. , MSc

-

2007

A-PDF Merger DEMO : Purchase from www.A-PDF.com to remove the watermark

. , MSc

-

,

, , , , , , ,

2007

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, ,

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, , . .

,

, .

, .

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. ,

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, .

, 2007

.

............................................................................... 1 ..................................................................................................... 9 I. ...................................................................................................................9

II. H .........................................10

. .....................................................................................13

V. ..............................................................................22

1- 1.1 ................................................................................................................25

1.2 .....................................................................................26

1.3 (ARTIFICIAL INTELLIGENCE).............................27

1.4 ..................................................................................................29

1.5 (CRISP SETS) ....................30

1.6 ..................................................................................................33

1.6.1 ........................................................36

1.6.2 .....................................................................36

1.6.3 (support of a fuzzy set) .............................38

1.6.4 (cardinality) .................................................38

1.6.5 (Normal fuzzy set) ................................................39

1.6.6 .........................................................................39

1.6.7 ......................................................................40

1.6.8 ...................................................................40

1.6.9 (inclusion) .......................................................41

1.6.10 - (-level set-credibility level) ...............................41

1.7 ...............................................................................................42

1.7.1 .....................................................................................43

1.8 ........................................................43

1.9 .....................................................49

1.10 (DECOMPOSITION PRINCIPLE).................50

- 1 -

1.11 .......................................................50

1.12 NORMS..................................................................................................................52

1.12.1 t-norm t-conorm............................................54

1.12.2 t-norm t-conorm..............................................................................54

1.12.3 norm ...........................................................55

1.13 (CARTESIAN PRODUCT).............................55

1.14 (EXTENSION PRINCIPLE).................................56

1.15 ...............................................57

1.16 (LINGUISTIC VARIABLES) ...............................59

1.16.1 (linguistic modifiers) ...............................................60

1.17 .........................................61

1.17.1 .........................................................................61

1.17.2 (possibility theory) .....................................................63

1.17.3 .....................................64

1.17.4 - ...............................................................................67

2- 2.1 ...............................................................................................................69

2.2 .........................................................................70

2.3 FRB ...........................................................................................72

2.4 ....................................................................73

2.5 ............................................................75

2.5.1 (completeness).............................................................................75

2.5.2 (redundancy) ..................................................................................75

2.6 ..............................................................76

2.6.1 ....................................77

2.7 (FUZZIFICATION) ............................79

2.7.1 ........................................80

2.7.2 .......................................................................................................81

2.7.3 ................................................................83

2.8 ...........................................................................83

2.9 .............................................................84

2.10 (DEGREE OF FULFILLMENT) .......84

2.10.1 ...............................85

- 2 -

2.11 .....................................................................................87

2.12 (AGGREGATION) .........................87

2.12.1 ..........................95

2.13 (DEFUZZIFICATION PROCEDURE).........96

2.13.1 .................................................................................97

2.13.2 ...............................................................108

2.14 ....................................................109

2.14.1 (training set) ..........................................................109

2.14.2 (rule verification-validation)......117

2.14.3 (Artificial neural networks) .............................119

3- - 3.1 .............................................................................................................123

3.2 ................................................................124

3.3 ................126

3.3.1 ............................................................................127

3.3.2 ...................................................................................127

3.4 ..............................................................................................................128

3.4.1 ....................................................................128

3.4.2 .......................................................129

3.5 ................................................................129

3.6 .......................................................................................................132

3.6.1 ...........................................................132

3.6.2 ...............................................................135

3.6.3 ..........................................................................................137

3.6.4 .....................................................................138

3.7 ..................................................139

3.7.1 .................................................................................................139

3.8 ...............................................................................................140

3.9 2 .........................147

3.9.1 ................................................................148

3.9.2 ....................................................................................................149

3.9.3 .....................................................................150

3.9.4 ...............................................................................................150

- 3 -

3.10 ............................................................................151

3.11 ............152

3.11.1 ...................................................................153

3.11.2 .........................................................................153

3.11.3 .............................................................................................153

3.12

..............................................................................................154

3.13

..............................................................................................162

4- 4.1 ..............................................................................................................169

4.2 ...........................................................................................170

4.3 .......................................................................................................171

4.4 .....................................................................................................173

4.5 ...................................................................................................................174

4.6 ...............................................................................................................174

4.7 .............................................................................................175

4.7.1 .......................................................................175

4.7.2 ............................................................176

4.8 .....................................................177

4.9 .................................................................................................177

4.10 - ...................177

4.10.1 ............................................................................177

4.10.2 .....................................................................................................178

4.10.3 ....................................................................................................181

4.10.4 ..........................................................................182

4.10.5 .............................................................................................183

4.10.6 ....................................................................................................184

4.11 ........................................................184

4.12 ..................................................................................................185

4.12.1 ........................................................................................186

4.12.2 ................................................................................................187

- 4 -

4.12.3 .......................................................................................187

4.12.4 ....................................................................................................190

4.12.5 ...............................................................................190

4.13 .....................................................................................191

4.13.1 ............................................................................................191

4.13.2 ..................................................................................192

4.13.3 ........................................................................................................192

4.14 .........................................................................................192

4.14.1 .......................................................................................................192

4.14.2 .........................................................................................................193

4.15 ...........................................................................193

4.16 .................................................................194

4.17 ............................................................194

4.18 ..................................................................................195

4.18.1 ........................................................................................................195

4.18.2 ..............................................................................196

4.18.3 ..........................................................................................................197

4.18.4 - .............................................................................198

4.19 .......................................................................................................199

4.19.1 ........................................................................................................199

4.19.2 ................................................................................................200

4.19.3 ...........................................................................................201

4.20 ..............................................................................201

4.20.1 .....................................................................201

4.20.2 - ..................................................................................201

4.20.3 ........................................................202

4.20.4 ......................................................................................................203

4.21 ...........................................203

4.21.1 .....................................................................................................203

4.21.2 ..................................................................................203

4.21.3 .........................................................................................204

4.21.4 ....................................................................................................206

4.22 ....................................................................206

4.22.1 .......................................................................207

4.23 ................................................................................210

- 5 -

4.24 .................................................................................211

4.24.1 ..................................................................................................211

4.24.2 .........................................................................................212

4.24.3 ..........................................................................................213

4.24.4 ..................................................................................................214

4.24.5 .......................................................................................................215

4.24.6 ...............................................................................215

4.24.7 .............................................................................216

4.25 ...............................................................................218

4.25.1 - .............................................................................219

4.26 ...........................................................................................219

4.27 .........................220

4.28 -..................................................................221

4.28.1 ..........................................................................................................221

4.28.2 Thornwaite.................................................................................222

4.28.3 Thornwaite-Mather....................................................................223

4.28.4 .....................................................................................225

4.28.5 ............................................................................225

4.28.6 ...................................................................................................226

4.28.7 ............................................227

4.29 ........................................................................................................228

4.29.1 ......................................................228

4.29.2 .......................................................................228

4.29.3 .......................................................................228

4.29.4 Blaney-Criddle.............................................................................229

4.30 ANFIS (Adaptive network fuzzy inference

system)..........................................................................................................................232

4.30.1 . .......................................................................................................234

. ........................................................................................................255

.

- ..255

- 6 -

.

......................................................................................................................................258

IV.

................................................259

V.

..............................260

.......................................................................................... 263 Summary ................................................................................................. 273

- 7 -

- 8 -

I.

,

,

,

.

, ,

.

,

.

,

.

,

, ,

.

,

.

,

.

.

,

, .

- 9 -

. -,

, , , -

(, 2003).

.

. ,

,

.

.

. II. H

.

.

,

.

,

.

. ,

,

(, 2005).

1965 Lotfi A. Zadeh .

, .

,

- 10 -

.

, , ,

. ,

.

.

(fuzzy rule-base, FRB) (Bardossy

and Duckstein,1995). ,

.

,

, .

,

.

, ,

,

. ,

.

,

.

(, 2006). :

1.

.

2. ,

.

3. ,

.

4. ,

.

