A new fuzzy interpolative reasoning method based on interval type-2 fuzzy sets

(1)

A New Fuzzy Interpolative Reasoning Method Based on Interval Type-2 Fuzzy Sets

Yu-Chuan Chang ¹ and Shyi-Ming Chen ^{1, 2}

1 Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R. O. C.

2 Department of Computer Science and Information Engineering, Jinwen University of Science and Technology, Taipei County, Taiwan, R. O. C.

Abstract—Fuzzy rule interpolation plays an important role in sparse fuzzy rule-based systems. In this paper, we present a new method for handling fuzzy rule interpolation in sparse fuzzy rule- based systems based on interval type-2 fuzzy sets. The proposed method handles fuzzy rule interpolation based on the principle membership functions and the uncertainty grade functions of interval type-2 fuzzy sets. The proposed method can handle fuzzy rule interpolation with polygonal interval type-2 fuzzy sets. It also can handle fuzzy rule interpolation with multiple antecedent variables and can generate reasonable fuzzy interpolative reasoning results in sparse fuzzy rule-based systems based on interval type-2 fuzzy sets.

Keywords—Fuzzy rule interpolation, interval type-2 fuzzy sets, principal membership functions, uncertainty grade functions, sparse fuzzy rule-based systems.

I. I NTRODUCTION

Fuzzy rule interpolation is an important research topic in sparse fuzzy rule-based systems and fuzzy rule interpolation techniques are used to solve the problems of sparse fuzzy rule bases which may occur in the following situations [13]: (1) Rule bases reduction, (2) incomplete knowledge in the rule bases, (3) tuning the rule bases, and (4) in order to avoid too high complexity in very large systems, “gaps” are defined intentionally in the rule bases. Some fuzzy rule interpolation methods [2]-[4], [6], [8], [13], [14] have been presented for sparse fuzzy rule-based systems based on type-1 fuzzy sets [15].

In recent years, interval type-2 fuzzy sets, a subclass of type-2 fuzzy sets [16], which are useful when it is difficult to determine the exact and precise membership functions. In [11], Mendal presented the introduction of uncertain rule-based fuzzy logic systems. In [12], Mendal et al. presented the introduction of interval type-2 fuzzy logic systems. Interval type-2 fuzzy sets play an important role in fuzzy rule-based systems [1], [5], [7], [10]. If we can handle fuzzy rule interpolation in sparse fuzzy rule-based systems based on interval type-2 fuzzy sets, then there is room for more flexibility. In [9], Lee and Chen presented a method for fuzzy interpolative reasoning using interval type-2 fuzzy sets.

In this paper, we present a new method for fuzzy rule interpolation based on interval type-2 fuzzy sets. The rest of this paper is organized as follows. In Section II, we briefly review some basic concepts of interval type-2 fuzzy sets [11], [12]. In Section III, we present a new method for handling fuzzy rule interpolation based on interval type-2 fuzzy sets. In

Section IV, we use two examples to show the fuzzy interpolative reasoning results of the proposed method. The conclusions are given in Section V.

II. B ASIC C ONCEPTS OF I NTERVAL T YPE -2 F UZZY S ETS

In this section, we briefly review some basic concepts of interval type-2 fuzzy sets [11], [12] and the definitions of the principle membership functions [11] and the uncertainty grade functions of interval type-2 fuzzy sets, respectively.

Definition 2.1 [12]: A type-2 fuzzy set A ~

in the universe of discourse X is characterized by a type-2 membership function

A ~

P (x, u), where

}, ,

| )) , ( ), , (

~ {(

~ x

A x u x X u J

u x

A P (1)

x X, u J x , J x denotes the primary membership of x, J x [0, 1], P _A ~ (x, u) denotes the secondary grade of (x, u) and 0 d

A ~

P (x, u) d 1. The type-2 fuzzy set A ~

also can be represented as:

, ) , ( ) ,

~ (

³ ³ ~

X x

A J u

u x u x A

x

P (2) where x X, u J x , J x [0, 1] and ³³ denotes the union over all admissible x and u. A type-2 membership function is three-dimensional and the third dimension (i.e., P _A ~ (x, u)) is provides a degree of freedom in handling uncertainties.

Definition 2.2 [12]: Let A ~

be a type-2 fuzzy set in the universe of discourse X represented by the type-2 membership function P _A ~ . If all the secondary grades P

~_A

( x , u ) of A ~

are equal to 1, then A ~

is called an interval type-2 fuzzy set, shown as follows:

, ) , (

~ 1

³ ³ X x u J

u x A

x

(3) where x X, u J x , J x [0, 1] and ³³ denotes the union over all admissible x and u.

