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The first of the above equations implies that the union of all subsets
A
g
contains
all the data. The second and third equations respectively suggest that the
intersection of the subsets must be a void set,
i.e.
subsets are disjoint, and none of
the subsets is empty or contains all the data contained in
Z
. In terms of membership
functions, a partition can be conveniently represented by the partition matrix:
ªº
U
¬¼
.
P
gs
cN
u
That is, the
g
th row of this partition matrix contains the values of the membership
function
P of the
g
th subset
A
g
of
Z
. Therefore, it can be represented as
"
"
# # #
"
PP P
PP P
ª
º
11
12
1
N
«
»
«
»
21
22
2
N
U
«
(4.18)
»
«
»
PP P
u
«
»
¬
¼
c
1
c
2
cN
cN
It follows from the above equation that the elements of the
U
partition matrix must
satisfy the following conditions:
P
{0,1}, 1
dd dd
g
c
; 1
s
N
;
(4.19a)
gs
c
P
dd
1, 1
s
N
;
(4.19b)
¦
gs
g
1
N
0
P
Ng
, 1
d d
c
.
(4.19c)
¦
gs
s
1
The space of all possible hard partition matrices for
Z
, called the hard partitioning
space (Bezdek, 1981), is thus defined by
^
`
c
N
^`
MU
\
cN
u
P
0,1 ,
g s
, ;
P
1,
s
; 0
P
Ng
,
.
¦
¦
hc
gs
gs
gs
g
1
s
1
In the following, let us illustrate the hard partitioning concept by an example with
the given data set
" , where
N
= 10. Suppose that the given data
set is hard partitioned into three clusters
A
1
,
A
2
and
A
3
. The partition matrix
U
in
this case may look like:
Zzz z
{, , , }
N
12
1, 1, 1, 0, 0, 0, 0, 0, 0, 0
0, 0, 0,1,1,1, 0, 0, 0, 0
0, 0, 0, 0, 0, 0,1,1,1,1
ª
º
«
»
U
«
»
«
»
¬
¼
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