Journal of Software - Academy Publisher

882 JOURNAL OF SOFTWARE, VOL. 6, NO. 5, MAY 2011
w
=
n n
∑∑
i= 1 k=
1
j p n n
∑∑∑
j= 1 i= 1 k=
1
| z − z |
ij kj
| z − z |
ij kj
where wj denotes weight of the jth index,
n
zij= xij ∑ x
i=
1 ij ( i = 1,2, ⋅⋅⋅ , n; j = 1,2, ⋅⋅⋅ , p)
To enhance the stability of clustering result, a
determinate initial clustering center is desired prior to
performing clustering algorithm. In this paper, the
relation matrix R is constructed by employing the
included angle cosine formula (4), and the samples are
classified by threshold partition method. Though the
classifying effect is not well, the classification result can
provide a decided initial clustering center for improved
FKCM clustering algorithm.
r
ij
=
p
∑
| x x |
k = 1
ik jk
p p
2 2
( ∑x )( )
ik ∑x
jk
k= 1 k=
1
(3)
(, i j = 1,2, � , n)
(4)
where r ij denotes the similarity between the ith and the
jth sample.
B. FKCM algorithm
However, the FCM clustering analysis is somewhat
limited in real world problems and nonlinear clustering
analysis would be highly desirable. An efficient method
of obtaining the nonlinear cluster algorithm is to first map
p
the patterns of input space Ω into some higher
q
dimensional feature space Ω using a kernel
function φ() ⋅ , the FCM can be performed in this feature
space. When the kernel function is chosen, the Euclid
distance between i x and x j in feature space
1/2
is dˆ ij( xi, x j) = [ K( xi, xi) − 2 K( xi, x j) + K(
xj, x j)]
,
i, j = 1,2, � , n.
Let V be clustering center matrix in input space,
V= ( v , v , , v ) v = ( v , v , �, v ),( i = 1,2, � , c)
.
1 2 � c i i1 i2 ip
Let Û be membership matrix in feature,
Uˆ= ( uˆ ˆ ˆ
1, u2, � , un)
,
uˆ ˆ ˆ ˆ
i = ( ui1, ui2, �, uic),( i = 1,2, � , n)
. Hence, Objective
function of FKCM clustering algorithm in feature space
is
m( ; ,
c n
) = ∑∑
j= 1 i=
1
m ˆ 2
ji ij
Jˆ XUD ˆ ˆ uˆd , 2 < c < n (5)
New clustering center vectors in feature space are
© 2011 ACADEMY PUBLISHER
j
j
n
ukj m
i
n
ukj
m
k= 1 i=
1
vˆ φ( v ) ( ˆ ) φ(
x )/ ( ˆ ) , j = 1, 2, � , c (6)
= =∑ ∑
The φ( xi ) is dropped in (6). Unfortunately, the mapping
function φ() ⋅ may not be known explicitly and if the
dimension of the feature space
q
Ω is very high or
infinite, it is difficult to solve for objective function by
(6). To get around this difficulty, the problem is
reformulated to involve only the dot product of the
patterns x i ( i= 1, 2, � , n)
in the feature space.
K
n
m
∑(
uˆkj ) K(
xk, xi)
( , ˆ i j) = φ( i) ⋅ φ ( k = 1
j) = n
m
∑ ( uˆ
kj )
k = 1
n n
∑∑ uˆkj m
uˆtj m
K xkxt j j = φ j ⋅ φ j
k= 1 t=
1 =
n
2
⎛ m ⎞
⎜∑( uˆ
kj ) ⎟
k = 1
x v x v (7)
K ( vˆ , vˆ ) ( v ) ( v )
uˆ
=
=
ij c
j=
1
(1 / dˆ
( x , vˆ
))
j=
1
2 1/( m−1)
ij i j
(1 / dˆ
( x , vˆ
))
2 1/( m−1)
ij i j
( ) ( ) ( , )
⎝ ⎠
(1 / ( K( x , x ) − 2 K( x , vˆ ) + K(
vˆ , vˆ
)))
c
∑
∑
i i i j j j
(1 / ( K( x , x ) − 2 K( x , vˆ ) + K(
vˆ , vˆ
)))
i i i j j j
1/( m−1)
1/( m−1)
When ˆ 2
d ( , ˆ ij xiv j ) = 0,
uˆ 1, ˆ ij = uit = 0, ( t∈[1, j) ∪ ( j, c])
.
ˆ 1/2
d ( , ˆ ) [ ( , ) 2 ( , ˆ ) ( ˆ , ˆ
ij xiv j = K xixi− K xiv j + K v j v j )] ,
i = 1, 2, �, n; j = 1, 2, �,
c
(10)
To acquire the optimization membership degree matrix
*
Û and corresponding distance matrix * ˆD , the equation of
ˆ() l ˆ(
l 1)
| J J |
−
− must be convergent, that is, equation of
lim Jˆ
( ; ˆ, ˆ
m XUD) comes into existence [18]. Hence, the
l→∞
variance ε can be set at a random small value, the initial
distance matrix and the initial membership matrix are
known, then the iterative algorithm can be performed by
() ( 1)
(5), (7), (8), (9), and (10), if ˆ l
| ˆ l−
J − J | < ε , the
iterative algorithm ceases, and the constraint optimization
membership matrix () ˆ l
U and the constraint optimization
()
distance matrix ˆ l
D can be acquired, finally, samples is
classified in terms of maximum membership principle.
When dot product operation is performed between pattern
and indexes weight vector in kernel function of the
FKCM clustering algorithm, the improved FKCM
clustering algorithm can be obtained. Furthermore, the
statistic F in (11) can be utilized for acquiring the optimal
classification amount. Statistic F is a conventional index
for evaluating the clustering validity and the nonlinear
factors of data set are not considered, hence the KF index
is constructed by using kernel function and shown in (12)
for evaluating more efficiently the clustering validity .
(8)
(9)