PhD Thesis Yasmina-Bashon 2013-06-18

7/24/2019 PhD Thesis Yasmina-Bashon 2013-06-18

http://slidepdf.com/reader/full/phd-thesis-yasmina-bashon-2013-06-18 1/230



•

•

•



•

•



Age

a

b

A

a

b

α

F 1

F 2



N 1 N 2

C 1

C 2

−2

α = β = γ = 1



W A

Min

Min/Max



α =

1, β = 0.5 γ = 0.8

α =

1, β = 0.8

γ = 0.5



Chapter1



{16, 17, 18, 19}

old

45

98%

0.7/middle

aged

•

•



•

•

•



•



•

•

•



Chapter2



•

a = 3

a ∈ (0.25, 0.75]

•



∗

∗

∗

∗



•

•



a

b

a

b

A

B

a

b

A − B

b

a

B − A

a b A B

S (a, b) = f (A ∩ B)

f (A ∩ B) + αf (A − B) + βf (B − A)



Chapter3

•

•



U



u U

0

5

A

U

A ⊆ U

x

A (x ∈ A)

A (x ∈ A)

A

A : U → {0, 1}

A(x) = 1,

x

AA(x) = 0,

x

A

ℵ = {1, 2, 3, . . .})

A = {x ∈ U |x ≥ 0}

µA(x) : U → {0, 1} x ∈ A

µA(x) =

1 if x ∈ A

0 if x ∈ A



1 0

A = {Ai| i ∈ I }

i

I

Ai

A = {A1, A2, . . . , An}

• φ : U → {0, 1}

φ(x) = 0

x ∈ U

• A

B A ⊆ B ⇔

A

B

• A B

A = B ⇔ A ⊆ B

B ⊆ A

• A

B

A = B

• A

B A ⊂ B ⇔ A ⊆ B

A = B

• A

B A ⊆ B

B ⊇ A

A

P (A)

A

2A

A |A|

A

2|A|

A = {1, 2, 3}

P (A) = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3},

{1, 2, 3

}}

|A

|= 3

|P (A)

|= 2|A| = 23 = 8



•

A

B

A − B = {x|x ∈ A

u ∈ B}

• B B − A = A A

¯A = A

φ = U

U = φ

• r

A

rA = {rx|x ∈ A}

•

A

B

A ∪ B = B ∪ A = {x|x ∈ A x ∈ B}

A ∪ U = U

A ∪ φ = A

A ∪ A = U

i∈I Ai = {x|x ∈ Ai

i ∈ I }

•

A

B

A ∩ B = B ∩ A = {x|x ∈ A

x ∈ B}



A ∩ U = A A ∩ φ = φ A ∩ A = φ

i∈I Ai = {x|x ∈ Ai

i ∈ I }

• A

B

A ∩ B = φ

A ⊆ U

B ⊆ U

A

A A

A

π(A) = {Ai|i ∈ I, Ai ⊆ A}

Ai = φ

A ↔ Ai ∩ A j = φ

i, j ∈ I, i = j

i∈I Ai = A

A

π(A)

F

U



¯A = A

A ∪ B = B ∪ AA ∩ B = B ∩ A

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∪ A = AA ∩ A = A

A ∪ (A ∩ B) = AA ∩ (A ∪ B) = A

U

φ

A ∪ U = U A ∩ φ = φ

A ∪ φ = AA ∩ U = A

A ∩ A = φ

A ∪ A = U

A ∩ B = A ∪ BA ∪ B = A ∩ B

x U

[0, 1]

µF (x) : U →

[0, 1]

x ∈ U

F = {µF (x)/x|x ∈ U, µF (x) ∈ [0, 1] ⊂ }



µF

µF (x) = 0,

x

F,

µF (x) = 1,

x

F,

µF (x) ∈ (0, 1),

x

F,

µF (x)

1

x

F

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

M e m b e r s h i p D e g r e e s

F (U ) = {F |F

U }

U

F

F = µF (x1)

x1

+ µF (x2)

x2

+ . . . + µF (xn)

xn

,



F =x∈U

µF (x)

x ,

F =

x∈U

µF (x)

x dx.

{0.1/1.4, 0.4/1.6, 0.78/1.8} {(1.4, 0.1), (1.6, 0.4), (1.8, 0.78)}

A

U

µA(x)

x

[0, 1]

[0, 1]



1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

0

0.2

0.4

0.6

0.8

1

Height

D e g r e e o f m e m b e r s h i p

tall

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

0

0.2

0.4

0.6

0.8

1

Height

D e g r e e o f m e m b e r

s h i p

tall

µA : x

→[0, 1], x

∈U



0 2 4 6 8 10 120

0.5

1

aU

a

m b

aa

00

1

m−margin m−margin

U

m

x

U

A

U

U

•

a

b

m

a < m < b

b − m

m − a

trimf (x) =

0,

x ≤ a x ≥ b

x−am−a ,

x ∈ (a, m]

b−xb−m ,

x ∈ (m, b)

