A new method for fuzzy rule interpolation based on the ratio of fuzziness of interval type-2 fuzzy sets

Abstract—Some fuzzy rule interpolation methods have been presented for sparse fuzzy rule-based systems based on interval type-2 fuzzy sets. However, they have the drawbacks that they can not guarantee the convexity of the fuzzy interpolated result and may generate the same fuzzy interpolated result with respect to different observations. In this paper, we present a new method for fuzzy rule interpolation for sparse fuzzy rule-based systems based on the ratio of fuzziness of interval type-2 fuzzy sets. It can overcome the drawbacks of the existing methods. We also use some examples to compare the fuzzy interpolated results of the proposed method with the results by the existing methods. The proposed fuzzy rule interpolation method gets more reasonable results than the existing methods.

I. I

NTRODUCTION

In recent years, some methods [1]-[3], [5]-[9], [11], [17]-[22] have been presented for fuzzy rule interpolation based on type-1 fuzzy sets [23]. Moreover, some methods [4], [13] have been presented to deal with fuzzy rule interpolation for sparse fuzzy rule-based systems based on interval type-2 fuzzy sets [15]. In [4], Chang and Chen presented a method for fuzzy rule interpolation based on principle membership functions and uncertainty grade functions of interval type-2 fuzzy sets. However, the method presented in [4] has the drawback that it can not guarantee the convexity of the fuzzy interpolated result. In [13], Lee and Chen presented a method for fuzzy rule interpolation based on the ranking values of interval type-2 fuzzy sets. However, the method presented in [13] has the drawback that it will generate the same fuzzy interpolated result with respect to different observations. Therefore, it is necessary to develop a new method for fuzzy rule interpolation for sparse fuzzy rule-base systems based on interval type-2 fuzzy sets.

In this paper, we present a new method for fuzzy rule interpolation based on the ratio of fuzziness of interval type-2 fuzzy sets. It can overcome the drawbacks of the methods presented in [4] and [13]. First, it calculates the weights of the closest fuzzy rules with respect to the observation to obtain an intermediate consequence fuzzy set. Then, it uses the ratio of fuzziness of interval type-2 fuzzy sets to infer the fuzzy interpolated result based on the intermediate consequence fuzzy set. We also use some examples to compare the fuzzy interpolated results of the proposed method with the results by the methods presented in [4] and [13]. The proposed fuzzy rule interpolation method gets more reasonable results than

S.-M. Chen and Y.-C. Chang are with the Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R. O. C. (e-mail:

[email protected]; [email protected]).

the methods presented in [4] and [13].

II. P

RELIMINARIES

In this section, we briefly review the definition of polygonal interval type-2 fuzzy sets [4].

Definition 2.1 [4]: A polygonal membership function A in the universe of discourse X can be represented by n characteristic points a

0

, a

1

, …, a

l

, a

r

, …, a

n-2

and a

n-1

based on the operations of α -cuts, as shown in Fig. 1, where A = (a

0

, a

1

,

…, al

, a

r

, …, a

n-2

, a

n-1

; h

A

), h

A

denotes the maximum membership value of A, 0 ≤ h

A

≤ 1, l =

⎥⎦⎥

⎢⎣⎢ − 2 1 n

, r =

⎥⎥⎤

⎢⎢⎡ − 2 1 n

and n

≥ 1.

−1

αl

α2

α1 hA

Fig. 1. A type-1 polygonal fuzzy set.

The representative value “Rep(A)” of the polygonal type-1 fuzzy set A shown in Fig. 1 is defined by [5], [8]

Rep(A) =

n a a

a

a₀+ ₁+ ₂+ ...+ _{n 1−}

. (1) Definition 2.2 [4]: A polygonal interval type-2 fuzzy set

A~

is an interval type-2 fuzzy set whose upper membership function

A~^U

and lower membership function

A~^L

are polygonal membership functions, respectively. A polygonal interval type-2 fuzzy set

A~

can be represented by

A~

= <

A~^L

,

U>,

A~

as shown in Fig. 2, where

A~^L

=

(a₀^L ,a₁^L, ...,a_l^L ,a_r^L, ),

; ,

..., L L

1 L

2 n A~

n a h

a_{− −} A~^U (a₀^U ,a₁^U, ...,a_l^U,a^U_r ,...,a_n^U₂ ,a_n^U₁ ;h_A^~U),

−

= − hA~L

and

hA~U

are the maximum membership values of

A~^L

and

AU

~

, respectively, and 0 ≤

^h_A^~L

≤

^h_A^~U

≤ 1.