Definition 2.3 [12]: Uncertainty of an interval type-2 fuzzy

set A ~ can be represented by the union of the primary

memberships in a bounded region called the “footprint of

uncertainty”, shown as follows:

(2)

,

~ ) (

FOU _x

X

x J

A

(4) ,

,

~ ) ( FOU )

~ (

X x A

x

A ^U (5) ,

,

~ ) ( FOU )

~ (

X x A

x

A ^L (6)

where ~ )

FOU( A denotes the footprint of uncertainty of the interval type-2 fuzzy set A ~

, A ~ ^U and A ~ ^L

denote the upper membership function (UMF) and the lower membership function (LMF) that bound ~ ),

FOU( A respectively, )

~ ( x

A ^U [0, 1], ~ ( ) x

A ^L [0, 1], ~ ( ) x

A ^L d ~ ( ) x

A ^U and x X. The interval type-2 fuzzy set A ~

can be denoted by < A ~ ^L , A ~ U

>. A type-1 fuzzy set is a special case of interval type-2 fuzzy sets whose membership function uncertainties (i.e.

~ )

FOU( A ) disappears.

Definition 2.4: A polygonal interval type-2 fuzzy set is an interval type-2 fuzzy set whose upper membership function and lower membership function are polygonal membership functions, respectively. A polygonal membership function can be characterized by (a 0 , a 1 , …, a l , a r , …, a n-2 , a n-1 ) in the universe of discourse X based on the operations of D -cuts, as shown in Fig. 1, where 0 d D d 1, l =

»¼

»

«¬

« 2 1 n , r =

»»

º

««

ª 2 1 n , a l

and a r denote the “left normal point” and the “right normal point” of the polygonal membership function, respectively, and the membership values of a l and a r are equal to 1. The

“composite normal point” a c of the polygonal membership function A shown in Fig. 1 is calculated as follows:

2 ,

r l c

a

a a (7)

where a c denotes the composite normal point of the polygonal membership function A; a l and a r denote the left normal point and the right normal point of the polygonal membership function, respectively. Fig. 2 shows an example of a trapezoidal interval type-2 fuzzy set A ~

= < A ~ ^L

, A ~ ^U >, where A ~ L

is denoted by ( a ₀ ^L , a ₁ ^L , a ₂ ^L , a ₃ ^L ; H ^L ) , A ~ ^U

is denoted by )

; , , ,

( a ₀ Û a ₁ Û a ₂ Û a ₃ Û H Û , H ^L and H Û are called the maximum membership values of A ~ ^L

and A ~ Û , respectively, and 0 d H ^L d H Û d 1. In Fig. 2, if a ₁ Û = a Û ₂ and a

₁^L

= a

₂^L

, then A ~ b e c o me s a t r i a n g u l a r i n t e r v a l t y p e - 2 f u z z y se t .

Fig. 1. A polygonal membership function [3].

Fig. 3 shows an example of a polygonal interval type-2 fuzzy set A ~

= < A ~ ^L , A ~ ^U

>, where A ~ ^L

is denoted by ),

; , ..., , , ..., , ,

( a ₀ ^L a ₁ ^L a _l ^L a _r ^L a _n ^L ₂ a _n ^L ₁ H ^L A ~ ^U is denoted by ),

; , ..., , , ..., , ,

( a ₀ Û a ₁ Û a Û _l a Û _r a Û _n ₂ a Û _n ₁ H Û and 0 d H ^L d H Û d 1.

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

.

Fig. 3. A polygonal interval type-2 fuzzy set A ~

.

Definition 2.5 [11]: The principal membership function A ^P of an interval type-2 fuzzy set A ~

in the universe of discourse X is defined as follows:

, 2 /

)

~ ( )

( ³

X x

U L

P A x A x x

x

A ⁽⁸⁾

where A ^P denotes the principal membership function of A ~ , 1

) (

0 d A ^P x d and x X. For example, the principal membership function of the interval type-2 fuzzy set A ~

shown in Fig. 2 is shown in Fig. 4.

Fig. 4. The principal membership function A ^P of the trapezoidal interval type-2 fuzzy set A ~

shown in Fig. 2.

Definition 2.6: The uncertainty grade function R _A ~ of an interval type-2 fuzzy set A ~

in the universe of discourse X is

defined as follows:

(3)

³

X x

L U

A A x A x x

x

R /

2 )

~ ( )

~ ( ) (

~ , (9) where R _A ~ ( x ) denotes the grade of uncertainty of x in the interval type-2 fuzzy set A ~

, 0 d R _A ~ ( x ) d 1 and x X.

Definition 2.7: The membership function normalization is to transform a non-normal type-1 fuzzy set A in the universe of discourse X into a normal type-1 fuzzy set A

^N

, whose maximal primary membership is equal to 1, shown as follows:

A A

A H

x x

N

) ) (

( P

P , (10)

where P _A ^and P A

N

denotes the membership functions of A and A

^N

, respectively, 0 d P _A ( x ) 1 , 0 d P A

N

( x ) d 1 ^{, x X, H} ^A denotes the maximal membership value of A and 0 d H A 1.