•



00

1

m

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

ba

sg(x) =

0,

x = m

1,

x = m

•

a

b

L(x) =

1,

x ≤ a

b−xb−a ,

a < x < b

0,

x ≥ b

•

k > 0

gamma(x) =

0,

x ≤ a

1 − ek(x−a)2,

a < x



gamma(x) =

0,

x ≤ a

k(x−a)21+k(x−a)2 ,

a < x

a

k

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

k=1/2

k=1

k=2

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

a b

•

a

b

m a < m < b



m = (a+b)2 a − b

S (x) =

0,

x < a

x−ab−a ,

a ≤ x < m

x−bb−a ,

m ≤ x ≤ b

1,

x > b

0

0.5

1

a m b

• a

d

b

c

trap(x) =

0,

x ≤ a x ≥ d

x−ab−a ,

x ∈ (a, b)

1,

x ∈ [b, c]

d−xd−c ,

x ∈ (c, d)



0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

X

b ca d

•

m

k > 0

k

gauss(x) = e−k(x−m)2

σ2

00

1

m



0 20 40 60 800

0.2

0.4

0.6

0.8

1

Age

M e m b e r s h i p D e g r e e s

young middle−aged old

Age

A

B

U

µA

µB

• A

B

A ∪ B

U



µA∪B

µA∪B(x) = f (µA(x), µB(x)),

f

• A

B

A ∩ B

U

µA∩B

µA∩B(x) = g(µA(x), µB(x)),

g

•

t : [0, 1]2 → [0, 1]

x t y = y t x

x t (y t z ) = (x t y) t z

x ≤ y

w ≤ z

x t w ≤ y t z

x t 0 = 0

x t 1 = x

•

s : [0, 1]

2

→ [0, 1]

x s y = y s x

x s (y s ) = (x s y) s z

x ≤ y

w ≤ z

x s w ≤ y s z

x s 0 = x

x s 1 = 1



0 2 4 6 8 1 00

0.2

0.4

0.6

0.8

1

U

A B

A B

0 2 4 6 8 1 00

0.2

0.4

0.6

0.8

1

U

A ∪ B

0 2 4 6 8 1 00

0.2

0.4

0.6

0.8

1

U

A ∩ B

a

b

• N : [0, 1] → [0, 1]

N (0) = 1

N (1) = 0

N (N (x)) = x

N

N

•

A

A

U

µA : U → [0, 1]

∀x ∈ U, µA(x) = 1 − µA(x)

A

∩ A

= φ



0 2 4 6 8 100

0.5

1

1.5

A

U

A

0 2 4 6 8 100

0.5

1

1.5

Not(A)

U

A

A

a

b

F

U

x1, x2 ∈ U

µF (λx1 + (1 − λ)x2) ≥ min[µF (x1), µF (x2)]

λ ∈ [0, 1]

¯A = A

A ∪ B = B ∪ AA ∩ B = B ∩ A

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∪ A = AA ∩ A = A

A ∪ φ = AA ∩ U = A

(A ⊂ B) ∩ (B ⊂ C )

(A ⊂ C )

A ∩ B = A ∪ BA ∪ B = A ∩ B



x1, x2 ∈ U

µF (λx1 + (1 − λ)x2) ≤ max[µF (x1), µF (x2)]

λ ∈ [0, 1]

F

U

x ∈ U

µF (x) = 1

A

•

F

F



supp(F ) = {x|x ∈ U µF (x) > 0}

• F

F

ker(F ) = {x|x ∈ U

µF (x) = 1}

• F

U

Card(F ) = |F | =

x∈U µF (x)

•

F

α

0 ≤ α < 1 (0 < α ≤ 1)

α F

F α+ = {x|x ∈ U

µF (x) > α}

F α = {x|x ∈ U

µF (x) ≥ α}

α



•

•

•

α



x ∈ U

COG =

x µA(x) × x

x µA(x)

U

COG =

x

µA(x) x dx x

µA(x) dx



A

B

X

Y

A

B

(a, b)

A

B

A × B = {(a, b)|a ∈ A, b ∈ B},

A × B = B × A

A = {a,b,c}

B = {1, 2}

A × B = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)}

B × A = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}

R

X

Y

X × Y R = {(x, y)|x ∈ X y ∈ Y }

A

B

X

Y

A × B

A

B

µA(x)

µB(x)

µA×B(x, y) = min[µA(x), µB(x)] = µA(x) ∧ µB(x),

µA×B(x, y) = µA(x) µB(x) ∀x ∈ X

y ∈ Y

X

Y

R

X

Y



R(x, y) = {µR(x, y)/(x, y)|(x, y) ∈ X × Y }

µR(x, y)

R(x, y)

x

y

d : U ×U → R



∀x,y,z ∈ U

d(x, x) = 0

d(x, y) = d(y, x)

d(x, y) > d(x, x)

d(x, y) + d(y, z ) > d(x, z )

n

U



S : U ×U → R

∀x, y ∈ U

S (x, x) = 1

S (x, y) = S (y, x)

S (x, x) ≥ S (y, x)

S (x, y) ≥ 0

[0, 1]

•

•

•



•

d : U × U → R+

d(x, y) =

ni=1(xi − yi)2

d(x, y) =

ni=1 α(xi − yi)2

d(x, y) =

(x − y)T C −1(x − y)

d(x, y) = p n

i=1(xi

−yi) p

x = (x1, x2, . . . , xn)

y = (y1, y2, . . . , yn)

n

n = 2

√ αi



x

y

C

p = 2

p = 1



δ (xi, yi) =

0 if xi = yi

1 if xi = yi

δ (xi, yi) =

0 if xi = yi

f (xi)(f (xi)

−1)

N (N − 1) if xi = yi

δ (xi, yi) =

0 if xi = yi

1

1 + log N f (xi)

× log N f (yi)

if xi = yi

δ (xi, yi) =

0 if xi = yi

n2i

n2i + 2

if xi = yi

f (aij) aij

a j

N

n2i /(n2

i + 2)

n j

a j



a

b

A

B

S (A, B)

s(a, b)

s(a, b) = S (A, B) = θf (A ∩ B) − αf (A − B) − βf (B − A).

s(a, b) = S (A, B) = θf (A ∩ B)θf (A ∩ B) − αf (A − B) − βf (B − A) .