- 11 -

,

. ,

,

. ,

,

.

,

, .

,

.

,

.

,

, .

, .

.

,

,

, ,

.

.

- 12 -

.

,

1996

:

Brdossy (1996)

.

.

,

. ,

,

, .

Brdossy

.

:

1,i

2,i

.

(training set).

(

) ,

.

(counting

algorithm),

(normed weighted sum combination).

(fuzzy

mean).

Pesti et al (1996)

.

(atmospheric circulation

- 13 -

patterns, CP),

(Palmer Drought Severity Indices, PSDI).

Shrestha et al (1996)

.

,

.

:

i,1 Ai,2

Ai,3 Ai,4

Bi

, (fuzzy mean)

b :

( )I

1i ii

1ii

ii

1b

=

=

=

I BM1

( )= dxxBi

+1 , i i, (i)

i ,

( ).

, (engineering sustainability)

(engineering risk). ,

(reliability), (vulnerability) (repairability),

, (reliability),

(incident period) (repairability) .

Chang F and Chen L., (1998)

.

(Real-Coded and Binary-Coded) ,

. ,

- 14 -

. (

)

a, b, c, d

. :

1. (R1t) = a *

2. (R2t) = b *

3. , (R3t) = c *

4. , (R4t) = d *

.

Pongracz et al (1999)

(SOI, Southern Oscillation Index)

El Nino La Nina.

, ,

.

(Palmer Drought Severity Index,

PDSI).

(Principal Component Analysis, PCA)

(cluster analysis) k- (k-Means).

(weighted counting algorithm).

AND

:

(X 1,j A(l1) X 2,j A(l2) X 10,j A(l10)) Yj B(l)

Abebe et al. (2000)

.

.

, ,

.

- 15 -

,

.

Panigrahi and Mujumdar (2000)

(single purpose).

.

.

( ) ( ) feasible1;i,kj,1fPBMini,kfj

1n1t

tijkilt

nt

+= +

( )ntf k, i t, Bkilt t, k

, i ,

i t j t+1 l

t. H

.

tijP

36 10 .

. ,

.

(10)

( ),

. AND.

Chang L. and Chang F. (2001)

.

,

.

- 16 -

Hundecha et al (2001)

,

.

.

, ,

.

.

:

( ) ( )( )

+

==

dtt

dtttBMb

B

B

+

t , (t) ()

.

Jolma et al (2001)

(case based).

.

,

,

,

.

.

,

, .

(product inference),

. t-norm

Lukasiewicz (bounded difference-sum) .

,

( t-

norm). t-norm

, .

- 17 -

Pongracz et al (2001)

, ,

.

CP (atmospheric Circulation

Pattern) SOI (Southern Oscillation Index)

,

.

63

(weighted counting

algorithm).

, 31 .

(standard deviation)

.

Xiong et al (2001)

,

(rainfall-runoff)

Takagi-Sugeno-Kang.

(pattern recognition) .

,

.

, :

( ) ( )r

2r i i r S

a Q exp Q = S (

) Qi .

(SAM),

(WAM) (NNM).

.

.

- 18 -

Dubrovin et al (2002)

(total fuzzy similarity).

,

.

(WREF)

,

.

, (

, , ).

(SWE)

, :

IF SWE is smaller than average/average/larger than average/much larger than

average, THEN WREF is high/middle/low/very low

(

WREF) (

).

Mahabir et al. (2003) 27

.

, .

:

,

.

:

,

- 19 -

.

Vernieuwe et al (2003)

Takagi-Sugeno-Kang

. Takagi-Sugeno-Kang

Mamdani . Mamdani

.

Takagi-Sugeno-Kang

. :

If P(k) is Ai AND Q(k) is Bi THEN Qi(k+1)=aiP(k)+biQ(k)+di

Nash-Suttcliffe

. Nash-Suttcliffe

:

( ) ( )( )( )( )

=

=

= N

1k

2

obsobs

1kobsm

QkQ1NS N 2kQkQ

,

mQ obsQ

obsQ

. NS ,

.

:

( ) ( )( )N

RMSE 1kmobs

==kQkQ

N2

ANFIS

(Adaptive Network-based Fuzzy Inference System)

.

Keskin et al (2004)

.

,

.

184

:

- 20 -

IF Ta is Ta(p) and Tw is Tw(p) and Rc is Rc(p) and Pa is Pa(r) THEN E is E(p)

p = 1, 2,..,8, r = 1,,4., , Tw

, Rc .

Kronaveter et al (2004)

.

,

.

.

.

, .

,

.

Nayak et al (2005)

.

Takagi-Sugeno-Kang.

(subtractive clustering algorithm)

(linear least squares).

Pao-Shan Yu and Shien-Tsung Chen (2005)

.

.

,

.

.

.

- 21 -

V. :

1.

,

.

,

.

, .

2.

Visual Basic.

.

,

.

,

,

.

.

,

. ,

. :

. ,

.

.

.

- 22 -

,

.

.

,

.

.

,

.

.

.

.

.

.

,

1980-2003 .

, , ,

.

,

.

.

- 23 -

.

ANFIS

.

.

.

,

.

- 24 -

1:

1 1.1

.

,

.

.

.

, , , ,

.

.

,

. H

,

.

,

,

.

.

1965 Zadeh ,

.

.

Zadeh,

,

- 25 -

1:

,

.

.

,

.

.

.

.

, .

,

.

,

.

1.2

.

,

.

:

;, ; .

, .

.

, . ,

.

,

- 26 -

1:

.

,

. ,

(logical language),

.

, ,

.

, .

.

. ,

. L.

Zadeh [1965]:

.

,

.

,

.

,

,

. ,

,

, .

1.3 (ARTIFICIAL INTELLIGENCE)

.

- 27 -

1:

, .

,

. ,

(Artificial Neural Networks, ANN),

(Genetic Algorithms, GA) (Intelligent

control theory).

. (Chang and Chang, 2001).

(expert systems),

,

.

(Decision Support Systems).

, : , ,

.

:

1.

2. .

3.

.

:

.

, .

. .

.

.

.

- 28 -

1:

.

:

.

.

.

,

.

1.4

. ,

. ,

,

, .

,

.

Zadeh (1965)

.

, , ,

..

,

.

, , , , .

.

- 29 -

1:

.

.

.

(causal)

, .

,

.

,

.

1.5 (CRISP SETS)

, (crisp sets), :

, R Z.

, ,

(characteristic function):

( ) { }1,0xEx A . ,

(1 0), (Kaufmann and Gupta,

1991).

( )A x 1 =

( )A x 0 =

1.1: (x) .

- 30 -

1:

, ,

, .

,

,

(union) ,

, : { AxxBA == }Bx 1.2.

1.2: .

,

, :

( ) ( )BAB,BAA (intersection) ,

, : { AxxBA == }Bx

1.3: .

,

, :

( ) ( ) BBA,ABA - 31 -

1:

(complement) A ,

,

, { AxxA = }

( 1.4):

__

1.4: A ,

.

:

XAA = , (The law of excluded middle)

OAA /= , (The law of contradiction), O/ .

:

1. (Idempotent law)

AAA = , AAA = 2. (Commutative law)

ABBA = , ABBA = 3. (Associative law)

C)BA()CB(A,C)BA()CB(A

==

4. (Distributive law)

)CA()BA()CB(A),CA()BA()CB(A

==

5. (The law of double negation) AA =

6. Morgan (De Morgans law)

BABA

,BABA

==

- 32 -

1:

(Ganoulis, 1994), (, 2005).

1.6

. Zadeh

[0, 1]. x

(membership function). Zimmermann (1985)

( , , 2006)

.

(1998)

.

,

.

1 .

x .

,

.

. (

2000 ;)

.

2001 1999 ,

.

.

: 1 , .

- 33 -

1:

( )[ ] ( ) [ ]{ } 1,0x,Xx; x,x A AA = (x) x .

,

.

x

. (x)

, x .

x

.

.

x

[0, 1] ,

.

,

,

, .

( ):

(universe of discourse)

.

