Information Technology Reference
In-Depth Information
Chapter 3
Methods of Comparative and Fuzzy Cluster
Analysis of Formalized Information
3.1 Definition of Comparative Indicators and Indicators of
Models of Expert Characteristic Evaluations Consistency
3
.1 Definit ion of Co mparative Indicators a nd Indicators
As mentioned in Chapter 1, one cannot exclude that while estimating the same
characteristic at a population of objects by several experts, the information obtained
from the experts will differ various. As models of expert evaluations of some
characteristic are constructed based on this information, they will obvious differ.
Thus, if for one characteristic several models can be constructed, then we have a
task of their comparative analysis with the subsequent building of a generalized model.
Let us denote with
a set the elements of which are
k
models of expert
evaluations of qualitative characteristic or expert description of physical values of
quantitative characteristic in linguistic terms (
k
COSS)
()
k
Ξ
{
}
()
(
)
il
il
il
L
il
R
X
= μ
x
,
l
=
1
m
μ
x
≡
a
,
a
,
a
,
a
i
=
1
k
;
;
i
il
il
1
2
[
]
a
,
il
L
a
il
R
il
il
where
a
1
,
a
is a tolerance interval;
are the left and right parameters of a
2
fuzziness, accordingly.
After determining of set
it is necessary to carry out comparative analysis of
its elements. In particular, it is necessary to find to what extent the elements are
various (similar) pairwise, to what extent all elements are various (similar) in
aggregate, whether there are essentially differing groups of elements, or structural
composition of set
k
Ξ
is homogeneous enough.
For this purpose comparative quantity indicators of expert characteristic
evaluation models are defined, based on which the fuzzy binary relations of
similarity and conformity are then constructed.
As is known (15), a fuzzy set
~
is referred to as intersection of fuzzy sets
~
and
~
,
k
Ξ
~
~
~
()
()
()
μ
x
=
μ
x
∧
μ
x
∧
, if
,
where
is an operator
C
=
A
∩
B
∀
x
∈
X
~
~
~
C
A
B
of triangular norm class.
The fuzzy set
~
is referred to as association of fuzzy sets
~
and
~
,
~
~
~
,
C
=
A
∪
B
()
()
()
∨
if
μ
x
=
μ
x
∨
μ
x
,
, where
is an operator of triangular conorm
∀
x
∈
X
~
~
~
C
A
B
class.