A ∩ B

A

B

A − B

A

B

B − A

B

A

θ

α

β

f

α = β = 1

S (a, b) = f (A ∩ B)

f (B ∪ A)



α = β = 12

S (a, b) = 2f (A ∩ B)

f (A) + f (B)

α = 1, β = 0

S (a, b) = f (A ∩ B)

f (A)

J



xi

v j

V j

E =C

j=1

ni=1,xi∈V j

(xi − v j)2.

•

• C

2 ≤ C ≤ N



N

µ

xi

{C i|i = 1, . . . , C }

µC j (xi) =

1

xi ∈ C j

0

xi ∈ C j

C

c

c



•

•

•



J (U, V ) =

C

j=1

N

i=1 (µij)

m

xi − v j 2

D = {X 1, X 2, . . . , X i, . . . , X n}

X i

p

X i = {xi1, xi2, . . . , xip}

c(1 < c < n)

m

A = {A1, . . . , Ac};

A(0)(t = 0)

c j=1

µij = 1,

i ∈ {1, 2, . . . , n}

ith

j ∈ {1, 2, . . . , c}

v j =

ni=1(µij)mX ini=1(µij)m

; ∀ j = 1, 2, . . . , c .



µt+1ij =

1

(c

l=1(

dtij

dtil )

2m−1

)

A(t + 1) − A(t) ≤

t ← t + 1



•

•

•



Chapter4



n



W A

min



F 1

F 2 r

atF 1 = {a1, a2, . . . , ar}

atF 2 = {b1, b2, . . . , br}

dis : F (U j) × F (U j) → [0, 1]

Aij, Bij ∈ F (U j)

jth

U j jth i = 1, 2, . . . , m j

a j

b j

µAij(x) = µBij

(x)

x ∈ U j

dis(Aij, Bij)

x1, x2 ∈ U j

dis(Aij, Bij) = |µAij(x1) − µAij

(x2)|

a j

b j

µAij

(x)

µBij(x)

x1, x2 ∈ U j

dis(Aij, Bij) = |µAij(x1) − µBij

(x2)|



dF : atF 1×atF 2 →

[0, 1]

⊕ j : ([0, 1])mj

→[0, 1]



dF (a j , b j) = ⊕ j(dis(A1 j , B1 j),dis(A2 j , B2 j), . . . , d i s(Amjj , Bmjj )) m j

jth

dF (a j , b j) =

(mj

i=1 dis(Aij , Bij)2√ m j

dF

√ m j

S F : atF 1 ×atF 2 → [0, 1]

a j

b j

S F (a j, b j) = 1 − dF (a j, b j)

1 + k jdF (a j, b j)

k j ≥ 0

j = 1, 2, . . . , r

r

k j

dF

k j

dF



k

F = {F 1, F 2, . . . , F k}

r

F s, F t ∈ F

FuzzySim : F × F → [0, 1]

FuzzySim(F s, F t) = ⊗((S (a1, b1), S (a2, b2), . . . , S (ar, br))

⊗ : [0, 1]r → [0, 1]

•

FuzzySim(F s, F t) =

r j=1 α jS F (a j, b j)r

j=1 α j

α j ∈ [0, 1]

•

FuzzySim(F s, F t) = min(S F (a1, b1), S F (a2, b2),ldots,S F (ar, br))

α j

jth



F 1 F 2

FuzzySim(F s, F t)

F s

F t



dis(Aij, Aij) = |µAij

(x) − µAij(x)| = 0

x ∈ U j

dF (a j, a j) =√ (mj

i=1 dis(Aij ,Aij)2√ mj

= 0

S F (a j, a j) = 1−dF (aj ,aj)1+kjdF (aj ,aj)

= (1 − 0)/(1 + k j(0)) = 1

dis(Aij, Bij) = |µAij

(x) − µBij(y)| = |µBij

(y) −µAij

(x)| = dis(Bij, Aij)

x, y ∈ U j dF (a j , b j) = dF (b j, a j)

S F (a j, b j) = S F (b j , a j)

FuzzySim(F s, F t) = FuzzySim(F t, F s)

a j = b j, dF (a j, b j) > 0 S F (a j , b j) < 1 = S F (a j, a j)

DQ = [0, 1]



F DQ = {Low, Regular, High}

DQ

m j = 3

Q(room1) = {0.0/Low, 0.198/Regular, 0.375/High}Q(room2) = {0.0497/Low, 0.667/Regular, 0.0/High}

a1

b1

A11, A21 A31

Low, Regular

High

dF (a1, b1) = (|µA11

(x)−µA11(y)|2+|µA21

(x)−µA21(y)|2+|µA31

(x)−µA31(y)|2

3 )