- 34 -

1:

0

1

(x)

1

(x)

0

1.5: (Goktepe et al., 2005)

: 1.6 1.7

, . (

) ( ) .

1

(x)

0 500 1000 (m)

1.6:

(x)

0

1

1000500 750 1250 (m)

1.7:

- 35 -

1:

. 800 m

.

800 0 1 0

800

.

800 m

0,9 , 0

0,1 .

800 0 0,9 0,1

1.6.1

,

.

.

1.6.2 ,

, .

,

:

.

.

- 36 -

1:

x1, x2, x3 =

A(x1)/x1 + A(x2)/x2 + A(x3)/x3 x1, x2, x3

A(x1), A(x2), A(x3) .

+ /

.

. :

( )( ){ }Xx x,x A A =

10. :

= 0.1/7 + 0.5/8 + 0.8/9 + 1/10 + 0.8/11 + 0.5/12 + 0.1/13

= {(7, 0.1), (8, 0.5), (9, 0.8), (10, 1), (11, 0.8), (12, 0.5), (13, 0.1)}

.

,

.

10C 30C

. ( )

:

1 x

1:

010 20 30 40

1

C

1.8:

1.6.3 (support of a fuzzy set)

x . :

( ) ( ){ } 0x;x Apsup A >=

.

1.6.4 (cardinality) I,

:

( )==

I

1iA

~xA (1.2)

.

- (relative cardinality) :

XA

A = (1.3) X .

:

( )= x A~

dxxA ~ (1.4)

- 38 -

1:

:

6/3.05/7.04/13/8.02/5.01/2.0A~ +++++=

5.3=3.0+7.0+1+8.0+5.0+2.0=A~

:

5833.065.3A ==

1.6.5 (Normal fuzzy set)

. 1.9

.

0,5

.

( ) 1xmax aXx =

1 1

0 0

0.5

(x) (x)

1.9: () ()

(x) supx(x).

.

1.6.6

- 39 -

1:

. [ ] ( ) Xx,x 1,0 21 , :

( )[ ] ( ) ( )[ 2A1A21A x,xminx1x ]+ (1.5)

(

).

.

:

f(x1 + (1-)x2) f(x1) + (1-) f(x2)

.

(x) (x) 1 1

0 0

1.10: () ()

1.6.7

1.5

.

:

1. (The law of excluded middle)

XAA 2. (The law of contradiction)

AA (Tanaka, 1996).

1.6.8 . :

( ) ( ) X x xxBA XBA == (1.6) .

- 40 -

1:

1.6.9 (inclusion)

,

, .

, ,

:

~A

~B

~A

~B

~A

~B

~A

~B

Xx),x()x(B~A~ BA (1.7)

1.6.10 - (-level set-credibility level)

-.

, 1 0 . -:

- - (strong -level set): ( ){ } xXx A Aa >= - - (weak -level set): ( ){ } xXx A Aa =

.

-

.

,

[0, 1].

- . H -

(Kaufmann and Gupta, 1991).

: = {(1, 0.2), (2, 0.5), (3, 0.5) (4, 1), (5, 0.7), (6, 0.3)}.

- :

0.2 = {1, 2, 3, 4, 5, 6} = 0.5 = {2, 3, 4, 5} 0.7 = {4, 5} 1 = {4}

- = 0.7 0.7 = {4}

- :

- 41 -

1:

1. ( ) = BABA ~~ 2. ( ) = BABA ~~ 3. = 0.5

4.

__

~AA

10 AA(Ross, 1995)

1.7

, :

1. .

2. Rx0 (x) .

3. .

4. (x) .

.

,

,

.

z (z)=1 x1,

x2, x3 x1< x2< x3 :

( )( ) ( ) ( )1A1A21A xxx1x +

( 1.5).

:

5 = {(3, 0.2), (4, 0.6), (5, 1), (6, 0.7), (7, 0.1)}

10 = {(8, 0.3), (9, 0.7), (10, 1), (11, 0.7), (12, 0.3)}

- 42 -

1:

1.7.1 -

x .

( ) 0x ,0x =

(flat fuzzy numbers)

.

, :

( )( ) [ ]21A

2121

x,xx 1xxx Rx,x

=

1:

( )

322322

211211

2312

x ,1 , x, x,0x ,1 , x, x,0

==

32

11A ,

xxx +

=

)

(1.8)

(1, 3),

.

, ,

( 1.12).

-

: ( T21 ,a,a +

( )

1:

( )

323233

432342

211121

2312

x ,1 ,x ,0x ,1 ,x ,0

x ,1 ,x ,0

1:

:

( ) b2a

cx1

1x += (1.13)

c , a, b

.

Gauss

Gauss

:

( ) ( )22

2cx

A ex

= (1.14)

c Gauss

.

Gauss,

.

L-R (Left-Right)

,

.

, (L)

(R). LR,

L ( ) R ( ) >0,

>0, :

~M

( )

=m xmxR

x~M

m xxmL

~M

(1.15)

m , ,

. :

=

= mxz ,xmz 1 :

- 46 -

1:

( ) ( )( ) ( ) 1z ,1z ,''1R1L

0z 0,z ,10R0L

1

1

========

L-R

: L(z) = 1-z, R(z1)=1-z1. .

L R ,

.

L-R

.

,

[0, 1].

L-R ,

.

. L-R

, ,

. :

1- (2004)

:

=

( )

( )

( )

+

=

=

=

==

mx-m xm

11expzL

, ,m

4

23122

mxm mx1

11expzR

1

x

41

A

(1.16)

x m- m+ (x)

.

- 47 -

1:

00.10.20.30.40.50.60.70.80.9

1

x

(x)

1.14: 1

2- (2004)

:

( )

+

=

mxm n

mx

1

x n2A

mx-m n

xm

1

n2

(1.17)

x 1 3 (x) ,

n

.

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

x

(x)

m- m

m+

m

m- m+

1.15: 2

- 48 -

1:

1.9

. Diamond (1988)

( )T321 a,a,aA = ( )T321 b,b,bB = : ( ) ( ) ( ) ( )( )2332222112 bababaB,Ad ++= (1.18)

, .

, . LR

.

0, .

L-R . Hagaman

(Brdossy and Duckstein, 1995)

( )LR321 a,a,aA = ( )LR321 b,b,bB = LA, LB, RA, RB L-R , :

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( )dqqfqRbbbqRaaaqLbbbqLaaa

f,B,AD1

021

B2321

A232

21B122

1A1222

+++=

(1.19)

f [0, 1],

:

( ) ( ) =>>1

0 21dqqf 0q 0qf (1.20)

f(q)

. f(q) .

, . :

( ) ( )f,B,ADf,B,AD 2= (1.21) L-R .

,

(Bardossy and Duckstein,

1995).

- 49 -

1:

1.10 (DECOMPOSITION PRINCIPLE)

(x),

, , (-

cut) .

)x()x( AA )x(A

A~ , - - , -

- .

, A~ ( 1.16):

[ ] [ ])x(max)x(max)x( A]1,0(A)1,0[ == (1.22)

1.16: A~ .

,

,

.

.

1.11

. ,

. ,

,

. Zadeh 1965 :

A~ 1

0

3

2

1

3

2

1

- 50 -

1:

1. C=AC

:

C(x) = 1- (x) (1.23)

2. BAD = :

D(x) = min ((x), B(x)) (1.24) B

3. BAE = :

(x) = max ((x), B(x)) (1.25) B

1.17: ,

,

.

