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where
[
]
B
1
,
B
=
are the weighed segments of t
he f
uzzy numbers making
in aggregate a reference pattern of objects, with
j
1
m
,
j
j
2
ρ
j
=
1
m
,
.
j
n
=
1
N
Let us obtain rating points of
n
-th object,
as follows [223, 224]:
m
∑
=
r
=
1
−
ω
ρ
.
(6.10)
j
j
j
1
The rating points obtained by the formula (6.10) will differ from the evaluations
obtained by a principle “the higher individual indexes are, the higher rating points
are” [225—226]. The rating points constructed on the basis of s reference pattern,
are quantity indexes of affinity of indexes of estimated objects and indexes of the
pattern constructed on the basis of real a posteriori data. Therefore, the analysis of
the objects' functioning monitoring data based on the obtained rating points
improve reliability of the prediction of final indexes (in our case, they are indexes
of characteristic
Y
) and control operations on their improving.
6.7 Examples of Fuzzy Regression Models Employment
Example 6.1.
Prediction of progress of the trainees. The comparative analysis of
combined (hybrid) and classical regression models. To construct classical and
hybrid linear regression models, data of progress of 30 trainees in four subjects
[227] are taken. From these data non-repeating results are selected and
summarized in Table 6.1.
Based on the data obtained by a method described in §2.2, four COSS's are
constructed, their membership function parameters are summarized in Table 6.2.
By the method described in §6.4, linear hybrid regression model is constructed
with definite coefficients:
~
~
~
~
Y
=
a
X
+
a
X
+
a
X
+
a
,
1
1
2
2
3
3
0
a
,
where
=
j
are unknown coefficients of regression model. The solution of
an optimization problem allows obtaining the model
0
~
~
~
~
Y
=
0
352
X
+
0
466
X
+
0
133
X
;
1
2
3
HRS
According to the method described in §6.4, linear hybrid regression model is
constructed with fuzzy coefficients
=
0
454
;
=
0
805
;
HS
=
0
239
.
~
~
~
~
~
~
~
~
Y
=
a
+
a
X
+
a
X
+
a
X
,
0
1
1
2
2
3
3
(
)
~
j
are unknown coefficients of regression model, and
also normal triangular numbers. The solution of an optimization problem allows
obtaining the model
a
≡
b
j
,
b
j
L
,
b
j
R
=
0
where
,
