Data Quality Enhancement Technology to Improve Decision Support - Efficient Decision Support Systems: Practice and Challenges from Current to Future - page 70

NC

2



m

J  u x  c ,1   (1)

m ij i j

i  11 j

Where m is any real number greater than 1, i u is the degree of membership of x in the
cluster j , x is the i th of d-dimensional measured data, c is the d-dimension centre of the
cluster, and * is any norm expressing the similarity between any measured data and the
centre.
Fuzzy partitioning is carried out through an iterative optimization of the objective function
shown above, with the update of membership

i u and the cluster centre c by:

1

u  , (2)

ij

2

 

xc
xc



m  1

C

i j



 

 



k  1 i k

 

N



m
ij

ux .

i

i  1

c  (3)

j

N



m
ij

u

i  1

 

 

k  1

() k

This iteration will stop when the max u  u  , where  is a termination criterion 

ij

ij ij

between 0 and 1, and k is the iteration step. This procedure converges to a local minimum or
a saddle point of J m .
The algorithm is composed of the following steps [17]:

1. Initialize U=[u ij ] matrix, U (0)

2. At k-step: calculate the centre vectors C (k) =[c j ] with U (k)

N



m
ij

ux .

i

i  1

c 

j

N



m
ij

u

i  1

3. Update U (k) , U (k+1)

1

u 

ij

2

 

xc
xc



m  1

C

i j



 

  

k  1 i k

 

If || U (k+1) - U (k) ||<  then STOP; otherwise return to step 2.

4.

The Fuzzy C-Mean clustering and statistic equation were implemented in the MATLAB
software.