D(x)

x 0

1

1

1

x

x

0

0

C(x)

(x)

A~C/

- 51 -

1:

= {(1, 0.2), (2, 0.5), (3, 0.5) (4, 1), (5, 0.7), (6, 0.3)}

={(3, 0.2), (4, 0.4), (5, 0.6), (6, 0.8), (7, 1), (8, 1)}.

~~~BAD = :

( ) ( ) ( ) ({ }3.0,6,6.0,5,4.0,4,2.0,3=D~ )

)

: ~~~BAE =

( ) ( ) ( ) ( ) ( ) ( ) ( ) ({ }1 0 2 2 0 5 3 0 5 4 1 5 0 7 6 0 8 7 1 8 1~E , . , , . , , . , , , , . , , . , , , ,=

AC ,

X.

1.12 NORMS

(. 1.11),

.

. (x) = B(x) = 0.1 C(x) =

0.9. :

( ) ( ) ( ) ( ) 1.09.0,1.0minx1.0,1.0minx CABA ==== .

norms, t-

norms t-conorms ( s-norm). t-norms ,

t-conorms

t-norm AND t-

conorm OR .

t-norm (triangular norm): : [ ] [ ] [ ]1,01,01,0:t

:

1. ( ) 00,0t =

- 52 -

1:

2. ( ) ( ) xx,1t1,xt == 3. () ( ) ( ) z v wu z,wtv,ut 4. () ( ) ( x,yty,xt = )5. ( )( ) ( )( )z,y,xttz,yt,xt = ()

.

U V ,

t-norm

.

, ,

.

H t-norm

. BAD = ( ) ( ) ( )( )x,xtx BAD =

t-norm :

. ,

.

, .

.

t-norm

.

t-conorm: :

[ ] [ ] [ ]1,01,01,0:c :

1. ( ) 11,1c =2. ( ) ( ) xx,0c0,xc ==3. () ( ) ( ) z v wu z,wcv,uc 4. () ( ) ( x,ycy,xc = )

)5. ( )( ) (( )z,y,xtcz,yt,xc = ()

- 53 -

1:

H t-conorm

. : BAE =( ) ( ) ( )( )x,xcx BAE =

t-conorm ( s-norm)

.

. t-conorm t-norm.

1.12.1 t-norm t-conorm t-norm t-conorm. t-norm

t-conorm.

( ) ( )y1,x1t1y,xc = (1.26)

( ) ( )y1,x1c1y,xt = (1.27) x y . t-norm

t-conorm .

1.12.2 t-norm t-conorm 1.1 t-norm t-conorm

t-norm t(x,y) t-conorm c(x,y)

Algebraic product-sum xy xyyx +

Hamacher product-sum xyyx

xy+ xy1

xy2yx+

Einstein product-sum ( )( )y1x11xy

+ xy1yx

++

Bounded difference-

sum

( )1yx,0max + ( )yx,1min +

min-max operators min(x, y) max (x, y)

Drastic product-sum min{x,y} if max{x,y}=1

0

max{x,y} if min{x,y}=0

1

1.1: t-norm t-conorm (Brdossy and Duckstein, 1995)

- 54 -

1:

Algebraic product-sum norm

(Brdossy and

Duckstein, 1995).

norm, , .

, ,

.

1.12.3 norm norm

(Zimmermann, 1985).

1. :

.

norm .

2. : norm

,

.

3. :

.

(product inference),

,

.

1.13 (CARTESIAN PRODUCT) n21 X

~,,X~,X~ K , .

n21 x,,x,x K

n21 X~,,X~,X~ K

, . ,

( x ) , ,

, :

n21 x,,x,x K

( ){ }= BAA:B,ABA iiii . n21 X

~,,X~,X~ K X~ , ,

- 55 -

1:

.

: n21 x,,x,x K

( ) ( ){ }kkkX~k

X~ Xx,n,,2,1k,xx kmin K== (1.28)

1.14 (EXTENSION PRINCIPLE)

Zadeh 1965.

.

f (, 1998).

, :

x f(x) y

f(x)

.

.

:

(Cartesian product) X = X1xxXr 1,,r r 1,,r, . f

: y = f(x1,.,xr).

:

( ) ( ) ( ) = = Xx,....,x,x,....,xfyy,yB r1r1B~

~ (1.29)

:

( ) ( ) ( ){ } ( )~ ~1 r~ -11 rA AB

supmin x ,..., x f y y =

0

(1.30)

- 56 -

1:

={(-1, 0.5), (0.5, 0.8), (1, 1), (2, 0.4)} f(x) = x2. A

:

= f(A)={0.25, 0.8), (1, 1), (4, 0.4)}

.

1.15 * () x1 > y1

x2 > y2:

x1 * x2 > y1 * y2 (x1 * x2 < y1 * y2)

f(x, y) = x + y

f(x, y) = x y + f(x, y) = - (x + y)

+, -, , : . + - :

1

~M

~N

[0, 1] * () , *

~M

~N

[0, 1].

2

( ) FN,M ~~ ( )xM ( )xN *: :

~M

~N

( ) ( ) ( ){ } y,x minsupz ~~~~NMy*xzNM

==

(1.31)

*

( )=

FN,M,MNM,Nf ~~~~~~

- 57 -

1:

:

1.

=

~~~~ NMNM2. (commutative) 3. (associative) 4. ,

( ) F0 ( )= FM,M0M ~~~

5.

( )

0MM:\FM ~~~

.

.

.

. ~

.

+ ~

M~N

~M

~~~N MNM

=

MNM

M~~

M~~~

M:\FM1~~~

:

1.

=

~~~~ N 2. (commutative)

3. (associative)

4.

( )= F1,M1 ( )= FM,M1

5. ( ) 1M

.

. :

~~NM

= ~~~~ NMNM

( ) ( ) ( ) = = y,xminsupz ~~~~ NMyxzNM( ) ( )( ) ( )

+=

+=

y,xminsup

y,xminsup

~~

~~

NMyxz

NMyxz

- 58 -

1:

. ~~NM ~M ~N

.

(

~M

~N ( ) 0xM = ( ) 0xN = 0x ) :

( ) ( ) ( )( )( ) ( )

=

=

y,xminsup

y1,xminsup

1

NMxyz

NMxyz

~~

~~

==

y,xminsupzNMy/xzNM~~~~

.

.

1~N

~M

~N

1.16 (LINGUISTIC VARIABLES)

, .

.

, , .

.

.

(, , G, M),

x

, X

, G

.

- 59 -

1:

0 100, U = [0, 100].

:

= { , , , ,

, }

.

( )( ) [ ]{ }100 ,0x x,x M = x

(( )

]( ]

+=

50,100x

550x1

x12

0,50x 0

.

1.16.1 (linguistic modifiers) (linguistic hedges)

.

.

: , , , .

. ,

:

( ) ( )2very xx = (1.32) :

( ) ( ) 2/1elyapproximat x=x (1.33)

.

x

, .

- 60 -

1:

1.17

.

.

.

.

.

.

:

1. , ,

, .

2.

(Zimmermann, 1985).

, .

1.17.1

,

. :

1. .

2. .

3. .

,

. .

,

.

.

- 61 -

1:

.

.

.

.

1,75 cm

.

.

.

1,75 cm.

. (1,75)

.

,

.

.

(

)

( ). ,

. .

.

95% 5% .

0,95 .

,

.

.

.

- 62 -

1:

1.17.2 (possibility theory) Zadeh 1978

.

,

.

.

.

, (Zadeh,

1978).

, 200 .

,

.

.

.

.

.

, .

,

.

.

(additive).

(possibility measure)

,

. :

- 63 -

1:

( ) ( ) 1X 0 .1 ==

)

( ) ( )( ) ( ) Ii ,AsupA .3

BA BA .2

iIiIi i=

U

:

( ) { }( = x supA Ax (1.34)

.

(density) (Brdossy and Duckstein, 1995).

(sup

{ }( ) (xx = )

x (x)=1) .

1.17.3 . ,

Borel B (Bass, 1971). -

:

, ,

, ,

. -

Borel B .

, Sugeno 1977, (Zimmerman, 1985),

(fuzzy measure) : , Borel , : B

( ) 00 = ( ) 1=X ,

, BA, B BA ( ) ( )BA , B , nA K 21 AA ( ) ( )nnnn AlimAlim = , Zadeh (1978),

(possibility measure),

, Sugeno :

- 64 -

1:

F(X) .

: F(X) [0, 1]

:

( ) 00 = ( ) 1X =

BA , ( ) ( )BA ( i )i AsupAU =

IiIi

.

