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Thus, to solve problems of fuzzy cluster analysis, possibility to transform an
initial intransitive relation to a transitive one is of great interest. Such
transformation is ensured with operation of transitive closure which was first
considered in [100—101].
The relation
R
=
R
∪
R
2
∪
R
3
∪
R
k
∪
...
, where relation
R
is defined
recursively as
2
k
k
−
1
R
=
R
R
;
R
=
R
R
,
k
=
3
4
,...
is referred to as transitive closure of a fuzzy binary relation.
Transitive closure
R
of any fuzzy relation
R
is transitive and is the least
transitive relation including
R
[15]. In [28] it is proved that a fuzzy relation is
transitive in the only case when
, and if the set
X
contains
n
elements,
R
=
R
then
R
=
R
∪
R
2
∪
R
3
∪
R
n
.
n
−
1
Besides, if fuzzy relation
R
is reflective, then
.
Application of fuzzy relations in the cluster analysis is discussed in
[102—103]. In [100] clusterization procedure based on transitive closure of an
initial relation of the similarity resulted from the expert survey, is offered. Experts
compared similarity between portraits of different people, and on the basis of
paired comparison the similarity matrix was built. Various methods of fuzzy
clusterization and their connections with traditional methods of the cluster analysis
and their practical applications are discussed in [104—120].
Let us suppose that some experts estimate expression of some characteristics at
a population of objects. Having in mind that each of these experts has his/her
individual evaluation criterion, we cannot exclude that the information obtained
from one expert differs from the information obtained from another. As models of
an expert evaluation of some characteristic are constructed rested upon this
information, it is obvious that the models based on information obtained from
various experts will differ.
Thus, if within the characteristic limits some expert evaluation models can be
constructed; there is a necessity of their comparative analysis versus the
subsequent building of the generalized model. The methods developed by authors
and allowing to carry out similar researches are discussed in Chapter 3.
R
=
R