12

= ( |0.0−0.0497|2+|0.1979−0.667|2+|0.3753−0.0000|2

3 )

12 ∼= 0.35

a1 b1isS (a1, b1) = 1−0.3480

1+k1(0.3480)

k1 ≥ 0

k1

k1 = 1

S (a1, b1) ∼= 0.4836

k1 = 2

S (a1, b1) ∼= 0.3844

DP = [0, 600]

F DP = {Cheap, Moderate, Expensive}

P (room1) = {0.2353/Cheap, 0.726/Moderate, 0.0169/Expensive}P (room2) = {0.0/Cheap, 0.2353/Moderate, 0.4868/Expensive}

P (room1)

P (room2)

a2

b2

A12, A22, andA32

Cheap, Moderate

Expensive

dF (a2, b2) ∼= 0.4151

k2 = 1

S (a2, b2) ∼= 0.4133

k2 = 2

S (a2, b2) ∼= 0.3196



FuzzySim(F 1, F 2) = FuzzySim(room1,room2) =

⊗(S (a1, b1), S (a2, b2))

α1 = 0.5

α2 = 0.8

k1 = k2 = 1

FuzzySim(F 1, F 2) =

2j=1 ajS (aj ,bj)2

j=1 aj= 0.5∗S (a1,b1)+0.8∗S (a2,b2)

0.5+0.8∼= 0.4403

k1 = k2 = 2

FuzzySim(F 1, F 2) ∼= 0.3445

k1 = k2 = 1

FuzzySim(F 1, F 2) = minn=2[0.4836, 0.4133] = 0.4133

k1 = k2 =



2

FuzzySim(F 1, F 2) = 0.3196

dF (a j, b j) =

mji=1 |µAij (x)−µBij (y)|2

mj

12

; x, y ∈ D

A11, B11

Low A21, B21

Regular

A31, B31

High

dF (a1, b1) ∼= 0.2469

a1

b1

k1 = 1

S F (a1, b1) ∼= 0.6039

S F (a1, b1) ∼= 0.5041

k1 = 2

a2

b2

A12, B12

Cheap A22, B22

Moderate

A32, B32

Expensive



dF (a2, b2) = 0.5979

k2 = 1

S F (a2, b2) = 0.2566

k2 = 2

S F (a2, b2) = 0.1871

FuzzySim(F 1, F 2)

α1 = 0.5

α2 = 0.8

k1 = k2 = 1

FuzzySim(F 1, F 2) =

0.3902 k1 = k2 = 2 FuzzySim(F 1, F 2) = 0.3090



k1 = k2 = 1 FuzzySim(F 1, F 2) = 0.2566 k1 =

k2 = 2

FuzzySim(F 1, F 2) = 0.1871

room1

room2



Q(room1) = 0.8 ≡ {0.0/Low, 0.1979/Regular, 0.3753/High}P (room2) = 420 ≡ {0.0/cheap, 0.2353/Moderate, 0.4868/Expensive}

k j

d



Chapter5



•

•

•

•

N 1

N 2

n

atN 1 = {a1, a2, . . . , an}

atN 2 = {b1, b2, . . . , bn}

(a j, b j) ∈ U j × U j U j

jth

d : U j × U j → [0, 1]

d



dN : U j × U j → [0, 1]

a j

b j

dN (a j, b j) = (d(a j, b j) − dm(a j, b j))

(dM (a j, b j) − dm(a j, b j))

dm

dM d

d = dm

d = dM

a j

b j dN : U j × U j → [0, 1]

dN (a j, b j) = min

max

d(a j, b j) − m

M − m , 0

, 1

dN = 0

d ≤ m

dN = 1

d ≥ M

[m, M ]

m

M

U j

jth

a j

b j U j



a j, b j ∈ U j

S N : U j × U j → [0, 1]

S N (a j, b j) = 1 − dN (a j, b j)

1 + k jdN (a j, b j); k j ≥ 0.

k j = 0

S N (a j , b j) = 1 − dN (a j , b j).

k j

N = {N 1, N 2, . . . , N k}

k

N s

N t ∈ N

NumSim : N × N → [0, 1]

NumSim(N s, N t) = ⊗(S N (a1, b1), S N (a2, b2), . . . , S N (an, bn))



N 1

N 2

⊗ : [0, 1]n → [0, 1]

NumSim(N s, N t)

N s

N t

d(x, x) = 0

∀x ∈ U j

dN (a j, a j) = 0

S N (a j, a j) = 1 − dN a j , b j) = 1

NumSim(N s, N s) = 1

s ∈

{1, 2, . . . , k

}

d(x, y) = d(y, x)

x, y

∈ U j

dN (a j, b j) = dN (b j, a j)



S N (a j, b j) = S N (b j, a j) NumSim(N s, N t) = NumSim(N t, N s)

s, t ∈ {1, 2, . . . , k}

∀a j

b j a j = b j

dN (a j, b j) >

0 = dN (a j, a j)

