Information Technology Reference
In-Depth Information
1
1
[
(
)
(
)
]
(
)
.
∫
B
=
b
−
1
−
α
b
+
b
+
1
−
α
b
α
d
α
=
b
+
b
−
b
L
R
R
L
6
0
(
)
In [93] definition of the weighed point is spread to tolerance
L
−
-numbers; and
the new method of defuzzification of fuzzy numbers is proposed which assigns to
them not a number, but a certain numerical segment called the weighed one.
According to [93], weighed segment of a tolerance
R
(
)
-number
L
−
R
(
)
[
]
A
≡
a
,
a
,
a
L
a
,
is the segment
A
1
,
A
:
1
2
R
2
1
1
()
()
∫
∫
A
=
a
−
la
;
A
=
a
+
ra
;
l
=
L
−
1
α
α
d
α
;
r
=
R
−
1
α
α
d
α
.
(1.10)
1
1
L
2
2
R
Before the weighed segment being defined, the known methods of defuzzification
of fuzzy numbers have allowed definition only pointwise aggregating indexes for
them. Thereby, for example, they made unimodal numbers with different
fuzziness coefficients or tolerance numbers indiscernible, i.e. their informational
singularities were lost, which were necessary to be outlined. In more details, the
method of defuzzification of fuzzy numbers based on the weighed segments is
discussed in Chapter 6.
0
0
1.9 Fuzzy Relations and Features of Fuzzy Cluster Analysis
One of the basic concepts of the fuzzy cluster analysis is the concept of the fuzzy
relation. Fuzzy relations play an essential role in the problems with the solutions
rested upon methods of the fuzzy set theory, on traditional methods and the theory
of clear ratios [94—99]. As a rule, the means of the theory of clear relations is
used in qualitative analysis of correlations between objects of investigated system
when they are of dichotomizing nature and, consequently, can be interpreted in
terms "there is no connection“, “there is a connection”, or when methods of
quantitative analysis of correlations are inapplicable for any reasons, and
correlations are derived to a dichotomizing form synthetically. For example, when
connection between objects takes values from a rank scale, then selection of
connection force threshold allows this connection to be transformed to a
demanded kind. However, despite the similar approach allows carrying out the
qualitative analysis of systems, it leads to loss of the information on connection
force between objects; therefore it is necessary to make evaluations at different
connection force thresholds. Methods of data analysis based on the theory of fuzzy
ratios do not suffer this shortage; they allow carrying out the qualitative analysis
of systems taking into account distinctions in connection force between objects of
the system [15].
Ordinary (definite)
n
-ary relation
R
between sets
X
,
X
,....,
X
is referred to
1
2
n
as a subset of Cartesian product:
R
⊆
X
×
X
×
....
×
X
.
1
2
n