[ ]1,0X:f : ( ) ( ) XA ,xf supA

Ax=

( ) { }( ) Xx xxf =

, [0, 1].

= {0, 1, 2, ...., 10} () x ( ),

8, :

Xx

x 0 1 2 3 4 5 6 7 8 9 10

({x}) .0 .0 .0 .0 .0 .1 .5 .8 1 .8 .5

() ,

8, ( ) { }( )xsupAXAAx

= , = {2, 5, 9} :

( ) { }( )xsupAAx

= = sup {({2}), ({5}), ({9})}= sup {0, .1, .8} = .8

,

.

, A~ . O de

Luca and Termini 1972, (Zimmermann 1985) ,

, :

, A~ ,

: ( )xA~( ) ( ) ( )A~CHA~HA~d /+= , (1.35) Xx

: (1.36) ( ) ( ) ( )( )iA~n1i

iA~ xlnxKA~H =

=

- 65 -

1:

n A~

.

de Luca Termini, Shannon,

( ) ( ) ( )x1lnx1xlnxxS = (1.37) :

d ( )( )x,xA~ A~= : ( ) ( )(

==

n

1iiA~ xSKA

~d ) (1.38)

1

A~

10, A~ = {(7, .1), (8, .5), (9, .8), (10, 1),(11, .8), (12, .5), (13, .1)}. =1 : ( )A~d = .325 + .693 + .501 + 0 + .501 + .693 + .325 = 3.038.

, B~ ,

10 B~ = {(6, .1), (7, .3), (8, .4), (9, .7),

(10, 1),(11, .8), (12, .5), (13, .3) (14, .1)}, : ( )B~d = .325 + .611 + .673 + .611 + 0 + .501 + .693 + .325 = 4.35. Yager (Zimmermann 1996),

A~

A~C/ .

( ) ( ) ( ) p1pn1i

iA~CiA~p xxA~C,A~D

=/

=/ p=1,2,3,

S= supp ( )A~ , ( ) p1p SSC,SD =/ . Yager :

( ) ( )( )A~pps A~C,A~D

1A~f pp /= , ( ) [ ]1,0A~fp (1.39)

p=1, ( A)~C,A~Dp / Hamming, : ( ) ( ) ( )

=/=/

n

1iiA~CiA~p xxA

~C,A~D (1.40)

, ( ) ( )x1x A~A~C = / , : ( ) ( )

==/

n

1iiA~p 1x2A

~C,A~D (1.41)

- 66 -

1:

p=2, ( A)~C,A~Dp / , : ( ) ( ) ( )( ) 212n

1iiA~CiA~p xxA

~C,A~D

=/

=/ (1.42)

( ) ( )x1x A~A~C = / , )( ) ( )( 212n

1iiA~p 1x2A

~C,A~D

=/

= (1.43)

2

1

:

p=1 ( A)~C~,A~D1 = .8 + 0 +.6+ 1+ .6 + 0 + .8 = 3.8 ( )A~pps =7, ( )A~f1 = 1-(3.8 / 7) = 0.457, ( )B~C,B~D1 / = 4.6 ( )B~pps =9 ( )B~f1 = 1-(4.6 / 9) = .489. p=2 ( A)~C~,A~D2 = 1.73

( ) 21A~pps =2.65= 7 , ( )A~f2 = 1-(1.73 / 2.65) = 0.347 ( )B~C,B~D2 / = 1.78

( ) 21B~pps =3 ( )B~f2 = 1-(1.78 / 3) = .407.

1.17.4 - , ,

. ()

() . :

- 67 -

1:

( ) ( )( ) ( ) 0P

1P====

(1.44)

A . :

( ) ( ) 1AA =+ (1.45) = BA

( ) ( ) (BABA += ) (1.46)

.

.

. :

( ) ( ) 1AA + (1.47)

- 68 -

2:

2 2.1

(fuzzy inference systems), (fuzzy models),

(fuzzy associative memories, FAM),

(fuzzy rule base, FRB) (Nozaki et al, 1997).

,

, .

,

.

, .

.

, .

.

, .

:

,

. 30C ,

. ,

.

- 69 -

2:

.

.

-.

.

.

.

, ,

. ,

.

, .

.

,

.

,

.

.

.

2.2

, ,

.

- 70 -

2:

, .

.

R

( ) i,k (fuzzy premises)

Bi,k i,k

(responses) .

iKi,Ki,22i,11 B thenA is a ...... A is a A is a If

.

-

(IF-THEN),

.

.

, .

AND, OR

.

,

.

.

.

0 1

. ,

.

:

- 71 -

2:

, .

.

2.1:

2.3 FRB

,

.

,

.

.

,

:

,

.

.

.

, ,

.

.

, ,

.

:

- 72 -

2:

1. (rule base)

-.

2. (database)

.

3. (decision-making unit).

.

4.

.

5. (defuzzification interface)

.

2.2: (Shu and Burn, 2004)

FRB :

1. .

2. (fuzzification).

3. (

).

4. .

5. (defuzzification).

.

2.4 (Xiong et al, 2001). :

- 73 -

2:

A. Mamdani

Mamdani ,

.

Mamdani .

, .

, .

B. Takagi-Sugeno-Kang

Takagi-Sugeno-Kang

,

.

,

.

Takagi-Sugeno-Kang :

( ) ( ) ( ) ( )p21rrprp2r21r1 x,....,x,xfy then ,A is x.......A is xA is xIf =

x

( )jrA

, p , fp r

, r y .

Mamdani.

Takagi-Sugeno-Kang, f

(x ,x ) y1 p r :

( ) ( ) ( ) ( ) ( ) (=

+=+++==p

1jjrrpr1rrK1rr xjb0bxpb.....x1b0bx,.....,xfy ) (2.1)

H y :

( )

k

r

k

1=rp1rr

k

r

k

1=rrr

x,....,xf=

y=y (2.2)

1=r1=r r o r .

Takagi-Sugeno-Kang

.

,

- 74 -

2:

. (p+1) (b(0),

b(1),,b(p)), (p+1)

. ,

.

2.5 ,

(completeness) (redundancy).

2.5.1 (completeness)

,

( , 1 2,....., )

B(1, 2,....., ) .

,

.

( ) Aa,....,a,a k21 ( ) 0>a,....,a,aD k21i , D (ai K) .

, .

2.5.2 (redundancy)

i, .

,

, .

. :

( ) ( ){ } ( ){ }(I1=i

K1K1iK1u a,...,aipsup1=0>a,...,aD;i=a,...,aO ) (2.3)

- 75 -

2:

.( ){ }( )K1 a,...,aipsup1

(a ,...a1 K) i. , (

).

R

:

( ) ( ) ( ){ } 1>Aipsupa,...,a;a,...,aOmin K1K1u (2.4)

, .

.

.

. , :

( ) ( ) ( ) Aipsupa,...,a 2 a,...,aO 0K1K1u i0.

(Brdossy and Duckstein, 1995).

2.6

,

.

A A A A1 2 3 mB C C C .. B1 1,1 1,2 1,3B C .. .. B2 2,1B .. B3

B CBm m,m

2.3:

- 76 -

2:

,

. 2.3

, ,

m ( , , ,., 1 2 3 m , , ,, 1 2 3 m).

:

IF A is A AND B is B THEN Ci j i,i

m2.

m m3

(Torra, 2002).

(curse of

dimensionality).

n ,

c ,

cn ,

. n-

( n )

.

cn

,

.

,

,

(Russell

and Campbell, 1996).

.

2.6.1

:

(Identification of Functional Relationships):

- 77 -

2:

,

.

(Sensory fusion): .

.

.

(Rule Hierarchy): (modules) .

.

(Hierarchical Fuzzy Systems).

(Interpolation). .

,

(Torra, 2002).

RB12 RB23 RB33

RB32

RB11 RB21 RB22 RB31

V2V1 V4V3V2V1V4 V4V3V3 V1 V2

2.4:

,

. ,

,

, .

- 78 -

2:

,

.

(Abebe et al, 2000).