AU

h~

0U

a a0^L a1^U a₂^U a_l^U₋₁ al^Uar^U a^U_r+1 an^U−3an^U−2a^Ln−1an^U−1 1L

a a2^L a_l^L₋₁ a_l^Lar^L ar^L+1 a^L_n−3a_n^L₋₂

A~U

A~L

α2

α1 AL

h~

Fig. 2. A polygonal interval type-2 fuzzy set A~ .

A New Method for Fuzzy Rule Interpolation Based on the Ratio of Fuzziness of Interval Type-2 Fuzzy Sets

Shyi-Ming Chen, Senior Member, IEEE, and Yu-Chuan Chang

978-1-4244-8126-2/10/$26.00 ©2010 IEEE

Definition 2.3: Given an interval type-2 fuzzy set

A~

, where

A~

= <

A~^L

,

A~^U

>, the representative value Rep(

A~

) of the interval type-2 fuzzy set

A~

is calculated as follows:

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

=

−

>

−

−

×

−

×

=

0, ) ( ) ( if ), (

0, ) ( ) ( if

) ,

( ) (

) ( ) ( ) ( ) (

) (

L U

U

L U

L U

L L

U U

A~ Area A~

Area A~

ep R

A~ Area A~

Area A~ Area A~

Area

A~ Area A~ ep R A~ Area A~ ep R A~ ep

R

(2)

where Rep(

A~^L

) and Rep(

A~^U

) denote the representative values of the lower membership function

A~^L

and the upper membership function

A~^U

, respectively, and

(A~^L)

Area

and

)

(A~^U

Area

denote the areas of

A~^L

and

A~^U

, respectively.

Let Rep(

A~

) and Rep(

B~

) be the representative values of the interval type-2 fuzzy sets

A~

and

B~

, respectively. If

Rep(A~

) ≤ Rep(

B~

), then

A~

is at the left-hand side of

B~

. Otherwise,

A~

is at the right-hand side of

B~

.

Definition 2.4: Given an interval type-2 fuzzy set

A~

, where

A~

= <

A~^L

,

A~^U

>, the fuzziness Fuzziness(

A~

) of the interval type-2 fuzzy set

A~

is defined as follows:

2

) ( ) ) (

( ^U A~^L

Area A~

A~ Area

Fuzziness = +

, (3) where

(A~^L)

Area

≤

^{Fuzziness(A~)}

≤

^Area(A~U)

,

(A~^L) Area

and

(A~^U)

Area

denote the areas of

A~^L

and

A~^U

, respectively,

)

(A~^L

Area

≥ 0 and

^Area(A~U)

≥ 0. The larger the value of

Fuzziness(A~

), the larger the fuzziness of the interval type-2 fuzzy set

A~

.

Definition 2.5: Let

A~

be an interval type-2 fuzzy set in the universe of discourse X, where

A~

= <

A~^L

,

A~^U

>. If both the lower membership function

A~^L

and the upper membership function

A~^U

are convex, then

A~

is called a convex interval type-2 fuzzy set.

III. A N

EW

M

ETHOD FOR

F

UZZY

R

ULE

I

NTERPOLATION

B

ASED ON THE

R

ATIOS OF

F

UZZINESS OF

I

^NTERVAL

T

^YPE

-2 F

^{UZZY SETS}

Let us consider the following multiple fuzzy rules interpolation scheme using polygonal interval type-2 fuzzy sets:

Rule 1: If X1 is A~_1,1

and X2 is A~_1,2

and … and Xm is A~_,m

1 Then Y is B~₁ Rule 2: If X1 is A~_2,1

and X2 is A~_2,2

and … and Xm is A~_,m

2 Then Y is B~₂ Rule t: If X1 is A~_t,1

and X2 is A~_t,2

and … and Xm is A~_t,m

Then Y is B~_t Observation: X1 is A~₁^∗

and X2 is A~₂^∗

and … and Xm is ∗

A~m

Conclusion: Y is B~^∗

where X

j

denotes the jth antecedent variable, Y denotes the consequence variable, the jth antecedent fuzzy set