Fig. 5 shows a non-normal type-1 fuzzy set A and its corresponding normalized type-1 fuzzy set A

^N

.

Fig. 5. The corresponding normalized type-1 fuzzy set A

^N

of the non-normal type-1 fuzzy set A .

III. A N EW M ETHOD FOR F UZZY R ULE I NTERPOLATION B ASED ON I NTERVAL T YPE -2 F UZZY S ETS

In this section, we present a method for fuzzy rule interpolation in sparse fuzzy rule-based systems based on polygonal interval type-2 fuzzy sets. The fuzzy rule interpolation scheme with multiple antecedent variables based on interval type-2 fuzzy sets is shown as follows:

Rule 1: If X 1 is ~ ₁₁

A and X 2 is ~ ₁₂

A and … and X M is A ~ ₁ _M Then Y is B ~ ₁

Rule 2: If X 1 is ~ ₂₁

A and X 2 is ~ A ₂₂ and … and X M is A ~ ₂ _M Then Y is B ~ ₂

Observation: X 1 is A ~ ₁ and X 2 is ~ ₂

A and … and X M is A ~ _M Conclusion: Y is B ~ ^*

where X 1 , X 2 , …, and X M are called the antecedent variables, Y is called the consequent variable, A ~ denotes the jth antecedent _ij fuzzy set of Rule i in the universe of discourse U j , B ~ _i

denotes the consequent fuzzy set of Rule i in the universe of discourse V, A ~ _j

denotes the jth observation fuzzy set (i.e.,

j j

j A A

A ₁ ~ ~ ₂

~ % % or A ~ ₂ _j A ~ _j A ~ ₁ _j

%

% , where “ % ” denotes a total ordering relationship), B ~

denotes the fuzzy interpolative reasoning result, A ~ , _ij B ~ _i

, A ~ _j and B ~

are polygonal interval

type-2 fuzzy sets, 1 d i d 2, 1 d j d M, and M denotes the number of antecedent variables in a fuzzy rule. The proposed method for fuzzy rule interpolation based on polygonal interval type-2-fuzzy sets is presented as follows:

Step 1: Based on Eq. (8), get the principal membership functions A _ij ^P , A _j ^P and B _i ^P of interval type-2 fuzzy sets A ~ , _ij

A ~ j

and B ~ _i

, where 1 d i d 2, 1 d j d M and M denotes the number of the antecedent variables.

Step 2: Based on Eq. (10), get the normalized type-1 fuzzy sets

N

A , ij A _j ^N and B of _i ^N A _ij ^P , A _j ^P and B _i ^P , respectively, where

N

A , ij A _j ^N and B are normal type-1 fuzzy sets, 1 d i d 2, 1 d _i ^N j d M, and M denotes the number of antecedent variables in a fuzzy rule. Then, we transform the fuzzy rule interpolation scheme into the following form:

Rule 1: If X 1 is A 11 ^N and X 2 is A ₁₂ ^N and … and X M is A ₁ ^N _M Then Y is B ₁ ^N

Rule 2: If X 1 is A ₂₁ ^N and X 2 is A ₂₂ ^N and … and X M is A ₂ ^N _M Then Y is B ₂ ^N

Observation: X 1 is A ₁ ^N and X 2 is A ₂ ^N and … and X M is A _M ^N Conclusion: Y is B ^N

where A denotes the jth antecedent fuzzy set of Rule i, _ij ^N B _i ^N denotes the consequent fuzzy set of Rule i, A _j ^N denotes the jth observation fuzzy set, B ^N denotes the fuzzy interpolative reasoning result, A , _ij ^N B , _i ^N A _j ^N and B ^N are normal type-1 fuzzy sets, A ₁ ^N _j d A _j ^N d A ₂ ^N _j , “ % ” denotes a total ordering relationship, 1 d i d 2, 1 d j d M, and M denotes the number of antecedent variables in a fuzzy rule. Assume that A is _ij ^N denoted by (a ij,1 , a ij,2 , …, a ij,l , a ij,r , …, a ij,n-2 , a ij,n-1 ), B is _i ^N denoted by (b i,1 , b i,2 , …, b i,l , b i,r , …, b i,n-2 , b i,n-1 ), and A _j ^N is denoted by ( a _j _, ₀ , a _j _, ₁ , ..., a _j _, _l , a _j _, _r , ..., a _j _, _n ₂ , a _j _, _n ₁ ) . Based on [4], we get the fuzzy interpolative reasoning result B ^N , where

