

“ “ over-generalover-general” ” happenhappen

BearBear

American gray bearAmerican gray bear bear of North bear of North PolePole

invoke invoke Object_Specialization Object_Specialization

procedureprocedure

An example


• Converting fuzzy table to the format of the tutoring strategy system shell (e.g., CLIPS format shown in the followings).

(deffacts initial-state (is boiling-point MORE-OR-LESS LOW) (is atom-radius NARROW) (is metalloid MORE-OR-LESS STRONG) (is negative-charge VERY WEAK))

(defrule Rule1 ?x1 <- (is ?X1 MORE-OR-LESS LOW) ?x2 <- (is ?X2 NARROW) ?x3 <- (is ?X3 MORE-OR-LESS STRONG) ?x4 <- (is ?X4 VERY WEAK) => (retract ?x1 ?x2 ?x3 ?x4) (assert (is Li -1-21-3)) (assert (CF 0.8)) (printout t ”Li is -1-21-3 with CF=0.8" crlf))

Knowledge base generator


Illustrative example• Eliminate redundancy and incompleteness of elements and attributes.


• Select or define a set of fuzzy values for each fuzzy variable.


• Three educators fill the fuzzy values with degree of certainty.• The system invokes knowledge analysis rules.


• Check conflict values and decide if invokes Object_Specialization procedure.• Generate fuzzy rules.


Applications of Fuzzy Logic

• Fuzzy Expert Systems– Fuzzy Inferences in Expert Systems– Learning Mechanisms for Fuzzy Expert Systems– Knowledge Acquisition for Fuzzy Expert Systems

• Fuzzy Database Systems– Fuzzy Query Language– Fuzzy Database Management


Case Study: ITED-An Intelligent Tutoring, Evaluation and Diagnostic System

www.ited.im.ncnu.edu.tw


Prerequisite relationships among concepts

• Effectively learning a scientific concept normally requires first learning some basic concepts

• Consider two concepts Ci and Cj. If Ci is prerequisite to efficiently performing the more complex and higher level concept Cj, then a concept effect relationship Ci Cj is said to exist


Addition of integers

Positiveintegers

Multiplicationof integers

Division ofintegers

Subtraction of integers

Negativeintegers

Zero

Primenumbers

This is an example of concept effect

relationships for Integers and the relevant

operations


Conceptual Effect Table (CET)

Cj C1 C2 C3 C4 C5 C6 C7 C8

Prerequisite

Zero Positive integers

Addition Subtrac-tion

Multipli-cation

Negative integers

Division Prime numbers

C1 0 0 0 1 0 0 0 0 C2 0 0 1 0 0 0 0 0 C3 0 0 0 1 1 0 0 0 C4 0 0 0 0 0 0 0 0 C5 0 0 0 0 0 0 0 0 C6 0 0 0 0 0 1 1 0 C7 0 0 0 0 0 0 0 1

Ci

C8 0 0 0 0 0 0 0 0 NPj 0 0 1 2 1 1 2 1

Those concept effect relationships can be

represented as a CET.

e.g. C3 C4


Test Item Relationship Table (TIRT)

Concept Cj Prerequisite

C1 C2 C3 C4 C5 C^ C7 C8 Q1 1 0.2 0 0 0 0 0 0 Q2 0 0.8 0.4 0 0 0 0 0 Q3 0 0 0.6 0.2 0 0 0 0 Q4 0 0 0 1 0 0 0 0 Q5 0 0 0 0 0 0 0 0 Q6 0.2 0 0 0 0.8 0.2 0 0 Q7 0 0 0 0 0 1 0 0

Q8 0 0 0 0 0 0 0.6 0.4 Q9 0 0 0 0 0.2 0 0 0

Test item

Qi

Q10 0 0 0 0 0.2 0 0.4 1

The relationships among each test item and each concept can be

represented as a TIRT.

O: Not relevant 1: Very strongly relevant


Student Answer Sheet table (AST)

Test item Student

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10

S1 0 0 1 0 0 1 1 0 0 0 S2 0 1 1 0 0 1 1 0 0 0 S3 0 0 0 1 0 1 1 0 0 0 S4 0 1 1 1 0 0 1 0 0 0 S5 0 0 1 0 0 0 1 1 0 0

An AST is used to record the answers of the students to each test items.

O: The student has correctly answered the test item

1: The student failed to correctly answer the test item


Performing Max-Min Composition

Error_Degree (Si, Cj) = AST 。 TIRT 。 CET

0000000000

0000000000

0000000010

0000000101

0000000100

0000100000

0001100000

0100000000

14.002.00000

0002.00000

4.06.0000000

00100000

002.08.00002.0

00000000

00001000

00002.06.000

000004.08.00

0000002.01

0011000100

0001001110

0001101000

0001100110

0001100100

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

5

4

3

2

1

C

C

C

C

C

C

C

C

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

S

S

S

S

S

04.006.06.1002.06.02.0

00000.1002.10.12.1

00002.1008.100.1

00002.1000.10.12.000002.1000.16.02.0

10987654321

5

4

3

2

1

CCCCCCCCCC

S

S

S

SS


Generate learning guidance

IF Learning_Status (Si, Cj) is Poorly-learned

THEN Arrange for Student Si to re-learn the unit containing Concept Cj

IF Learning_Status (Si, Cj) is Partially-learned

THEN Arrange more practice concerning Concept Cj for Student Si

IF Learning_Status (Si, Cj) is Well-learned

THEN Record that Student Si has passed the study of Concept Cj


Membership functions for Learning Status

Error_degree (Si, Cj)0 0.5 1.0

Well-Learned Poorly-LearnedPartially- Learned

Lea

rnin

g_st

atus

(S

i, C

j)

1.0


Learning guidance generated by ITED


Case Study: 網路學習行為分析–學習效率 (Efficiency of Learning)

–學習意願 (Willingness)

–耐心度 (Patience)

–專心度 (Concentration)

–閒置 (Idleness)

–理解度 (Comprehension)

–聊天 (Chat)


學習意願分析• 學生用心學習的意願• 分析依據：有效登入時間 /登入時間

模糊推理法則If willingness is low Then insert INT(T×0.5) corresponding willingness frames.

If willingness is average Then insert INT(T×0.25) corresponding willingness frames

If willingness is high Then keep the current status.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

low average highdegree

ELT/LT


專心度分析• 學生集中精神於瀏覽教材的程度• 分析依據：回應時間

模糊推理法則If concentration is lowThen insert a corresponding concentration frame.

If concentration is highThen keep the current status.

If concentration is noresponseThen keep the current status.

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 1

low no responsehigh

RT

degree


聊天狀態分析• 學生利用線上討論區來閒聊而不是討論課程• 分析依據：學習相關比率

模糊推理法則If chat is highThen record this status and warn the student.

If chat is averageThen keep the current status.

If chat is lowThen keep the current status

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

lowaveragehigh

degree

LR

chapter 8 fuzzy inference 模糊推論

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