Information Technology Reference
In-Depth Information
It is assumed that the ev
alua
tion was carried out within the limits of a verbal
scale with levels
≥
m
ordered on increase of characteristic
appearance intensity degree. Levels of a used verbal scale unambiguously specify
term-set
X
,
l
=
1
m
;
2
(){
}
[]
T
X
=
X
,
X
,...,
X
.
is selected as universal COSS set. The
U
=
0
1
2
m
x
corresponds to a total absence of characteristic
X
appearance, and
consequently it is considered as a typical point of the term
=
0
point
x
=
1
X
. The point
corresponds to total presence of characteristic
X
appearance, and consequently it
is considered as a typical point of term
X
.
Let us denote relative frequencies of objects' occurrence for which
characteristic
X
intensity is estimated by levels
a
, accordingly
X
by
m
∑
=
a
=
1
l
Let us assume that fuzzy numbers corresponding to terms
l
1
X
with membership
()
μ
x
functions
belong to population
and satisfy a side condition (1*):
Λ
l
()
()
If
R
are nonlinear functions, they have a central symmetry versus the
point of inflexion.
Building of membership functions of COSS term-set will be carried out so that
squares of the figures limited to graphs of functions
L
x
,
()
and an axis of abscissas
be equal to
a
. It is obvious that the there is an infinite number of membership
functions meeting such requirements, therefore it is necessary to limit building to
logical requirements for fuzziness areas between the adjacent terms (or to
parameters of fuzziness of the fuzzy numbers corresponding to terms). On the one
hand, there is a desire to make this area as small as possible, then the degree of
fuzziness (1.7) constructed as a COSS model will be less accordingly. On the
other hand, it is necessary to rest upon substantial sense of fuzziness area, so we
propose to calculate a potency (length) of this area for extreme terms as
(
μ
x
l
(
)
)
, accordingly, and for mean terms to calculate the
same base the relations between numbers
or
min
a
,
a
min
a
1
,
a
m
−
1
m
2
a
,
a
,
a
ml
. Graphs of the
constructed membership f
unc
tions will be in a form of curvilinear trapezoids with
midlines equal to
;
=
3
−
2
l
−
1
l
l
+
1
a
,
.
Let us construct membership function of term
l
=
1
m
:
X
m
a
≤
a
1.
If
, then
m
m
−
1