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1
1
() ()
() ()
∫
−
1
−
1
∫
−
1
−
1
m
=
L
α
R
α
α
d
α
,
p
=
L
α
R
α
α
d
α
.
2
1
1
2
0
0
The proposition 6.3 is proved.
Propositions 6.1 - 6.3 are true for
Λ
-unimodal numbers with the following
a
=
a
replacements
,
accordingly.
b
=
b
1
2
1
2
6.3 Linear Hybrid Fuzzy Least-Squares Regression Model
In §2.1 the set of
-numbers, subdivided into
-tolerance and
-unimodal
Λ
Λ
Λ
numbers, is described.
Let us define an affinity measure for two
-tolerance numbers
~
,
~
, with the
Λ
[
]
[
]
weighed segments
,
A
1
,
A
B
1
,
B
2
2
(
)
~
~
(
)
(
)
2
2
f
A
,
B
=
A
−
B
+
A
−
B
.
(6.5)
1
1
2
2
Let
~
⎛
⎞
Y
⎜
⎜
⎟
⎟
1
~
~
(
)
i
i
i
L
i
R
Y
=
...
,
Y
≡
y
,
y
,
y
,
y
,
i
=
1
n
i
1
2
⎜
⎜
⎟
⎟
Y
⎝
⎠
n
be output
-tolerance numbers, and
Λ
~
⎛
1
⎞
X
⎜
⎟
j
~
~
(
)
i
j
ji
ji
ji
L
ji
R
X
=
⎜
...
⎟
,
X
≡
x
,
x
,
x
,
x
,
j
=
1
m
,
i
=
1
n
j
1
2
⎜
⎜
⎟
⎟
n
j
X
⎝
⎠
(
)
~
j
j
L
j
R
a
≡
b
,
b
,
b
be input
-tolerance numbers, and
unknown coefficients of
Λ
j
regression model be
-unimodal numbers.
Relations of input and output data will be in the form [222]
Λ
~
~
~
~
~
~
Y
=
a
+
a
X
+
...
+
a
m
X
.
(6.6)
0
1
1
m
According to the definition of operations for fuz
zy n
umber
s a
nd to the proposition
2.2, the multiplication of
~
~
i
j
and
,
j
=
1
m
,
i
=
1
n
gives
-tolerance
Λ
j
~
(
)
X
i
j
≡
x
ji
,
x
ji
,
x
ji
L
,
x
ji
R
j
=
1
m
i
=
1
n
numbers. If, for example,
,
,
, and
1
2
(
)
~
(
)
j
j
L
j
R
a
≡
b
,
b
,
b
ji
ji
L
j
j
L
are positive fuzzy numbers
x
−
x
>
0
b
−
b
>
0
, then the
j
1
Λ
-tolerance numbers with parameters
multiplication of these numbers results in