2.7 (FUZZIFICATION)

.

.

(Shrestha et al, 1996).

.

, , ,

. .

,

,

.

,

.

.

.

.

.

[0, 1]

.

. 2.5

300

0,66 .

- 79 -

2:

350 ,

200 500 .

1

2.5:

2.7.1

.

,

:

1. x*

.

2. x-

.

3. x+

.

[x-, x+]

(Brdossy and Duckstein, 1995).

,

.

x* .

x*, x-, x+ .

0 500300 200

0,66

350

- 80 -

2:

,

.

.

.

. Civanlar and Trussell (1986)

(probability density function)

. Trcksen (1991)

.

, .

,

.

.

.

2.7.2

.

.

.

,

,

( 2.6)

- 81 -

2:

2.6:

Kosko (1992)

25%. ,

.

, Input 1 Input 2.

Low, Medium High

. x1 x2

. x1

Medium High x2

Low Medium.

.

,

,

.

2.7:

(x)

1

0

- 82 -

2:

2.7.3

.

,

,

.

, .

(rdossy and Duckstein, 1995).

[(x1,

A(x )), (x , 1 2 A(x )),, (x , 2 n A(xn))], Lagrange (Langrage

interpolation), (Least squares curve fitting),

(neural networks) (Klir and Yuan, 1995).

2.8

. ,

. ,

2.7.

, (x)

(x)

DOF :

( ) ( ) ( ) ( )( )x=x xmax=a aaax (2.5)

:

= (175, 185, 210). = (174, 179, 184)

0,625

. 0,625 DOF

0,625.

- 83 -

2:

00.10.20.30.40.50.60.70.80.9

1

170 180 190 200 210 220

(cm)

2.8:

2.9

,

(operators),

.

AND OR. AND

(t-norms).

OR (t-conorms).

norms .

2.10 (DEGREE OF FULFILLMENT,

DOF) .

(0-1).

. ,

,

.

.

( ,.....1 ) ,

- 84 -

2:

D ( ,.....i 1 ).

(AND, OR ),

( ).

. DOF

, ,

.

, , , Then.

2.10.1

. DOF.

- (min-max inference)

(product inference).

1. - (min-max operators)

AND,

DOF

( 2.9).

:

( )kAK,...,1=ki amin= k,i (2.6)

1

0 1 2

(x) A

x x

2.9: AND

- 85 -

2:

1

0 1 2

(x) A

x x

2.10: OR

OR, DOF

( 2.10)

:

( )kAK,...,1=ki amax= k,i (2.7)

2. - (Algebraic product-sum)

DOF

AND

:

(K1=k

kAi a= k,i ) (2.8)

OR (probor, probabilistic OR):

( OR A ) = ( ) + (1 2 1 1 2 2)) - ( ) (1 1 2 2)) (2.9)

-

- ,

. (t-norm, t-conorm)

AND OR .

:

If (1, 2, 3) AND ((1, 2, 6)T OR (4, 5, 7)T) AND (2.5, 4, 4.5)T then (0, 1, 2)T

DOF (1, 2, 3, 4 )= (1.5, 4, 4.8, 3)

- 86 -

2:

0.5, 0.5, 0.8 0.333 .

:

1. - :

min(0.5, max(0.5, 0.8), 0.333) = 0.333

2. :

0.5 (0.5 + 0.8 - 0.50.8)0.333 = 0.15

DOF

. , - DOF

.

2.11

(DOF) .

, .

,

.

DOF

.

, .

,

.

2.12 (AGGREGATION) ,

,

.

- 87 -

2:

,

(DOF)

( , ,....1 2 ).

,

DOF.

,

.

( , ,....1 2 )

i . DOF i i =

D ( , ,....i 1 2 ) i.

, (B , ). i i

B = C((B , ),.........,( , )) 1 1 i i

C .

.

(

).

,

(1, ,....2 ).

C((B , ),.....,( , )) = B (1 1 i i 1, ,....2 )

.

(Mahabir et al, 2003).

, .

1. (minimum combinations)

i ,

i .

:

- 88 -

2:

( ) ( )xmin=xi

iBi0>B

(2.10)

(x) x Bi i

i.

,

, :

( ) ( )( )x,minminxi

iBi0B

= > (2.11)

,

.

,

.

,

, .

i j

( ) ( ) 0>a,...,a,aD,a,...,a,aD K21jK21i

.

0BB ji

:

:

IF A1,1 = (1, 2, 3)T THEN B = (1, 2, 3)1 T

IF A2,1 = (2, 3, 4)T THEN B = (3, 4, 5)2 T

[2, 3]

, :

min ( (x), (x)) = 0 1 2

2x3.

2. (maximum combinations)

, .

- 89 -

2:

,

DOF. ( , i i)

:

( ) ( )xmax=xiBiI,...,1=iB

(2.12)

(x) x Bi i

i.

, ,

(x) , :

( ) ( )( )x,minmaxxiBiI,...,1iB

==

(2.13)

,

.

3. (additive combinations)

.

. (weighted sum

combination) (normed

weighted sum combination).

A. (weighted sum combination)

O ( , i i)

:

( )( )

( )

I1=i Biu

1=i BiB

umax=x

i

iI x

(2.14)

.

,

2.14. :

- 90 -

2:

( ) ( )( )( )( )

I 1i Biu1i Bi

Bu,minmax

xi

i

=

=

= I x,min (2.15)

. (normed weighted sum

combination)

H ,

.

(

) i. O

( , i i)

:

( )( )

( )

I

1=i Biiu

1=i BiiB

umax

x=x

i

i

I

(2.16)

:

( )+= dxx

1iB

i

(2.17)

={(b , (1),.,(b , (J))} : i 1 i j i

( ) ( ) j=Bcar=1

iii

(2.18)

:

( )( )(

( )( ))I

1=i Biiu

1=i BiiB

u,minmax=x

i

iI x,min

(2.19)

.

- 91 -

2:

: .

1 1 = 0.4

1 = (0, 2, 4).

= 0.5 2 2 = (3, 4, 5).

B 1 2

[3, 4]

.

[3, 4]

2.10:

( )

3x5.0,2

x44.0min [3, 4].

:

( ) 723=x3-x5.0=

2x-4

4.0

:

( )( )

2:

00.10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6

x

(x

)

1 2

2.11: 1 2 ( )

( )

,

, .

, 2.12. :

( )( )( )

2:

,

.

00.10.20.30.40.50.60.70.80.9

1

(x) 1 2

0 1 2 3 4 5 6x

2.12: 1 2 ( )

( )

1.

2.14 :

( )( )

( )

2:

2.

( 2.17).

1 2.

1/ = 2 1/ = 1. 2.16 : 1 2

( )( )

( )

2:

1

2.15:

2.13 (DEFUZZIFICATION PROCEDURE)

, .

, ,

.

,

. ( )

B

(crisp) b,

. H :

2

2 1

1 1 1

1

1

1 1 1

1 2

2

- 96 -

2:

B=D (B) f

(1,......)

( ,......1 ) .

.

.

2.13.1

. ,

.

, , ,

, (,

2006).

1.

,

, .

( ) ( )xmax=b BxB (2.20)

:

A.

,

.

( , 1 2, , ) [ , 3 4 2 3]

.

- .

B.

.

(0, 3, 3) (2, 3, 9) ,

- 97 -

2:

3

.

2. - (fuzzy mean-center of gravity)

, .

:

( )( ) ( ) ( )( ) (( )( ) + =BM- BM BB dttBM-tdttt-BM )

( )

(2.21)

:

( ) ( )

+

==- B

- B

dttBMb

+ dttt (2.22)

.

(1, 2, +).

L-R .