A~_{k j}

,

of the fuzzy rule Rule k is represented by

( ..., L ; L ),

1 - L ,

1 L ,

0

, kj kjn A~k,j

kj ,a , a h

< a

, ) ; ...,

( U U

1 - , U

1 , U

0

, >

j , Ak n ~ kj kj

kj ,a , a h

a

the consequence fuzzy set

B~_k

of the fuzzy rule Rule k is represented by

<(b_k^L_,0,b_k^L_,1, ...,b_k^L_,n-1 ;

),

L,j

B~k

h (b_k^U_,0 ,b_k^U_,1, ...,b_k^U_,n-1; U )>

j ,

B~k

h

, the observation fuzzy set

∗j

A~

is represented by

(a^*Lj_,0 ,a^*Lj_,1 ,a^*Lj_,2 ,... ,a^*Lj_,n_-1 ;hA~*L),( a^*Uj_,0,

< k

, ) ; ,

, *U *U

1 -

*U , 1

, jn _A^~_k >

j ...a h

a

the fuzzy interpolated result

B~^∗ is

represented by

( , , ..., *L ; *L),

1

*L 1

*L

0 b bn hB~k

b ₋

< (b₀^*U ,b₁^*U, ...,b^*U_n−1; ,

)

*U >

B~k

h

1 ≤ k ≤ t, 1 ≤ j ≤ m, t denotes the number of fuzzy rules and m denotes the number of antecedent variables of the fuzzy rules. Based on the multiple fuzzy rules interpolation scheme using polygonal interval type-2 fuzzy sets, the proposed method for fuzzy rule interpolation is now presented as follows:

Step 1: Choose p closest fuzzy rules with respect to the observation for fuzzy rule interpolation, where each antecedent fuzzy set

A~_kj

,

of the closest fuzzy rule Rule k is either the left nearest fuzzy set or the right nearest fuzzy set of the observation fuzzy set

A~^∗_j

, 1 ≤ j ≤ m and 1 ≤ k ≤ p.

Step 2: Assume that Rule 1, Rule 2, …, and Rule p are the p closest fuzzy rules obtained in Step 1, then based on Eq. (2), calculate the representative values

(A~_{k, j})

ep

R

and

Rep(A~^∗_j)

of

j

Ak

~

,

and

A~^∗_j

, respectively, where 1 ≤ k ≤ p and 1 ≤ j ≤ m.

Step 3: Calculate the weight W

k

of the fuzzy rule Rule k, where 1 ≤ k ≤ p. Assume that

A~l_j j

,

and

A~r_j j

,

are the left nearest antecedent fuzzy set and the right nearest antecedent fuzzy set with respect to the observation fuzzy set

A~^∗_j

, respectively, where 1 ≤ l

j

≤ p, 1 ≤ r

j

≤ p and 1 ≤ j ≤ m, then let

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≤

≤

<

≤

≤

−

− −

=

∗

∗

∗

, 1 where ), ( ) ( if 1,

), ( ) ( ) ( if

), ( ) (

) ( ) 1 (

,

, ,

, ,

,

,

p k A~

ep R A~ ep R

A~ ep R A~ ep R A~ ep R

A~ ep R A~ ep R

A~ ep R A~ ep R

j j

k

j r j

j l

j l j r

j l j

j l

j j

j j

j

λj

(4)

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≤

≤

<

≤

≤

−

− −

=

∗

∗

∗

, 1 where ), ( ) ( if 1,

), ( ) ( ) ( if

), ( ) (

) ( ) 1 (

,

, ,

, ,

,

,

p k A~

ep R A~ ep R

A~ ep R A~ ep R A~ ep R

A~ ep R A~ ep R

A~ ep R A~ ep R

j k j

j r j

j l

j l j r

j j

r

j r

j j

j j

j

λj

(5)

where 1 ≤ j ≤ m. Let

, min min

1 , 2 1 2, , 1

∑

= =

= _p=

k

j m k ..., , , j

j m k ..., , j

Wk

λ

λ (6)

…

where W

k

denotes the weight of the fuzzy rule Rule k, 0 ≤ W

k

≤

1,

1

1

∑

=

= p

k

Wk

and 1 ≤ k ≤ p. For each antecedent fuzzy set

A~kj ,

of the closest fuzzy rule Rule k, we can see that 0 ≤ λ

k ,j

≤ 1 and

0 ≤

kj

m ..., ,

j ,

2, 1min

λ

=

≤ 1, where 1 ≤ k ≤ p and 1 ≤ j ≤ m. Therefore, we can see that

1

1

∑

=

= p

k

Wk

.

