State of the Art and Development Trend - Computational Intelligence in Time Series Forecasting - page 342

From the generated fuzzy partition matrix U ¬ ¼ ª º P that contains the

gs

cN u

> @

membership degrees of the data object , in

the cluster group g = 1, 2, ..., c , the one-dimensional antecedent fuzzy sets are
constructed from the point-wise projection of the rows of matrix U . Thereafter, the
Takagi-Sugeno (TS) rule's consequents are estimated from the training data, using
the antecedent fuzzy sets, by the least squares error method. After validating the
time series model with the validation data, the future values of the time series can
be predicted easily by applying the generated Takagi-Sugeno rules.

s z , s = 1, 2, ..., N , and

T ZXy

,

10.5.1 Fuzzy Clustering Using Kohonen Networks

A Kohonen network is a self-organizing neural network, usually trained in
unsupervised competitive mode. It is very well suited for data clustering. The
network is closely related to the c -means clustering algorithm (Huntsburger and
Ajjimarangsee, 1989). This was demonstrated by Bezdek et al. (1992) in their
proposal of a data clustering algorithm that was based upon the Kohonen networks.
The ideas from the fuzzy c -means (FCM) algorithm are basically integrated
into the learning rate and weight-updating strategies of the Kohonen-type
networks, while implementing the fuzzy Kohonen clustering network ( FKCN ). The
new algorithm can be viewed as a Kohonen-type fuzzy c -means ( FCM ) algorithm.
A Kohonen network (Kohonen, 1982) basically performs on some specific
heuristic procedures, the termination of which does not represent the optimization
of any model. In this kind of network, the final weight vectors depend on the input
sequence. As a consequence, different initial conditions usually lead to different
results.
Bezdek et al. (1992) introduced a new class of networks called FKCNs. In
FKCNs, fuzzy membership values of output categories are incorporated into
learning rates. In addition, FKCNs are self-organizing networks, since the size of
the update neighbourhood is automatically adjusted during the learning process.
Also, FKCNs usually terminate in such a way that the FCM objective function is
approximately minimized. An FKCN is non-sequential and, therefore, it is
independent of the sequence of feed of the input data.

The learning algorithm of an FKCN can be described as follows.

A data set that consists of observations of n measured variables ( e.g. pressure,
temperature, flow, etc. of a process) grouped into n -dimensional column vectors
>

@

T

n
z \ and a set of N such observations ( e.g. at time
instants 1, 2, ..., N etc .) can be denoted as Z = { z s | s = 1, 2, ..., N } and represented
by the n u matrix

z z , z , " , z ,

,

s 1 s 2 s ns

> @ ,

where the rows and columns are indicated
respectively by r = 1, 2, ..., n and s = 1, 2, ..., N . The rows and columns of this Z
matrix are called features (attributes) and patterns (objects) respectively. For a
given data set Z , c fuzzy clusters (groups)

Zz u

rs

nN

^ `

> @

are fuzzy partitions of P : Z o 0,1

g

data Z in the c u values of , with 1 dd and 1 dd , that satisfy

PP z g c s N

gs g s