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(a) Choose n
q
, the cardinality of data points to be contained in C
k
, randomly.
(b) For each dataset
uð
k
Þ;
y
ð
k
Þ
, build C
k
containing
f
uð
k
Þ;
y
ð
k
Þ
g
and its
ð
n
q
1
Þ
nearest neighbors satisfying:
2
2
^
uð
k
Þu
uð
k
Þu
k
k
; 8ðu;
y
Þ 62
C
k
:
ð
7
Þ
(c) Determine
h
k
for each data in C
k
;
k
¼
1
; ...;
N using the least square
method.
T
k
/
k
Þ
1
T
k
Y
k
:
h
k
¼
ð/
/
ð
8
Þ
where
T
t
n
q
k
t
k
Þ...u ð
/
k
¼
u ð
Þ
;
T
t
n
q
k
t
k
Þ...
Y
k
¼
y
ð
y
ð
Þ
:
t
n
k
) are the indexes of the elements belonging in C
k
3. Cluster the local parameter vectors (
and (t
k
; ...;
; ...;
N) into s disjoint clusters
while determining the value of s by using a suitable classi
h
k
;
k ¼
1
cation technique.
s
i¼1
:
5. Estimate the polyhedral partitions Hif
fg
h
fg
4. Identify the
final models
s
i¼1
i.e. estimate the hyperplanes sepa-
rating Hi
i
from Hi
j
, i
cation
problem that can be solved by several established techniques. The most com-
mon technique is the Support Vector Machines (SVM) (Wang
2005
; Duda et al.
2001
).
6
¼ j. This is a standard pattern recognition/classi
The classi
cation of data represents the main step for PWARX system identi-
fication because a successful identi
fication of models
parameters and hyperplanes
'
depends on the correct data classi
cation. For the sake of simplicity, the early
approaches use classical clustering algorithms for the data classi
cation such as
k-means algorithms.
However, these algorithms present several drawbacks. In fact, they may con-
verge to local minima in the case of poor initializations because they are based on
the minimization of non linear criterion. Furthermore, their performances degrade in
the case of the presence of outliers in the data to be classi
ed. In addition, most of
them assume that the number of sub-models is a priori known.
4 The Proposed Clustering Techniques
In order to improve the identi
cation results we propose the use of other classifi-
-
cation algorithms such as Chiu
s alogorithm and DBSCAN algorithm.
'
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