Step 4: Construct an intermediate interval type-2 fuzzy set

B~′ , where B~′ = ^{( , ,..., , ,...,} L ^{, L}1 ^{; L),}

L 2 L 1L

0L b bl br bn bn hB~

b′ ′ ′ ′ ′₋ ′_{− ′}

<

>

′

′

′

′

′

′ , ,..., , ,..., ₋ , ₋ ; _′ )

( U U

1 U

2 U U U 1 U

0 b bl br bn bn hB~

b

, where

∑

=

×

′ =

p

k

i, k k

i W b

b

1 L

L

, (7)

∑

=

×

′ =

p

k

i, k k

i W b

b

1 U

U

, (8)

1

L

L W h ,

h

p

k

B~ B~ k

∑

k

=

′ = ×

(9)

1

U

U W h ,

h

p

k

B~ B~ k

∑

k

′ = = ×

(10) where 0 ≤ W

k

≤ 1,

¹

1

∑

=

= p

k

Wk

and 1 ≤ k ≤ p.

Step 5: Calculate the average of the fuzziness

“Average_Fuzziness(

A~₁^∗, A~₂^∗, …, A~_m^∗ )” of the observation

fuzzy sets

A~₁^∗

,

A~₂^∗

, … and

A~_m^∗

and calculate the average of the fuzziness “Average_Fuzziness(

A~k,1

, A~k,2

, …, A~k,m

)” of

the antecedent fuzzy sets

A~k,1

,

A~k,2

, … and

A~k,m

of the fuzz rule Rule k, where

) , ) (

...

(

1 2

1

∑

=

∗ ∗

∗

∗ =

m

j

j

m m

A~ Fuzziness A~

, , A~ , A~ zziness

Average_Fu

(11)

), ) (

..., (

1 2

1

∑

=

=

m

j

j , k m

, k , k ,

k m

A~ Fuzziness A~

, A~ , A~ Fuzziness _

Average

(12)

and 1 ≤ k ≤ p. Calculate the ratio RF of the average of the fuzziness of the observation fuzzy sets to the weighted average fuzziness of the antecedent fuzzy sets of the p closest fuzzy rules, shown as follows:

, ) ..., , , (

) ..., , , (

1

2 1 2 1

∑

=

∗

∗

∗

×

= _p

k

m , k , k , k k

m

A~ A~ A~ Fuzziness _

Average W

A~ A~ A~ Fuzziness _

Average

RF

(13)

where RF ≥ 0, 0 ≤ W

k

≤ 1,

¹

1

∑

=

= p

k

Wk

and 1 ≤ k ≤ p.

Step 6: Get the fuzzy interpolated result

B~^∗, _∗ B~

=

..., , , ..., , , ( ), ; , ..., , , ..., , ,

( ₀^L ₁^{L L L L}₂ ^L₁ L ₀^U ₁^{U U} *^U

* r

* l

* B~

*n

*n

*r

*l

*

* b b b b b h b b b b

b − − *

<

, )

;

, U U

1 U

2 ₋ >

− * B~^* n

*

n b h

b

where the values of the characteristic points

b₀^*L ,b₁^*L ,...,b_l^*L ,b^*_r^L ,..., b^*_n−^L₂

and

b^*_n^L₋₁

are calculated as follows:

cL

b′

=

2

L

L r

l b

b′ + ′

, (14)

∗L

bl

=

2

L

L rL l

c

b RF b

b ′ − ′

×

′ −

, (15)

∗L

br

=

2

L

L rL l

c

b RF b

b ′ − ′

×

′ +

, (16)

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

−

≤

≤

′ +

′ −

× +

−

≤

′ ≤

′ −

×

−

=

∑

∑

−

=

+

−

=

+

, 1 1 if ,) (

, 1 0 if ,) (

1

L L

1 L

1

L L

1 L

L

n i r b b RF b

l i b

b RF b

b

i

r t

t t

* r

l

i t

t t

* l

i*

(17)

where 0 ≤ i ≤ n – 1; the values of

U U U ^U2 U 1

0 , ^* ,..., _l^* , ^*_r ,..., ^*_n

* b b b b

b ₋

and

b^*_n^U₋₁

are calculated as follows:

cU

b′

=

2

U

U r

l b

b′ + ′

, (18)

