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In-Depth Information
Unknown parameters
j
j
L
j
R
b
,
b
,
b
j
=
0
m
,
of membership functions of coefficients
~
j
=
0
m
of regression models
,
are found from the system of normal equations:
j
⎧
∂
F
=
0
j
=
0
m
;
⎪
j
∂
b
⎪
⎪
∂
F
=
0
j
=
0
m
;
⎨
j
L
∂
b
⎪
F
∂
⎪
=
0
j
=
0
m
,
⎪
j
R
∂
b
⎩
(6.9)
Being transformed (6.9), the system turns out
-
-
½
n
1
1
m
ª
§
1
·
§
1
1
·
º
¦
¦
2
nb
+
−
b
0
−
y
i
+
y
i
L
+
b
j
x
ji
−
x
ji
L
−
b
j
L
x
ji
−
x
ji
L
+
°
¨
©
¸
¹
¨
©
¸
¹
®
¾
«
¬
»
¼
0
L
1
1
1
6
6
6
6
12
°
¯
¿
i
=
1
j
=
1
°
-
½
n
1
1
m
ª
§
1
·
§
1
1
·
º
¦
¦
°
+
b
0
−
y
i
−
y
i
R
+
b
j
x
ji
+
x
ji
R
+
b
j
R
x
ji
+
x
ji
L
=
0
¨
©
¸
¹
¨
©
¸
¹
®
¾
«
¬
»
¼
R
2
2
2
6
6
6
6
12
°
¯
¿
i
=
1
j
=
1
°
-
ª
º
½
n
1
m
§
1
·
§
1
1
·
¦
°
0
i
i
L
j
ji
ji
L
j
L
ji
ji
L
nb
−
b
+
−
y
+
y
+
b
x
−
x
−
b
x
−
x
=
0
¨
©
¸
¹
¨
©
¸
¹
®
¾
«
¬
»
¼
0
L
1
1
1
6
6
6
6
12
°
¯
¿
j
=
1
°
-
½
n
1
m
ª
1
1
1
º
§
·
§
·
°
¦
0
i
i
R
j
ji
ji
R
j
R
ji
ji
R
nb
+
b
+
−
y
−
y
+
b
x
+
x
+
b
x
+
x
=
0
®
¨
©
¸
¹
¨
©
¸
¹
¾
«
¬
»
¼
0
R
2
2
2
°
6
6
6
6
12
¯
¿
j
=
1
®
-
½
1
1
n
1
1
m
ª
1
1
1
º
§
·
§
·
§
·
°
¦
¦
ji
ji
ji
L
ji
R
0
i
i
L
j
ji
ji
L
j
L
ji
ji
L
¨
©
x
+
x
−
x
+
x
¸
¹
b
−
b
−
y
+
y
+
b
¨
©
x
−
x
¸
¹
−
b
¨
©
x
−
x
¸
¹
+
®
¾
«
¬
»
¼
°
1
2
0
L
1
1
1
6
6
6
6
6
6
12
¯
¿
i
=
1
j
=
1
°
-
½
1
1
n
1
1
m
ª
1
1
1
º
°
§
·
§
·
§
·
¦
¦
ji
ji
ji
L
ji
R
0
i
i
R
j
ji
ji
R
j
R
ji
ji
R
+
¨
©
x
+
x
−
x
+
x
¸
¹
b
+
b
−
y
−
y
+
b
¨
©
x
+
x
¸
¹
+
b
¨
©
x
+
x
¸
¹
=
0
®
¾
«
¬
»
¼
°
1
2
0
R
2
2
2
6
6
6
6
6
6
12
¯
¿
i
=
1
j
=
1
°
ª
º
°
1
1
n
1
1
m
ª
1
1
1
º
§
·
§
·
§
·
¦
¦
x
ji
x
ji
L
b
b
0
y
i
y
i
L
b
j
x
ji
x
ji
L
b
j
L
x
ji
x
ji
L
=
0
j
=
1
m
;
¨
©
−
¸
¹
−
−
+
+
¨
©
−
¸
¹
−
¨
©
−
¸
¹
«
¬
»
¼
«
¬
»
¼
°
1
0
L
1
1
1
6
12
6
6
6
6
12
°
i
=
1
j
=
1
°
ª
º
§
1
1
·
n
1
1
m
ª
§
1
·
§
1
1
·
º
¦
¦
x
ji
+
x
ji
R
b
+
b
0
−
y
i
−
y
i
R
+
b
j
x
ji
+
x
ji
R
+
b
j
R
x
ji
+
x
ji
R
=
0
j
=
1
m
.
¨
©
¸
¹
¨
©
¸
¹
¨
©
¸
¹
«
¬
»
¼
°
«
¬
»
¼
2
0
R
2
2
2
6
12
6
6
6
6
12
¯
i
=
1
j
=
1
The obtained system is the system of simple equations relating to variables
j
R
b
j
,
b
j
L
,
b
and is solved with well-known methods.
6.5 Nonli near Hy brid f uzzy Least-Squares Regression Mo del
6.5 Nonlinear Hybrid Fuzzy Least-Squares Regression Model
Based on Nonnegative
T
-Numbers
~
6.
5 Nonli near Hy brid f uzzy Least-Squares Regression Mo del
Let us consider nonnegative
T
-numbers
(
)
,
x
−
x
≥
0
,
X
≡
x
,
x
,
x
L
x
,
1
1
2
R
L
~
(
)
(
)
, and let us prove
propositions, related to characteristics of the weighed segments of results of
operations with these fuzzy numbers.
Z
≡
z
,
z
,
z
L
z
,
,
z
−
z
≥
0
, a triangular number
a
≡
b
,
b
,
b
1
2
R
1
L
L
R