Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 180

FCM, since the inverse and the determinant of the cluster covariance matrix must
be calculated in each iteration.

4.7.3.1 Gustafson-Kessel Clustering Algorithm
Given the data set

^ `

the
weighting exponent or fuzziness exponent parameter m > 1, the termination
tolerance

ZZZ , , " , select the number of clusters 1 , Z cN ,

12 N

H! and the cluster volumes 0

U . Initialize the partition matrix

randomly, such that

l . 0

U

M fc

Repeat for iterations l = 1, 2, 3, ...

x Step 1 compute the cluster prototypes or cluster centres (means)

m

N

§

l
gs

1

·

Z

¦ ¨ P

¸

s

© ¹

l

s 1

;1 d d g c . v

g

m

N

l
gs

1

§ ·

P

¦ ¨

¸

© ¹

s 1

x Step 2 compute the cluster covariance matrices

m

N

§

l
gs

1

·

T

Z

() l

Z

() l

¦ ¨ P

v v

¸

g g

s s

© ¹ s 1

F ;1 d d g c .

g

m

N

§

l
gs

1

·

P

¦ ¨

¸

© ¹

s 1

x Step 3 compute the distances

T

§

l

·

ª

1/ n

º

1

l
g

2

det Z ,

Z U F

D ¨ v ¸ F v

g

s g g

« »

gsA

¬

g

¼

s

© ¹

g

1 dd dd g c ; 1 s N ;

x Step 4 update the partition matrix:

for 1 dd s N

if ! 0 for all g " 1, 2, , c ;

D

gsA