∗U

bl

=

2

U

U lU r

c

b RF b

b′ − × ′ − ′

, (19)

∗U

br

=

2

U

U lU r

c

b RF b

b ′ − ′

×

′ +

, (20)

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

−

≤

≤

′ +

′ −

× +

−

≤

′ ≤

′ −

×

−

=

∑

∑

−

=

∗ +

−

=

∗ +

∗

, 1 1 if ,) (

, 1 0 if ,) (

1

U U1 U

1

U U

1 U

U

n i r b

b RF b

l i b

b RF b

b

i

r t

t t r

l

i t

t t l

i

(21)

where 0 ≤ i ≤ n – 1; and the values of

^h_B^~*L

and

hB~*U

are calculated as follows:

∑

=

×

=

p

k

B~ B~ k

k

* W h

h

1

L

L

, (22)

∑

=

×

=

p

k

B~ B~ k

* W h k

h

1

U

U

, (23) where 0 ≤ W

k

≤ 1, 1

1

∑ =

= p

k

Wk

and 1 ≤ k ≤ p. Therefore, we can get the fuzzy interpolated result

B~^∗,

where

B~^∗ =

), ; , ..., , , ..., , ,

( L L

L 1 L 2 L 1L

0L * B~^*

* n

* n

* r

* l

* b b b b b h

b _{− −}

< (b₀^*U ,b₁^*U ,...,b_l^*U ,b^*_r^U,

,

..., b^*_n−^U_{2 −}U; U)>

1 B~^*

*

n h

b

.

Because each observation fuzzy set

A~^∗_j

has at least one

nearest neighboring antecedent fuzzy set and has at most two

nearest neighboring antecedent fuzzy sets, where 0 ≤ j ≤ m,

we can see that 1 ≤ p ≤ 2

^m

, where p denotes the number of closest fuzzy rules with respect to the observation.

Furthermore, the proposed fuzzy rule interpolation method also can guarantee the convexity of the fuzzy interpolated result.

Property 3.1: If the consequence interval type-2 fuzzy sets of the closest fuzzy rules with respect to the observation are all convex interval type-2 fuzzy sets, then the fuzzy interpolated result

B~^∗

of the proposed fuzzy rule interpolation method is also a convex interval type-2 fuzzy set.

IV. E

XPERIMENTAL

R

ESULTS

In this section, we use some examples to compare the fuzzy interpolated results of the proposed fuzzy rule interpolation method with the results by using the methods presented in [4]

and [13].

Example 4.1: Let us consider the following fuzzy rule interpolation scheme using polygonal interval type-2 fuzzy sets shown in Fig. 3:

Rule 1: If X is

A~₁

Then Y is

B~₁

, Rule 2: If X is

A~₂

Then Y is

B~₂

, Observation: X is

A~^∗

Conclusion: Y is

B~^∗

Fig. 3 shows a comparison of the fuzzy interpolated fuzzy sets of Example 4.1 for different methods. From Fig. 3, we can see that Lee and Chen’s method [13] and the proposed method generated a convex polygonal interval type-2 fuzzy set, whereas Chang and Chen’s method [4] generated a nonconvex polygonal interval type-2 fuzzy set. We also can see the fuzziness of the fuzzy interpolated result

B~^∗

of the proposed method is smaller than the fuzziness of the consequence interval type-2 fuzzy set

B~1 (i.e., (B~^∗)

Fuzziness

<

Fuzziness(B~₁),

whereas the fuzziness of the fuzzy interpolated result

B~^∗

by Lee and Chen’s method [13] is larger than the fuzziness of the consequence interval type-2 fuzzy set

B~₁ (i.e., ( ) (B~₁)).

Fuzziness B~

Fuzziness ^∗ >

Because the fuzziness of the observation

A~^∗

is smaller than the fuzziness

of the antecedent interval type-2 fuzzy set

A~₁

(i.e.,

)),

( )

( A~₁

Fuzziness A~

Fuzziness ^∗ <

we can see that the proposed method gets a more reasonable result than Chang and Chen’s method [4] and Lee and Chen’s method [13].

