Complete Orthogonal Semantic Spaces in Problems of the Fuzzy Regression Analysis - Expert Fuzzy Information Processing - page 203

~ ~ ~ ~ ~ ~

~ ~ ~ ~

2 2

i i

m

i i i

m

i
m Y = a X + ... + a X + a X X + ... + a X X +

( )

i 1 1 m m + 1 1 2 m m + 1 − 1

2

~ ~

~ ~ ~

i i

m

+ a X + ... + a X + a ;

( )

2

1 m m + 3 0

m + m + 2

2

2

1

m

( ) ( ) +

∑
=

1 k k

L

k
R

0 0 1 j j

L

j
R

θ b , b , b = b − b + θ b , b , b

L ~

2

Y ~

6

i
j

a X

i

j

j 1

m

( ) +

∑
=

1 m + j − 1 m

L

+ j − 1 m

R

+ j − 1

+ θ b , b , b

~ ~

~ i j

a X X

m + j − 1 1

j 2

m

( )

∑
=

1 2 m − j + 3 2 m − j + 3 2 m − j + 3

+ θ b , b , b + ... +

~ ~

~

L R

i i

j

a X X

2 m − j + 3 2

j 3

⎛

2 2 2

⎞

m + m m + m m + m

⎜ ⎜
⎝

⎟ ⎟
⎠

~

θ b

2

, b

2

, b

2

+

a L R

2

m + m ~ ~

i m i m

X X

− 1

2

⎛

2 2 2

⎞

m + m m + m m + m

m

+ j + j + j

∑
=

⎜ ⎜
⎝

⎟ ⎟
⎠

1

+ θ b , b , b ;

2 2 2

~

~

i
j

L R

a X

2

m + m

j 1 + j

2

where

( ) ;

m m + 3

k = 0

2

1

m

( ) ( ) +

∑
=

2 k k

L

k
R

0 0 2 j j

L

j
R

θ b , b , b = b + b + θ b , b , b

R ~

2

Y ~

6

i

i

a X

j j

j 1

m

( ) +

∑
=

2 m + j − 1 m

L

+ j − 1 m

R

+ j − 1

+ θ b , b , b

~ ~

~

i i

j

a X X

m + j − 1 1

j 2

m

( )

∑
=

2 2 m − j + 3 2 m − j + 3 2 m − j + 3

+ θ b , b , b + ... +

~ ~

~

i i

L R

a X X

2 m − j + 3 2 j

j 3

⎛ ⎞

2 2 2

m + m m + m m + m

⎜ ⎜
⎝

⎟ ⎟
⎠

θ

2
~

b , b , b +

2 2 2

a L R

2

m + m ~ ~

i m i m

X X

− 1

2

⎛ ⎞

2 2 2

m + m m + m m + m

m

+ j + j + j

⎜ ⎜
⎝

⎟ ⎟
⎠

∑
=

2

+ θ b , b , b .

2 2 2

~

~

L R

i j

a X

2

m + m

j 1

+ j

2

where

( ) .

m m + 3

k = 0

2

Let us consider a functional