S N (a j , b j) < 1

•

a j

xij, yij ∈ U j

xij = yij

xij = yij



•

a j

{0, 1}

•

•

C 1

C 2 m

atC 1 =

{a1, a2, . . . , am}

atC 2 = {b1, b2, . . . , bm}

a j, b j ∈ U j = {u1 j, u2 j, . . . , umj j}



j = 1, 2, . . . , m m j

jth

δ 1 : U j × U j → {0, 1}

δ 1(aij, bi j) = 0

aij = bij

δ 1(aij, bi j) = 1

δ 2 : U j × U j → {0, 1}

δ 2(aij, bij) = |aij − bij|

|L − U |

L

U

a j

p

dm(a j, b j) = 0

dC : atC 1 × atC 2 → [0, 1]

a j

b j

dC (a j, b j) = δ 1(aij, bij)



a j, b j

dC (a j, b j) = δ 2(aij, bij)

a j, b j

S C : atC 1 × atC 2 → [0, 1]

S C (a j, b j) = 1 − dC (a j, b j)

1 + k jdC (a j, b j); k j ≥ 0,

k j = 0

S C (a j, b j) = 1 − dC (a j , b j)

C = {C 1, C 2, . . . , C k}

k

C s, C t

∈ C

CatSim :

C × C → [0, 1]

CatSim(C s, C t) = ⊗(S C (a1, b1), S C (a2, b2), . . . , S C (am, bm)),

⊗ : [0, 1]m → [0, 1]



C 1

C 2

CatSim(C s, C t)

C s C t

δ (aij, aij) = 0 ; ∀aij ∈ U j

dC (a j, a j) = 0

S C (a j, b j) = 1

δ (aij, aij) = δ (bij, aij)

aij ∈ U j

dC (a j, b j) = dC (b j, a j)

S C (a j, b j) = S C (b j, a j)

CatSim(C s, C t) =

CatSim(C t, C s) ∀a j, b j ∈ U j

a j = b j dC (a j , b j) > 0 =

dC (a j, a j)

S C (a j, b j) < 1

CatSim(C s, C t) < 1



O = {O1, O2, . . . , Ok}

k

Os

Ot

O

p

atOs = {a1, a2, . . . , a p}

atOt = {b1, b2, . . . , b p}

p = r + n + m

r, n

m



Os Ot Sim : O × O → [0, 1]

Sim(Os, Ot) = k1ClassicalSim(Os, Ot) + k2FuzzySim(Os, Ot)

ClassicalSim(Os, Ot) = αNumSim(Os, Ot) + βC atSim(Os, Ot)

Sim(Os, Ot) = k1 (αNumSim(Os, Ot) + βC atSim(Os, Ot))+k2FuzzySim(Os, Ot)

Os, Ot ∈

O

Simn : O × O → [0, 1]

Simn(Os, Ot) = NCF Sim(Os, Ot) = Sim(Os, Ot)

k1(α + β ) + k2

k1, α , β

k2

k1

k2

k1 + k2 > 0



Simn(Os, Ot)

Os

Ot

NumSim(Os, Os) = CatSim(Os, Os) = FuzzySim(Os, Os) =

1

Sim(Os, Os) = k1(α + β ) + k2 Simn(Os, Os) = (k1(α +



β ) + k2)/(k1(α + β ) + k2) = 1 Os Ot

Sim(Os, Ot) =

Sim(Ot, Os)

Simn(Os, Ot) = Simn(Ot, Os)

NumSim(Ot, Os) < 1

CatSim(Ot, Os) < 1

FuzzySim(Ot, Os) < 1

Ot = Os Sim(Ot, Os) < k1(α +β )+k2

Simn(Ot, Os) < (k1(α+β)+k2)

(k1(α+β)+k2) = 1 = S imn(Os, Os)



P 1

P 2

•

Dq = [0, 1]

CDq = {1 − low, 2 − average, 3 − high}

•

D p = [100, 700]

•

Dnob =

{1, 2, 3, 4, 5, 6

}



• D pt =

{house, f lat, bungalow, land, commercialproperty}

•

D pa = [0, 300]

atP 1 ={a1, a2, a3, a4, a5}

atP 2 = {b1, b2, b3, b4, b5}

atC 1 = {a1, a2}

atC 2 = {b1, b2}

a1

b1

ai1, bi1 ∈ CDq i = 1, 2, 3

a1 = a11

b1 = b21

dC (a1, b1) = δ 2(ai1, bi1) =

|a11

− b21

|/|3 −

1|

= |

3 −

2|/|3 −

1|

= 1/2

S C (a1, b1) = 1 − dC (a1, b1) = 1/2

a2

b2 a2 = a12 = house

b2 = b32 = bungalow

dC (a2, b2) = δ 1(ai2, bi2) = 1

S C (a2, b2) = 1 − dC (a2, b2) = 0

α1 = 0.8

α2 = 0.5

a1

a2

CatSim(P 1, P 2) = sum2

j=1α jS C (a j, b j)2 j=1 α j

= α1S C (a1, b1) + α2S C (a2, b2)

(α1 + α2)

= 0.8(0.5) + 0.5(0)

1.3

≈ 0.31



atN 1 = {a3, a4, a5} atN 2 = {b3, b4, b5}

d(a3, b3) = |450 − 250| = 200

dN

(a3, b

3) = min(maxd(a3, b3) − m

M − m , 0 , 1

= min

(max

200 − 100

700 − 100, 0

, 1

= 1/6

S N (a3, b3) = 1−dN (a3, b3) = 5/6

S N (a5, b5) =

0.83

d dN (a4, b4) = (1−0)/(5 −0) = 1/5 S N (a4, b4) = 4/5

α3 = 0.8, α4 = 0.9

α5 = 0.25

a3, a4

a5

NumSim(P 1, P 2) =

5 j=3 α jS N a j , b j)5

j=3 α j

= 0.8(0.83) + 0.9(0.8) + 0.25(0.83)

1.95 )

= 0.82.

