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From the
U
matrix it is seen that the elements
z
1
,
z
2
, and
z
3
possibly belong to
cluster
A
1
(as the first three entries in the first row are 1), and
z
4
,
z
5
, and
z
6
belong
to cluster
A
2
, whereas the remaining data elements
z
7
to
z
10
belong to cluster
A
3
.
Here, note that the sum of each column of the partition matrix
U
is always 1.
4.7.1.3 Fuzzy Partition
A fuzzy partition can be considered as a generalization of the hard partition and
this follows directly by allowing
g
P to attain any real values within [0,1]
(Babuška, 1996). Similar to hard partitioning the conditions for a fuzzy partition
matrix are described by Ruspini (1970):
> @
P
0,1 , 1
dd dd
g
c
; 1
s
N
;
(4.20a)
gs
c
P
dd
1, 1
s
N
;
(4.20b)
¦
gs
g
1
N
0
P
Ng
, 1
d d
c
.
(4.20c)
¦
gs
s
1
Similar to hard partitioning, the
g
th row of the partition matrix
U
contains the
values of the membership function
P of the
g
th subset
A
g
of Z. The fuzzy
partitioning space for Z is the set
^
`
c
N
>@
MU
\
cN
u
P
0,1 ,
g s
, ;
P
1,
s
; 0
P
Ng
,
.
¦
¦
fc
gs
gs
gs
g
1
s
1
Let us now illustrate the fuzzy partitioning concept by an example with the same
data set
" , where
N
= 10, as used in the hard partitioning example.
Suppose that the given data set is fuzzy 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
ª
0.82, 0.90, 0.96, 0.20, 0.10, 0.02, 0.03, 0.05, 0.1, 0.02
0.05, 0.06, 0.02, 0.75, 0.85, 0.90, 0.17, 0.25, 0.3, 0.08
0.13, 0.04, 0.02, 0.05, 0.05, 0.08, 0.80, 0.70, 0.6, 0.90
º
«
»
U
«
.
»
«
»
¬
¼
Here, the elements in the first row of the matrix correspond to the degrees of
membership of the elements
z
1
,
z
2
, ...,
z
10
respectively in the cluster or subset
A
1
.
Similarly, entries in the second row and third row of the
U
matrix represent the
degrees of membership of the data elements
z
1
, z
2
, ...,
z
10
in the clusters
A
2
and
A
3
respectively. In addition, the entries in the
U
matrix are not restricted to 0 and 1 but
can take any real value within 0 and 1. Moreover, the sum of each column of the
U
matrix is also equal to 1 in this case. If this restriction is relaxed,
i.e.
the sum of
degrees of membership of any particular data element in the various clusters need
not be 1, then we have
possibilistic partition
, a special case of fuzzy partition and
very useful in identifying
outliers
. Outliers are data points that are neither a
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