Example 4.2: Let us consider the following fuzzy rules interpolation scheme with two cases of observations shown in Fig. 4 and Fig. 5, respectively:

Rule 1: If X is

A~₁

Then Y is

B~₁

, Rule 2: If X is

A~₂

Then Y is

B~₂

, Observation: X is

A~^∗

Conclusion: Y is

B~^∗

A~1

A~2

A*

~

[4]

Method s Chen' and Chang

B~1

B~2

B~*

[13]

Method s Chen' and Lee

B~1

B~2

B~*

Method Proposed The

B~1

B~2

B~*

Fig. 3. A comparison of the fuzzy interpolated results of Example 4.1 for different methods.

Fig. 4 shows a comparison of the fuzzy interpolated results of Example 4.2 for different methods with respect to the observation

A~^∗

= <(7.5, 7.5, 9.5, 9.5; 0.875), (7, 7, 10, 10;

1)>. From Fig. 4, we can see that Lee and Chen’s method [13]

and the proposed method generated a trapezoidal interval type-2 fuzz sets, whereas Chang and Chen’s method [4]

generated a polygonal interval type-2 fuzzy set. By comparing the shapes of the membership functions of the fuzzy interpolative reasoning results, we can see that Lee and Chen’s method [13] and the proposed method get more reasonable results than Chang and Chen’s method [4]. Fig. 5 shows a comparison of fuzzy interpolated results of Example 4.2 for different methods with respect to the observation

A~^∗

= <(8, 8, 9, 9; 0.75), (7, 8, 9, 10; 1)>. From Fig. 5, we can see that Chang and Chen’s method [4], Lee and Chen’s method [13] and the proposed method generated a trapezoidal interval type-2 fuzzy set. We also can see that the fuzziness of the fuzzy interpolated results

B~^∗ by Chang and Chen’s method

[4] and the proposed method are smaller than the fuzziness of the consequence interval type-2 fuzzy sets

B~₁

and

B~₂

, whereas the fuzziness of the fuzzy interpolated result

B~^∗ by

Lee and Chen’s method [13] is larger than the fuzziness of the consequence interval type-2 fuzzy sets

B~₁

and

B~₂

. Because the fuzziness of the observation

A~^∗ is smaller than the

fuzziness of the antecedent interval type-2 fuzzy sets

A~₁

and

A2

~

, we can see that the proposed method gets a more reasonable result than Lee and Chen’s method [13]. From Fig.

4 and Fig. 5, we can see that the proposed method generated

two different fuzzy interpolated results with respect to the two

different observations, whereas Lee and Chen’s method [13]

generated the same fuzzy interpolated result with respect to the two different observations. Therefore, we can see that the proposed method gets a more reasonable result than Chang and Chen’s method [4] and Lee and Chen’s method [13].

A~1

A~2

A~*

[4]

Method s Chen' and Chang

B~1

B~2

B~*

[13]

Method s Chen' and Lee

B~1

B~2

B~*

Method Proposed The

B~1

B~2

B~*

Fig. 4. A comparison of the fuzzy interpolated results of Example 4.2 with the observation A~^∗

= <(7.5, 7.5, 9.5, 9.5; 0.875), (7, 7, 10, 10; 1)>

for different methods.

A~1

A~2

A*

~

[4]

Method s Chen' and Chang

B~1

B~2

B~*

[13]

Method s Chen' and Lee

B~1

B~2

B~*

Method Proposed The

B~1

B~2

B~*

Fig. 5. A comparison of the fuzzy interpolated results of Example 4.2 with the observation A~^∗ = <(8, 8, 9, 9; 0.75), (7, 8, 9, 10; 1)>

for different methods

.

V. C

ONCLUSION

In this paper, we have presented a method for fuzzy rule interpolation based on the ratio of fuzziness of interval type-2 fuzzy sets. It can deal with fuzzy rule interpolation with polygonal interval type-2 fuzzy sets and can guarantee the convexity of the fuzzy interpolated result. Moreover, it can produce different fuzzy interpolated results with respect to different observations. First, it calculates the weights of the closest fuzzy rules with respect to the observation to obtain an intermediate consequence fuzzy set. Then, it uses the ratio of fuzziness of interval type-2 fuzzy sets to infer the fuzzy interpolated result based on the intermediate consequence fuzzy set. The experimental results show that the proposed fuzzy rule interpolation method gets more reasonable results than Chang and Chen’s methods [4] and Lee and Chen’s method [13].

A

CKNOWLEDGMENT

This work was supported in part by the National Science Council, Republic of China, under Grant NSC 97-2221-E-011-107-MY3.

R

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