α = β = 1

Sim(P 1, P 2) = ClassicalSim(P 1, P 2) =

αNumSim(P 1, P 2) + βC atSim(P 1, P 2) = 0.31 + 0.82 = 1.13



Simn(P 1, P 2) = NCSim(P 1, P 2) = S im(P 1, P 2)/2 = 0.56.



• F Dq = {low, average, high}

• F D p = {cheap, moderate, expensive}

•

Dnob = {1, 2, 3, 4, 5, 6}

•

D pt = {house, f lat, bungalow, land, commercialproperty}

•

F D pa = {small, medium, big}

atP 1 = {a1, a2, a3, a4, a5}

atP 2 = {b1, b2, b3, b4, b5}

atF 1 = {a1, a2, a5}

atF 2 =

{b1, b2, b5}

atN 1 = {a3}

atN 2 = {b3}

atC 1 = {a4}

atC 2 = {b4}

m j = 3

j = 1, 2, 5

a1

b1

a1 = {0.00/Low, 0.32/Regular, 0.46/High},

b1 = {0.01/Low, 0.61/Regular, 0.14/High}.



x = 0.75

y =

0.60

d(a1, b1) =

m1

i=1 |µAi1(x) − µAi1(y)|2m1

1/2

=

|0.0 − 0.01|2 + |0.32 − 0.61|2 + |0.46 − 0.14|23

1/2

= 0.25.



a1 b1 k1 ≥ 0

S (a1, b1) = 1 − 0.25

1 + ka1(0.25).

ka1

ka1 = 1

S (a1, b1) = 0.60 a2 =

{0.00/cheap, 0.84/moderate, 0.25/expensive

}

b2 =

{0.25/cheap, 0.61/moderate, 0.01/expensive}

a5 = {0.00/small, 0.41/medium, 0.75/big}

b5 = {0.02/small, 0.88/med−ium, 0.33/big}

S (a2, b2) = 0.67

S (a5, b5) = 0.47

FuzzySim(P 1, P 2) =

(0.8(0.60) + 0.8(0.67) + 0.25(0.47))/1.85 = 0.28 NumSim(P 1, P 2) = 0.9(0.8) =

0.72

CatSim(P 1, P 2) = 0.5(0) = 0

α = β = 1

ClassicalSim(P 1,

P 2) = αNumSim(P 1, P 2)+βCatSim(P 1, P 2) = 0.72

k1 = k2 = 1

Sim(o1, o2) = k1(αNumSim(o1, o2) + βC atSim(o1, o2)) + k2FuzzySim(o1, o2)

= 0.72 + 0.28 = 1

Simn(P 1, P 2) = S im(P 1, P 2)/3 = 0.33.



H 0

α = 0.05

H 1



•

p −value = 0.00046 < α

•



p − value = 0.21891 > α

•

p − value = 0.75099 > α

ith

ith



Chapter6

a

b

s(a, b)

A

B

s(a, b)

s(a, b) = S (A, B) = θf (A ∩ B) − αf (A − B) − βf (B − A).



s(a, b) = S (A, B) = θf (A ∩ B)

θf (A ∩ B) − αf (A − B) − βf (B − A).

A∩B

A

B

A − B

A

B

B − A

B

A

θ

α

β

f

f

f

f (A ∪ B) = f (A) + f (B)

A

B



A

B

U

A

B



s : F (U ) × F (U ) → [0, 1]

s(A, B) = F s(f (A ∩ B), f (A − B), f (B − A))

F s : R+ × R+ × R+ → [0, 1]

f

f : F (U )

→ R+

U

A ∈ F (U )

f (A) = card(A) = |A| =x∈U

µA(x)



A

f (A) =

x∈U

µA(x)dx

A

N

σ : F (U ) → R+ ∩ 0

a, b ∈ [0, 1]

x, y ∈ U

A, B ∈ F (U )

σ(1/x) = 1

a ≤ b

σ(a/x) ≤ σ(b/x)

A ∩ B = φ

σ(A ∪ B) = σ(A) + σ(B)

A ∈ F (U )

f (A)

A

1/x

x

x

f (1/x) =

|1/x

|= x

∈U µA(x) = 0+ . . .+0+1+0+



. . .+0 = 1, x ∈ U a ≥ b

f (a/x) =

x∈U µA(x) =

x∈U a ≥

x∈U b = f (b/x)

A

B

f (A ∪ B) =

x∈U µ(A ∪ B)(x) =

x∈U µA(x) +

x∈U µB(x) = f (A) + f (B)

A, B ∈ F (U )

Ae ∈ F (U )

e ∈ J

f

f (

e∈J Ae) =

e∈J f (Ae)

