Methods of Expert Information Formalization Based on Complete Orthogonal Semantic Spaces - Expert Fuzzy Information Processing - page 51

( )

a

− ≤

min a , a 4. If , then

m 1 m m − 2

⎧ 3 a

⎛ ⎞

0 0 ≤ x ≤ 1 − a −

m − 1

;

⎜ ⎝ ⎟ ⎠

⎪

m

2

⎪

a

⎛ ⎞

⎪

1 − a −

m − 1

− x

⎜ ⎟

⎪ 3 a a

m

⎛ ⎞ ⎛ ⎞

2

⎜ ⎟

m − 1 m − 1

R , ⎜ ⎝ 1 − a − ⎟ ⎠ < x ≤ ⎜ ⎝ 1 − a − ⎟ ⎠ ;

⎪

m m

⎜ ⎜

a

⎟ ⎟

2 2

⎪ ⎪

m − 1

⎝ ⎠

()

μ x =

⎨ m − 1

a

⎛ ⎞

⎪

x − 1 + a +

m − 1

⎜ ⎟

⎛ a ⎞ ⎛ a ⎞

m

⎪ 2

⎜ ⎟

m − 1 m − 1

L , ⎜ ⎝ 1 − a − ⎟ ⎠ < x ≤ ⎜ ⎝ 1 − a + ⎟ ⎠ ;

⎪

m m

⎜ ⎜

a

⎟ ⎟

2 2

m − 1

⎪

⎝ ⎠

⎪

a

⎛ ⎞

⎪

0 1 − a +

m − 1

< x ≤ 1

⎜ ⎝ ⎟ ⎠

⎪

m

2

⎩

()

()

μ x

μ x Similarly to membership functions ; are constructed.

l = 2 m − 2

m 1 −

l

Let us construct membership function for term

X with even number of terms.

a ≤ a

1. If , then

1 2

a

⎧

1 0 ≤ x ≤

1

;

⎪

2

⎪

a

⎛ ⎞

⎪ ⎪

x −

1

⎜ ⎟

a 3 a

2

()

⎜ ⎟

μ x = L ,

1

< x ≤

1

;

⎨ 1

⎜ ⎜

a

⎟ ⎟

2 2

⎪

1

⎪ ⎝ ⎠

⎪ 3 a

0

1

< x ≤ 1 .

⎪

⎩ 2

a > a

2. If , then

1 2

⎧ a

⎛ ⎞

1 0 ≤ x ≤ a −

2

;

⎜
⎝

⎟
⎠

⎪

1

2

⎪

a

⎛ ⎞

⎪

x − a +

2

⎜ ⎟

⎪ ⎛ a ⎞ ⎛ a ⎞

1

2

()

⎜ ⎟

μ x = L , a −

2

< x ≤ a +

2

;

⎜ ⎝ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠

⎨

1 1 1

a 2 2

⎜ ⎜ ⎟ ⎟

⎪

2

⎪ ⎝ ⎠

⎪

a

⎛ ⎞

0 +

2

< ≤ 1 .

⎜ ⎝ a ⎟ ⎠ x

⎪

1

2

⎩

With odd number of terms we obtain: