X = {x1, x2, ..., xn} C = {1, 2, ..., k}k C

530978-1-4577-1676-8/11/$26.00 c©2011 IEEE

kk

2d

2011 11th International Conference on Intelligent Systems Design and Applications 531

X = {x1, x2, ..., xn}d k

X kZ = {z1, z2, ..., zk}

X k × nU = [μij ] μij

jth ith 1 ≤i ≤ k, 1 ≤ j ≤ n

k∑i=1

μij = 1, 1 ≤ j ≤ n

μij ∈ [0, 1], 1 ≤ j ≤ n, 1 ≤ i ≤ k

aVa a : U → Va a ∈ A

DS (U, A∪{d}) d /∈ AA

B ⊆ AB

INDIS(B)INDIS(B) = {(x1, x2) ∈ U2|a(x1) = a(x2) ∀a ∈ B}

x1 x2

B (x1, x2) ∈ INDIS(B)[x]B

X ⊆ U BB − lower BX = {x|[x]B ⊆ X}

B − upper BX = {x|[x]B ∩ X = φ}B − boundary region BX −BX

BUi BUi BUi − BUi

ith Ui

ith ωi

ith W = [ωir]k×d ωir

rth ith

d∑r=1

ωir = 1, 1 ≤ i ≤ k,

ωir ∈ [0, 1] , 1 ≤ i ≤ k, 1 ≤ r ≤ d

JPRFCM

k

JPRFCM

⎧⎪⎨⎪⎩

aA + bB BUi = φ ∧ BUi − BUi = φ

A BUi = φ ∧ BUi − BUi = φ

B otherwise.

A =∑

xj∈BUi

∑ki=1

∑dr=1 μα

ijωβird

2ijr

∑xj∈(BUi−BUi)

∑ki=1

∑dr=1 μα

ijωβird

2ijr

a b

d2ijr = (xjr − zir)

2

ith jth

rth α ∈ (1,∞) , β ∈ (1,∞)

μij ωir

μij ωir

μij = 1

/k∑

l=1

[∑dr=1(ωir)

βd2ijr∑d

r=1(ωlr)βd2ljr

]1/(α−1)

532 2011 11th International Conference on Intelligent Systems Design and Applications

ωir = 1

/d∑

l′=1

[ ∑nj=1(μij)

αd2ijr∑n

j=1(μij)αd2ijl′

]1/(β−1)

zir =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

∑xj∈(BUi−BUi)

μαijxjr∑

xj∈(BUi−BUi)μα

ij

BUi �= φ ∧ BUi − BUi = φ

a ×

∑xj∈BUi

μαijxjr∑

xj∈BUiμα

ij

+b ×

∑xj∈(BUi−BUi)

μαijxjr∑

xj∈(BUi−BUi)μα

ij

BUi �= φ ∧ BUi − BUi �= φ∑xj∈BUi

μαijxjr∑

xj∈BUiμα

ij

, otherwise.

a ≈ 1

a b o < a < b < 1 a + b = 1

zi, 1 ≤ i ≤ k, k

μij k nμij μij

μij μij ε xj ∈

BUi xj ∈ BUi xj

xj ∈ BUi

μij kωir k d

zi

ε10−3

ε

a = .85 b = .15

k

α βm

α βα β

2011 11th International Conference on Intelligent Systems Design and Applications 533

mm

α β

r =∑c

i=1 si/n si

i n

xj cl

l = arg max1≤i≤k μij

534 2011 11th International Conference on Intelligent Systems Design and Applications

•

PC = 1/n∑k

i=1

∑nj=1 μ2

ij

1/k

•

CE = −1/n∑k

i=1

∑nj=1 μij log(μij)

μij 1/k

2011 11th International Conference on Intelligent Systems Design and Applications 535

o o

u

u

o

536 2011 11th International Conference on Intelligent Systems Design and Applications