Ae ∩ Ae = φ;

e = e

A ∈ C (U ) ⇒ f (A) = |supp(A)

A ⊂ B ⇒ f (A) ≤ f (B)

|core(A)| ≤ f (A) ≤ |supp(A)|

f (A) + f (B) = f (A ∪ B) + f (A ∩ B)

f (

e∈J (Ae) ≤e∈J f (Ae)

A ⊂ B ⇒ supp(A) ⊂ supp(B) ⇒A ∩ B = A = 0

|supp(A)| ≤ |supp(B)| ⇒ f (A) = x∈supp(A) µA(x) ≤ f (B)

core(A) ⊂ A ⊂ supp(A)

A ∪ B

A ∩ B

f (A ∪ B) + f (A ∩ B) =

x∈U µ(A ∪ B)(x) +

x∈U µ(A ∩ B)(x)

=

x∈U [maxx∈U (µA(x), µB(x))] +

x∈U [minx∈U (µA(x), µB(x))]

=

x∈U [maxx∈U (µA(x), µB(x)) + minx∈U (µA(x), µB(x))]

= f (A) + f (B)



f (A) + f (B) = f (A∪ B) + f (A∩ B) f (A∪B) ≤ f (A) + f (B)

A

B

A ∩ B

µA∩B = minx∈U

[µA(x), µB(x)]

A − B

A

B

U

A ∩ Bc

U

A − B = {x ∈ U |x ∈ A

x ∈ B}

A, B ∈ F (U )

A − B = {x ∈ U |x ∈F A and x ∈F B}

∈F : U × F (U ) → [0, 1]

∈F (x, A) = µA(x)

∈F (x, A) = 1 − µA(x)

A − B

A

B

µ(A −1 B)

µA−1B(x) = minx∈U

[µA(x), 1 − µB(x)] ∀x ∈ U

A

B



µA−2B(x) = maxx∈U

[0, µA(x) − µB(x)]

µA−3B(x) =

µA(x) µB(x) = 0

0

µB(x) > 0

µA−1B(x) > µA−2B(x)

µA−3B(x) < µA−2B(x)

x ∈ U

−2

A

B

−2

A

B

A −2 B

A

B

B ⊂ A

A ⊂ B

A = B



A B

A−1 B

A−2 B

A−3 B



0 2 4 6 8 100

0.5

1Inputs: Two fuzzy sets A and B

ABmin(A,B)

0 2 4 6 8 100

0.5

1

Output: Fuzzy difference: FD(A,B)=A−B

A−B

0 2 4 6 8 100

0.5

1

Inputs: Two fuzzy sets A and B

A

Bmin(A,B)

0 2 4 6 8 100

0.5

1


A−B

0 2 4 6 8 100

0.5

1


0 2 4 6 8 100

0.5

1Output: Fuzzy difference: FD(A,B)=A−B

A

Bmin(A,B)

A−B

0 2 4 6 8 100

0.5

1


0 2 4 6 8 100

0.5

1


AB

min(A,B)

A−B

0 2 4 6 8 100

0.5

1


0 2 4 6 8 100

0.5

1


ABmin(A,B)

A−B

0 2 4 6 8 100

0.5

1


A

Bmin(A,B)

0 2 4 6 8 100

0.5

1


A−B

0 2 4 6 8 100

0.5

1


0 2 4 6 8 100

0.5


AB

min(A,B)

A−B

0 2 4 6 8 100

0.5

1


0 2 4 6 8 100

0.5

1


AB

min(A,B)

A−B

0 2 4 6 8 100

0.5

1


AB

min(A,B)

0 2 4 6 8 10−1

0

1


A−B

0 2 4 6 8 100

0.5

1


0 2 4 6 8 10−1

0


A

Bmin(A,B)

A−B

−2



−2 −

A, B ∈ F (U )

−

F (U )

A ⊆ B

A − B = φ

A ⊆ A

A − B ⊆ A − B

A

⊆ B ⇔ µA(x) ≤ µB(x) ∀ x ∈ U

µA(x) − µB(x) ≤ 0

A − B

µA−B(x) = max[0, µA(x) − µB(x)]; ∀ x ∈ U ⇒ µA(x) −µB(x) = 0 ⇒ A− B = φ

A ⊆ A ⇒ µA(x) ≤ µA(x) ⇒ µA(x)−µB(x) ≤ µA(x)−µB(x) ⇒

max[0, µA(x)] − µB(x)) ≤ max[0, µA(x) − µB(x)] ⇒ µA−B(x) ≤ µA−B(x) ⇒A − B ⊆ A − B

A, B ∈ F (U )

f (A ∩ B) = card(A ∩ B) = |A ∩ B|

A

B

f (A − B)

f (B

−A)

f (A ∩ B) = card(A ∩ B) = |A ∩ B| =x∈U

µA∩B(x) =x∈U

minx∈U

[µA(x), µB(x)]



f (A−B) = card(A − B) = |A− B| =x∈U

µA−B(x) =x∈U

(maxx∈U

[0, µA(x) −µB(x)])