=( , 1 2, ) : 3

( )3

a+a+a=AM 321 (2.23)

2 :

( )

=

2

1

dtttM AL

1 dt 2 3 t

t

M(A) (x)

- 98 -

2:

( ) ( =+== ddtt 12112121 )t

( )( ) ( ) =+= 10

12112L dM

( )

:

( ) ( ) ( ) ( ) =+=+= 10

112

22

12

1

011212 dd

( )

+=

+=232

131

31 112

12

1

0

21

31

3212

: 2

( )

=

3

2

dtttM AR

( ) ( ) === ddttt3

( )( ) ( ) == 0 dM( )( ) ( ) ( )

( ) ( ) ( )

: 2323323

1

23233R

( )

=

32

332

23323

233230

23323 = +=+=

=+==

111d

dd

3321

22

1

023323

1

0233

:

( ) ( ) ( ) ( )22

121ddE 132312

0

123

1

012

=+== () :

( ) { } RL MMEAM +=

( )( )

( )=

=2

3223AM13

2312 + + 233112

- 99 -

2:

332

32

32

23212132

213

13

++=++++=

3322

2323

212113

213

2

21

23

=++++

=

L-R

:

( ) +

+

= 3

2

2

1

a

a23

2a

a12

2A dtaa

attRdtaatatLdttt (2.24)

:

( ) ( ) ( ) ( )( ) ( ) *23*12

2323212122A RL

M +=**2***2* RRLL ++

( ) ( ) ( ) ( ) ==== 1**1*1**1* dtttRR ,dttRR ,dtttLL ,dttL

L 0000

2.

=

12

2 tLL

( ) ( === )

1221212

2 tddtt

:

1 dt 2 3 t

t

M(A) (x)

- 100 -

2:

( ) ( )[ ] ( )( )( ) === 0

dLdtttLM2

( ) ( ) == 10

**1

0

* dLL ,dLL

( ) ( ) ( ) ( ) =

1

0

212

1

0122

121

122L

dLdL

1

( ) ( ) **212*122L LLM =

2

=

23

2tRR

( ) ( ) =+== ddtt 23232232

t

( ) ( )[ ] ( )( ) =+== 1

dRdtttLM3

( ) ( ) == 1**1* dRR ,dRR

11

( ) ( ) ( ) ( ) +=

1

0

223

1

0232

230

232R

dRdR

2

00

( ) ( ) **223*232R RRM = :

( ) ( ) ( )( )( ) ( ) ( ) =+=+=

0

1

1

02312 dRdLdttRdttLE

3

2

2

1

( ) ( ) ( ) ( ) ( ) ( ) *23*120

230

12 RLdRdL +=+= () :

( ) { } RL MMEAM +=

( ) ( ) ( ) ( )( ) ( ) *23*12

**223

*232

**212

*122RL

A RLRRLL

EMM

M +++=+=

,

.

- 101 -

2:

,

. (

)

.

(weighted sum combination)

:

( )( )

( ) ( )

I1=i i

i

1=i i

I

1=iBi

1=i

1

=dtt

=BM

i

i I iiI Bi BM1dttt

(2.25)

i , i i 2.18

( ) i . i

2.22:

( )( )( )

+

=dtt

BM

B

B+ dttt

( ) ( )( )

== I

I

1i BiB

umax

tt

i

i

( )( )

=1i Biu

( 2.14). :

( ) ( )( )

( )( )( )

+

=

=

=+

= ===dtt

BM

dtumax

tBM

I

1i BiI

1i Biu

I

1i Bi

1i

i

i

i

i

+ ==

+

dtttdt

umax

tt I

1i BiIBiu

I

1i Bi

i

i

- 102 -

2:

( ) ( ) (( ) )( )( ) ( ) ( )( ) =+++=

dtt...ttBM

I21

I21

BIB2B1

BIB2B1 +++ dtt...ttt

( ) ( )

( )( ) ( ) ( ) +++= dtt...dttdtt I21

I21

BIB2B1

+++ dttt...dtttdttt BIB2B1

( ) ( ) ( )iB i B

t t dt = M B t dt i( ) ( ) ( )

:

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( )

=dt

dtBMdt

i

i

I21

Bi

iBi

BIB2B1 =+++

+++=dtt...dttdtt

dttBM...dttBMdttBMBM I21 BIIB22B11

( )

(normed weighted sum combination)

.

( )( )

( )

I1=i

i

1=iii

I

1=iBii

1=iBii

=

dtt=BM

i

i II BMdttt

(2.26)

2.22:

( )( )( )

+

=dtt

BM

B

B+ dttt

(t) 2.16:

( ) ( )( ) =

=

=

dtt

dtttBM

i

i

BI

1i ii

B1i iiI

- 103 -

2:

( ) ( ) ( ) ( )( ) ( ) ( )

++++++=

dtt...dttdtt

dttt...dtttdtttBM

I21

I21

BIIB22B11

BIIB22B11

( ) ( ) ( ) = dttBMdttt BB ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) +++ dttBM...dttBMdttBM I21 BIIIB222B111

( ) ( ) ( )

+++= dtt...dttdttBMI21 BIIB22B11

( 2.17) ( ) =1dtt B( ) ( )

=

==+++= I1i i

1i

I21

II2211

...BM +++

Iii BMBM...BMBM

,

, ().

-

.

3. (fuzzy median)

,

.

:

( ) ( )( )

( ) Am- +Am AA dtt=dtt (2.27) ,

.

= ( , 1 2, )3 :

( ) ( )( ) 213121 2aaaaaAm

+=

1

2a+a

>a 312 (2.28)

( ) ( )( ) 21

2313 aaaaaAm += 3 2 2231 a>

a+a (2.29)

- 104 -

2:

. :

1 :

, :

(x)

1

1 2 30 x m 1 2 3

213

213

1223

22 >+>+

>:

=

+

+23

332

23

312

321

m2

m2

m1m

2

=+ EEE

( ) ( )( )( ) ( )( ) ( ) =+ 232232312

2323

mmm2

=

++

23

2233

12m

mm

=+++=++++

23213132

23

2233213132

23

23

2222

23232131

2232

m2m4

m2m4

mm2mmm2m2

=++ :

02

m2m 21313223

32 =+++

2 :

=

24

m2

2-++

= 21313223, = 1, = -2 , 3

- 105 -

2:

( )( )21313223

21313223213132

23

23

2

8

888824164

+==+=++=

( ) ( )( ) (( )( ))( )( ) ( )( )

222 1323

31323

3

==4

844

84m 132332312333 ===

2 :

(x)

1

:

22 132132

+>+>2312 >

m :

+

+

=

22m1mm

2m 232

12

1

12

11

+= EEE 321

( )

( ) +

+=

232

1212

1 mmm

( ) ( )( ) ( )( )++= 1223212211212

m2mm

++++=+ 21223123121222222112 m22mmmm2m0m4m2 323121

211

2 =+++

1 2 30

x m 1 2 3

- 106 -

2:

2

=

24

m2

= 2, = -41, = 12+1 + - 2 1 3 2 3

( )32312121212 24164 =++= 323121

21323121

21

21 8888888816 +=+=

( ) ( ) ( )( ) =+=4

84

84m 3123111

323121211

( )( ) ( )( )22

m 213111 == 2 2131

( )( )

2m 12131 =

( )

4. (center of sums)

. ,

.

.

:

( )

r duuuU r

r

U r

duu=b (2.30)

5. (bisection)

.

6. (middle of maximum)

.

, :

- 107 -

2:

2u+u

=u maxmin (2.31)

7. (Largest of maximum)

.

8. (Smallest of maximum)

.

.

,

.

2.13.2

, .

.

.

( .).

.

.

- 108 -

2:

(Mahabir et al, 2003).

2.14

,

.

2.7.1.

:

1. . ,

.

.

2. ,

.

3. ,

.

4.

.

.

, .

.

2.14.1 (training set)

,

.

- 109 -

2:

.

.

,

.

. S

b.

={[ (s),.., 1 k(s), b(s)];s=1,..,S} (2.32)

(Brdossy and Duckstein, 1995).

1. (Counting algorithm)

, ,

( )+k,i- k,i a,a i,k.

. ,

, .

( )+k,i1 k,i- k,i a,a,a .

