Information Technology Reference
In-Depth Information
2
b
−
b
≥
0
⎩
⎨
⎧
⎩
⎨
⎧
L
,
r
=
1
L
r
=
M
=
r
1
b
+
b
<
0
R
,
r
=
2
R
The proposition 6.6 is proved.
It is worth mentioning that if many nonlinear classical regression models can be
reduced to linear models by means of corresponding replacements, nonlinear
fuzzy regression models are more complicated to reduce.
The matter is that, for example, while multiplying fuzzy numbers it is not
always possible to set an analytical form for membership function of a fuzzy
number which is a result out of the multiplication. Since all known linear fuzzy
regression models assume such possibility, it becomes clear why it is impossible
to reduce nonlinear models to linear fuzzy regression models on its own.
Let
~
⎛
⎞
Y
⎜
⎟
1
~
~
(
)
i
i
i
L
i
R
i
i
L
Y
=
⎜
...
⎟
,
Y
≡
y
,
y
,
y
,
y
,
y
−
y
≥
0
i
1
2
1
⎜
⎜
⎟
⎟
Y
⎝
⎠
n
are output
T
-numbers,
i
=
1
n
~
⎛
1
⎞
X
⎜
⎜
⎟
⎟
j
~
~
(
)
i
j
ji
ji
ji
L
ji
R
ji
ji
L
X
=
...
,
X
≡
x
,
x
,
x
,
x
,
x
−
x
≥
0
j
1
2
1
⎜
⎜
⎟
⎟
n
j
X
⎝
⎠
are input
T
-numbers.
Let us search relation between input and output data in the form
,
j
=
1
m
i
=
1
n
~
~
~
~
~
~
~
~
~
~
~
2
1
2
Y
=
a
X
+
...
+
a
X
+
a
X
X
+
...
+
a
X
X
+
(
)
1
m
m
m
+
1
1
2
m
m
+
1
m
−
1
m
2
~
~
~
~
~
+
a
X
+
...
+
a
X
+
a
,
(
)
2
1
m
m
+
3
m
0
m
+
m
+
2
2
2
(
)
(
)
m
m
+
3
~
k
k
L
k
R
where
a
≡
b
,
b
,
b
;
k
=
0
are unknown coefficients of regression
k
2
model, and also triangular numbers.
Let us determine the weighed segments
1
1
⎡
⎤
y
i
−
y
i
L
,
y
i
+
y
i
R
,
i
=
1
n
⎢
⎣
⎥
⎦
1
2
6
6
for observable output data
~
.
Let us denote the weighed segment of product of numbers
~
~
with
and
i
k
j
[
]
(
)
(
)
θ
1
b
k
,
b
k
L
,
b
k
R
,
θ
2
b
k
,
b
k
L
,
b
k
R
~
~
~
~
i
i
j
a
X
a
X
k
j
k
where
