Διαχείριση υδατικών πόρων με χρήση των ασαφών...
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ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΑΓΡΟΝΟΜΩΝ ΚΑΙ ΤΟΠΟΓΡΑΦΩΝ ΜΗΧΑΝΙΚΩΝ ΤΟΜΕΑΣ ΣΥΓΚΟΙΝΩΝΙΑΚΩΝ ΚΑΙ ΥΔΡΑΥΛΙΚΩΝ ΕΡΓΩΝ ΛΕΩΝΙΔΑΣ Γ. ΜΠΑΛΛΑΣ ΔΙΠΛΩΜΑΤΟΥΧΟΣ ΑΓΡΟΝΟΜΟΣ ΚΑΙ ΤΟΠΟΓΡΑΦΟΣ ΜΗΧΑΝΙΚΟΣ, MSc ΧΡΗΣΗ ΤΩΝ ΑΣΑΦΩΝ ΚΑΝΟΝΩΝ ΣΤΗ ΔΙΑΧΕΙΡΙΣΗ ΥΔΑΤΙΚΩΝ ΠΟΡΩΝ-ΕΦΑΡΜΟΓΗ ΣΤΗΝ ΥΔΡΟΛΟΓΙΚΗ ΛΕΚΑΝΗ ΒΟΛΒΗΣ ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ ΘΕΣΣΑΛΟΝΙΚΗ 2007 A-PDF Merger DEMO : Purchase from www.A-PDF.com to remove the watermark
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Transcript of Διαχείριση υδατικών πόρων με χρήση των ασαφών...
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. , MSc
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2007
A-PDF Merger DEMO : Purchase from www.A-PDF.com to remove the watermark
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2007
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............................................................................... 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 -
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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 -
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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