1k,ia

k(s)

i, ( )+k,i- k,i a,a ( )

iRsk

i

1k,i saN

1=a (2.33)

Ri

i, Ri :

( ) ( ) ( )( ) ( ) ( ){ }K1,...,=k a,asa ;Tsb,sa,....,sa=R +k,i- k,iKK1i R . i i

.

.

- 110 -

2:

( )+i1i-i ,, b(s) R-i , : i

( )sbmin=iRs

-i (2.34)

1i b(s) i

( )iRsi

1i sbN

1= (2.35)

b(s) R+i , i

( )sbmax=iRs

-i (2.36)

:

( ) ( ) ( )Ti1i-iTK,i1 K,i- Ki,T1,i1 1,i-i,1 ,, THEN a,a,a ... AND a,a,a IF +++

. ( )+k,i- k,i a,a

.

.

(Brdossy and Duckstein, 1995).

2. (Weighted counting algorithm)

.

DOF

DOF.

,

.

, .

.

- 111 -

2:

>0

.

.

( )T+i1i-i ,, , DOF . :

-i

( )( )sbmin=

>s

-i

i

(2.37)

1i b(s) i DOF

( ) ( )( )

( )( )

>s i

>s i1i

i

i

s

sbs= (2.38)

b(s) R+i i DOF

( )( )sbmax=

>s

-i

i

(2.39)

.

,

DOF

. ( )T+i-i , .

.

.

3. (Least squares algorithm)

O

.

.

- 112 -

2:

, ,

:

( ) ( )( ) ( )( )[ ]S

2K1 Sb-saR,......,saR (2.40)

,

, (s) (s),.., i 1 k(s)

i .

:

( ) ( )( )( ) ( ) ( )( )I 1i iI

1i iiK1

s

BMssaR,.....,saR

=

=

= (2.41) ( ) Bi i

: 2( ) ( )

( ) ( )

=

=s

I

1i i

I

1i ii sbs

BMs (2.42)

( ) i : i

( ) ( )( ) ( )

( )( )

=

==

=s

I

1i i

jI

1i i

1i ii 0s

ssb

s

BMsI (2.43)

((1)....(), :

( ) ( ) ( )( )( )

( ) ( )

( )

S

I

1=i i

j

S2I

1=i i

1=i iji

s

sbs=

s

BMssI

(2.44)

,

( ) (Abebe et al, 2000).

(B ) i s :

s = 1,.., S

( 2.42):

- 113 -

2:

( ) ( )( ) ( )

2Ii ii=1I

s ii=1

s M BJ = - b s

s

( ) ( )( ) ( )

( ) ( )( ) ( )

2I I2i i i ii=1 i=1

I Is i ii=1 i=1

s M B s M BJ = - 2b s + b s

s s

( ) ( )( ) ( )

( ) ( )( ) ( ) ( )

( )2I I

i i i ii=1 i=1I I

sj j ji ii=1 i=1

s M B 2b s s M BJ = -M B M B M B s s

2

:

( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )

2Ii i 1 1 2 2 I Ii=1

1 1 2 2 I I 1 2 1

s M B = s M B + s M B +...+ s M B

s M B + s M B +...+ s M B = F F F = F

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 2 1 22 1 j i i j i

i ij j j

j i i ji

F F F F= F + F = i s s M B + s s M B =M B M B M B

= 2 s s M B = 1,2,..., I

( )( ) ( ) ( )

( )( ) ( )

( )ji i

i ij

s b sb s s M B=

s sM B

( )( ) ( ) ( )

( )( )( ) ( )

( )I

j i i ji=12 IIsj ii=1ii=1

2 s s M B 2 s b sJ = -M B s s

= 0

( ) ( ) ( )( )( )

( ) ( )( )

Ij i i ji=1

2 IIs s ii=1ii=1

s s M B s b s=

s s

s = 1,.., S

( ) ( ) ( )( ) ( )2I I 2i ii=1 i=1 s = A s s = A s :

( ) ( ) ( )( )( )

( ) ( )( )

Ij i i ji=1

2s s

s s M B s b s=

A sA s (( )=X ): i i

- 114 -

2:

( ) ( )( )( )

( ) ( )( )

I1 i 1i=1

2s s

s s X s b s=

A sA si 1.

( ) ( )( )( )

( ) ( )( )

I2 i 2i=1

2s s

s s X s b s=

A sA si 2.

..

..

( ) ( )( )( )

( ) ( )( )

II i Ii=1

2s s

s s X s b s=

A sA si I.

( ) ( )( )

( )( ) ( ) ( ) ( ) ( )( )

( )( )

( ) ( )( )

( ) ( )( )

( ) ( )( )

I1 i i 1i=1

1 1 2 2 3 3 I I2 2s s

I I I I21 1 2 1 3 1i=1 i=1 i=1 i=1

1 2 3 I2 2 2

s s X s= s X + s X + s X +...+ s X =

A s A s

s s s s s s s= X + X + X +...+ X

A s A s A s A s

I

2

( )( )

( ) ( )( )

( ) ( )( )

21 1 2 1

11 12 1I2 2s s s

s s s s s = , = ,......, =

A s A s A s I2

( ) ( )( )

( ) ( )( )

( ) ( )( )1 21 2 Is s s

I s b s s b s s b sb = b = b =A s A s A s

, , .......,

( )( )

( )

111 12 1I 1

221 22 2I 2

II1 I2 II I

M B . . . bM B . . . b

.. . . . . . .

.. . . . . . .M B . . . b

=

( ). i

- 115 -

2:

4. ANFIS (Adaptive Network-Fuzzy Inference System)

ANFIS

Jang 1993.

,

,

.

.

,

,

.

,

(gradient vector)

.

(Mathworks, 1995). MATLAB

(back-propagation technique)

.

Takagi-Sugeno-Kang

x y .

-:

1: x A y B 1 1 f = p x + q y + r1 1 1 1 2: x A y B 2 2 f = p x + q y + r2 2 2 2

1 : : ( )x=O

iA1,i

x , A i Ai

. 1

. , :

( )i

2

i

i

A

ba

cx1

1xi

+

=

- 116 -

2:

{a , b , c } . i i i 2 :

, .

, t-norm. t-

norm Algebraic product : ( ) ( ) 1,2=i ,yx=w=O

ii BAi2,i

3 : i i

2.

1,2=i ,w+w

w=w=O

21

ii3,i

4 : i :

( )iiiiii4,i r+yq+xpw=fw=O {p , q , r } . i i i 5 : .

i

i

ii

i

ii

i5,i w=fw=O fw

2.14.2 (rule verification-validation)

(training set),

.

.

,

. ,

.

. : ( ) ( ) ( )( ){ }S,...,1=s;sb,sa,...,sa=P K1

- 117 -

2:

.

.

V.

TV=P =TV

R( (s),1 (s)) .

:

( ) ( )( ( )[ )Vs

K1 sb,sa,.....,saRDV1

=E ] (2.45) D R

b.

. :

(mean error)

( ) ( )( ) ( )[Vs

K1 sb-sa,....,saRDV1

=E ] (2.46)

(maximum absolute deviation) ( ) ( )( ) ( )sb-sa,....,saRmax=E K1 (2.47)

(sum of absolute deviations)

( ) ( )( ) ( )Vs

K1 sb-sa,....,saRV1

=E (2.48)

(sum of squared errors) ( ) ( )( ) ( )[

Vs2

K1 sb-sa,....,saRDV1

=E ]

( ) ( )

(2.49)

O (correlation coefficient)

( ) ( ) ( ( ) ( )) ( )

( ) ( )( ) ( ) ( )( ) ( ) ( ) 222K12K12

Vsb

Vsb

Vsa,...,saR

Vsa,...,saR

VV

= K1K1 sbsa,...,saRsbsa,...,saR

(2.50)

,

, ,

.

.

- 118 -

2:

,

(Demico and Klir, 2004).

2.14.3 (Artificial neural networks)

(...) .

...

,

.

...

.

,

.

.

.

...

( ) . ...

.

.

2.16

u , j j.

uj

1

2

uii

j

j

2.16:

- 119 -

2:

.

..

.

. :

1.

2.

3. .

...

(t) uj j