S (A, B) = α|A ∩ B|

α|A ∩ B| + β |A − B| + γ |B − A|

α > 0

β, γ ≥ 0

α

β = γ



o1

o2 ato1 =

{a1, a2,...,an} ato2 = {b1, b2,...,bn}

n

n

ai = {Ai1, Ai2,...,Aimi}

bi = {Bi1, Bi2,...,Bimi}

mi

i

th

i = 1, 2, . . . , n

Age = {(0.2/young), (0.75/middelage)}

ai = {Aimai

}

bi = {Bimbi

} mai, mbi ∈ {1, 2,...,mi}

young

young



S

S (a, b) = S (a, b)

s(A, B) = S (B, A)

β = γ

s

β = γ

f (A − B) =

f (B − A)

f

S (A, B) =

f (A∩B, A−B, B−A)

A∩B

A − B

B − A

m

OF = {o1, o2,...,om}

n

os, ot ∈ OF

Sim : OF × OF → [0, 1]

Sim(os, ot) = ⊗Sim (S (a1, b1), S (a2, b2),...,S (an, bn))



⊗Sim : [0, 1]n → [0, 1]

ai

wi

∀i; wi ∈ [0, 1]

wi

Sim(os, ot) =

ni=1 wiS (ai, bi)ni=1 wi

; wi ∈ [0, 1]

Sim(os, ot) = min[S (a1, b1), S (a2, b2),...,S (an, bn)]

Sim(os, ot) = min[S (a1, b1), S (a2, b2),...,S (an, bn)]

max[S (a1, b1), S (a2, b2),...,S (an, bn)]



Do =

[0, 1]

Do = [0, 600]

DdfUni = [0, 10]

F Do = {low, average, high}

Do F DP = {cheap, moderate,

expensive} dfUni

F DdfUni = {close − to,far − from}



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

(a) quality

low average

high

0 100 200 300 400 500 600

0

0.5

1

(b) price

D e g r e e o f m e m b e r s h i p

cheap moderate expensive

0 1 2 3 4 5 6 7 8 9 10

0

0.5

1

(c) dfUni

close−to far−from



α = β = γ = 1

s

s

s



0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Student Rooms

S i m i l a r i t y D e g r e e s

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Student Rooms


0 5 10 15 20 25 30 35 40

0

0.2

0.4

0.6

0.8

1

Student Rooms


α = β = γ = 1



wquality = w price = wdfUni = 1

wquality = 0.8, w price = 0.5, wdfUni = 1

α, β

γ

W A

Min

Min/Max

α

β

γ

s(average, high) = 0.1325 s(high, average) = 0.1107

α = 1

β = 0.5

γ = 0.8

s(average, high) = 0.1107 s(high, average) = 0.1325

α = 1 β = 0.8

γ = 0.5



0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Student Rooms


0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Student Rooms


0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Student Rooms


0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Student Rooms


W A

Min

Min/Max



α = 1, β = 0.5

γ = 0.8

s

α = 1, β = 0.8

γ = 0.5

s



Chapter7



•

•

•

W A



D = {o1, o2, . . . , on}

p

p

D

D

f

c

Df

V = {v1, v2, . . . , vc}

1/c

ni=1

c j=1

FuzzySim(oi, v j)

µij =

cl=1

FuzzySim(oi, vl)

FuzzySim(oi, v j)2/m−1



v j = ot

nji=1

sim(ot, oi) = max1≤r≤nj

njq=1

sim(oq, or)

n j

jth

1 ≤ r ≤ n j

min(FuzzSim(centres, newCentres) > .

[0, 1]

•

•

•



sw(oi)

oi

C j

C j

oi

a(i) =

oi C j.

C l

C j



d(oi, C l) =

oi C l.

C j = C l

oi

b(oi) = minC l=C j

d(oi, C l)

oi

oi

sw(oi)

sw(oi) = a(oi) − b(oi)

max[a(oi), b(oi)]

max a(oi), b(oi)

sw(oi) =

1 − a(oi)

b(oi)

a(oi) < b(oi)

0 a(oi) = b(oi)

b(oi)

a(oi) − 1

a(oi) > b(oi)

−1 ≤ sw(oi) ≤ 1

b(oi)



b(oi) = maxC l=C j

d(oi, C l)

sw(oi)

sw(oi) =

1 − b(oi)

a(oi)

a(oi) > b(oi)

0

a(oi) = b(oi)

a(oi)

b(oi)− 1

a(oi) < b(oi)



C = 3 m = 2

90%



0 10 20 30 40 50

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

Index

f c l Q u 1 $ m m e m b e r s h i p [ , 1 ]

0 10 20 30 40 50

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

Index

f c l Q u 1 $ m m e m b e r s h i p [ , 2 ]

0 10 20 30 40 50

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

Index

f c l Q u 1 $ m m e m b e r s h

i p [ , 3 ]



0 10 20 30 40 50

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

Index

f c l P r 1 $ m m e m b e r s h i p [ , 1 ]

0 10 20 30 40 50

0 . 1

0 .

2

0 . 3

0 . 4

0 . 5

0 . 6

Index


0 10 20 30 40 50

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

Index




Chapter8



•

•



•

•

[0, 1]

•

•



W A

min



min/max

W A

Min

Min/Max



•



•

•

•



•

•



AppendixA



AppendixB



dfUni



AppendixC

PhD Thesis Yasmina-Bashon 2013-06-18

Documents

Transcript of PhD Thesis Yasmina-Bashon 2013-06-18