Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 178

cluster means are all equal to the mean of Z . These limit properties of fuzzy c -
means functionals are independent of optimization method used (Pal and Bezdek,
1995). Usually, m is selected as 2.

Termination criterion
The FCM algorithm stops iterating when the norm of the difference between U in
two successive iterations is smaller than the termination tolerance parameter. The
usual choice of a termination tolerance is 0.001. The termination tolerance of 0.01
also works well in most cases, while it drastically reduces the computing times.

Norm-inducing matrix
The shape of the clusters is dependent on the choice of the norm-inducing matrix A
in the distance measure. A common choice of the norm-inducing matrix A is the
identity matrix I , which gives the standard Euclidean norm:

T

2

.

D Z v Z v

s g

gs s g

Another choice of the norm-inducing matrix A is a diagonal matrix that
accounts for different variances in the directions of the coordinate axes of Z :

2

ª "

"
# # #
"

00 º

V 1

« »

0

2

0

V

« »

2

A «

»

« »

00

2

« V »

¬ ¼ n

This matrix induces a diagonal norm on \ . Finally, A can be defined as the
inverse of the covariance matrix of Z: A = R -1 , with

1

N

T

R ZZZZ .

¦

s s

N

s 1

Here, Z denotes the mean of the data. In this case A induces the Mahalanobis
norm on \ . The norm influences the clustering criterion by changing the
measure of dissimilarity. The Euclidean norm induces hyperspherical clusters
(hyperspheres are surfaces of constant memberships). Both the diagonal and the
Mahalanobis norm generate hyperellipsoidal clusters. With the diagonal norm, the
axes of the hyperellipsoids are parallel to the coordinate axes, while with the
Mahalanobis norm the orientation of the hyperellipsoids is arbitrary. A common
limitation of clustering algorithms based on a fixed distance norm is that it forces
the objective function to prefer clusters of a certain shape even if they are not
present in the data.

Initial partition matrix
The partition matrix is usually initialized at random, such that

U . A simple
approach to obtain such U is to initialize the cluster centers v g at random and

fc