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In-Depth Information
2
⎡
⎤
n
m
(
)
∑
∑
0
0
i
i
R
2
j
j
L
j
R
+
b
+
rb
−
y
−
ry
+
θ
b
,
b
,
b
.
⎢
⎣
⎥
⎦
~
R
2
~
i
j
a
X
j
i
=
1
j
=
1
The optimization problem is set as follows:
(
n
(
~
)
)
∑
=
j
j
L
j
R
2
F
b
,
b
,
b
=
f
Y
,
Y
→
min,
i
i
i
1
j
L
j
R
b
≥
0
b
≥
0
j
=
0
m
.
(
)
(
)
2
j
j
L
j
R
θ
b
,
b
,
b
As
θ
1
b
j
,
b
j
L
,
b
j
R
and
are piecewise linear functions in the field
~
~
~
~
i
j
a
X
j
a
X
j
j
, then
F
is piecewise differentiable function, and solutions
of an optimization problem are determined by means of known methods [152].
Let initial output data
b
j
≥
0
,
j
,
j
=
0
m
b
≥
0
~
(
)
i
i
i
L
i
R
Y
≡
y
,
y
,
y
,
y
,
i
=
1
n
be formalizations of
i
1
2
~
(
)
k
k
k
L
k
R
Y
≡
y
,
y
,
y
,
y
,
k
=
1
p
of linguistic values
Y
of characteristic
Y
. After
k
1
2
obtaining of model output data
a problem of prediction the linguistic
values of characteristic
Y
or identification of each fuzzy numbers with one of
Y
,
i
=
1
n
~
,
fuzzy numbers
occurs.
Let us denote the weighed segments of output model data
k
=
1
p
Y
,
i
=
1
n
⎡
⎤
m
m
(
)
(
)
∑
∑
b
0
−
lb
0
+
θ
1
b
j
,
b
j
L
,
b
j
R
,
b
0
+
rb
0
+
θ
2
b
j
,
b
j
L
,
b
j
R
,
i
=
1
n
⎢
⎣
⎥
⎦
~
~
~
~
L
i
j
R
i
a
X
a
X
j
j
j
j
=
1
j
=
1
[
]
i
i
(
)
with
A
1
,
A
,
, and the weighed segments
k
k
L
k
k
R
of
i
=
1
n
y
−
ly
,
y
+
ry
2
1
2
(
)
formalizations
Y
≡
y
k
,
y
k
,
y
k
L
,
y
k
R
,
of linguistic values
Y
,
of
k
=
1
p
k
=
1
p
k
1
2
[
]
B
1
,
k
B
k
characteristic
Y
with
,
.
k
=
1
p
2
~
(
)
(
)
(
)
2
f
2
Y
,
Y
=
A
i
−
B
k
+
A
i
−
B
k
,
i
=
1
n
,
k
=
1
p
.
Let
Output model value
i
is
Y
i
k
1
1
2
2
identified with linguistic value
Y
, if
(
~
~
)
(
)
2
2
=
(6.8).
The developed model is hybrid because it includes elements of fuzzy and classical
regression models. The similar combination allows defining analogue of a
standard deviation for observations, analogue of determination coefficient and
analogue of an evaluation of a standard error for quality assurance of regression
models with fuzzy input data.
f
Y
,
Y
min
f
Y
,
Y
.
i
s
i
k
k
n
~
∑
Y
(
)
i
1
n
~
~
~
∑
S
=
f
2
Y
,
Y
,
Y
=
i
=
1
.
i
n
−
1
n
i
=
1