B N is denoted by ( b ₀ , b ₁ , ..., b _l , b _r , ..., b _n ₂ , b _n ₁ ) . Step 3: Based on Eq. (7), get the composite normal points a _ij _, _c ,

c

a j _, and b _i _, _c of A , _ij ^N A _j ^N and B , respectively, where 1 d i d _i ^N 2, 1 d j d M and M denotes the number of antecedent variables in a fuzzy rule. Calculate the weight w ij of each antecedent fuzzy set A with respect to the corresponding observation _ij ^N

N

A j and the weight W i of Rule i, respectively, shown as follows [4]:

c j c j

c ij c j

ij a a

a w a

, 1 , 2

,

1 ,

, (11)

(4)

¦¦

¦

2 1 1

1 i M j

ij M j

ij i

w w

W , (12)

where w ij denotes the weight of the antecedent fuzzy set A _ij ^N with respect to the observation fuzzy set A _j ^N , 0 d w ij d 1, W i

denotes the weight of Rule i, 0 d W i d 1, W 1 + W 2 = 1, 1 d i d 2, 1 d j d M, and M denotes the number of antecedent variables in a fuzzy rule.

Step 4: Calculate the left normal point b _l and the right normal point b _r of the fuzzy interpolative reasoning result B ^N as follows [4]:

,

2 1 ,

¦

i i i c

c W b

b (13)

° °

¯

° °

®

u

!

¸¸

¸ ¸

¸

¹

· ¨¨

¨ ¨

¨

©

§

u u

¦

¦ ! ¦

, 0 ) ( , if , ) (

, 0 ) ( , if

, ) (

) ) (

( ) (

2 1

2 0 ) ( i

, 1

1 N ij i

N i i

N ij A

j

i M

j N ij N i i N

j N

A ij B

W

A ij

A W B

A

B

^ij^PN

S S

S S S S

S ^S ⁽¹⁴⁾

2 , ) ( ^N

c l

b B

b S (15)

2 , ) ( ^N

c r

b B

b S (16)

where b _c , b _l and b _r denote the composite normal point, the left normal point and the right normal point of the fuzzy interpolative reasoning result B ^N , respectively, S ( A _ij ^N ) ⁼

r ij l

ij a

a _, _, ^, S ( B _i ^N ) ⁼ b i _, l b i _, r , S ( A _j ^N ) ⁼ a j _, l a j _, r , )

( B ^N

S = b l b r , l =

»¼ »

«¬ « 2 1 n , r =

»» º

«« ª 2 1

n , 1 d i d 2, 1 d j d M, and M denotes the number of antecedent variables in a fuzzy rule. From Eqs. (13)-(16), we can see that S ( B ^N ) t 0 and b l d b c d b r .

Step 5: Divide the membership function of each polygonal type-1 fuzzy set A appearing in the fuzzy rules and the observations into the areas ( )

0

A

S _L , S _L

₁

( A ) , …, ( )

1

A

S L

t

, )

1

( A S _R

_t

, …, S _R

₁

( A ) and S _R

₀

( A ) , t =

»¼

»

«¬

« 2 1

n , as shown in Fig. 6, where S _L

_p

(A ) denotes the area between a

_p

and a _p ₁ ,

)

(

1

(

A

S R

nq

denotes the area between _a

_q₁

and a

_q

, 0 d p d t – 1 and t + 1 d q d n – 1. Assume that the membership function of

the fuzzy interpolative reasoning result B ^N is divided into the

areas ( )

0

N

L B

S , S _L

₁

( B ^N ) , …, ( )

1

N

L B

S

t

, S R

_t₁

( B ^N ) , …, )

1

(

N

R B

S and ( )

0

N

R B

S , t =

»¼

»

«¬

« 2 1

n , as shown in Fig. 7, where S _L ( B ^N )

p

denotes the area between b and _p b _p ₁ , )

)

(

1 (

N

R B

S

nq

denotes the area between b and _q ₁ b , 0 d p d t – _q 1 and t + 1 d q d n – 1. Calculate the area S _K ( B ^N ) of the fuzzy interpolative reasoning result B ^N as follows [4]:

° °

¯

° °

®

!

¸¸

¸ ¸

¸

¹

· ¨¨

¨ ¨

¨

©

§

u

¸ u

¸

¹

· ¨ ¨

©

§

¦

¦ ¦

¦

!

, 0 ) ( , if ,

) 1 (

, 0 ) ( , if

, ) (

) ) (

( )

(

1

2

0 ) ( ,

, 1

1 1

N ij K M

j N j K

N ij K A

S j

i M

j N ij K

N i K i M

j N j K N

K

A S ij A

M S

A S ij

A S

B W S A

S B

S

^K ^ij^N

(17)

where K {L 0 , L 1 , …, L , _t ₁ R , …, R _t ₁ 1 , R 0 }, t =

»¼

»

«¬

« 2 1 n , 1 d i d 2, 1 d j d M, and M denotes the number of antecedent variables in a fuzzy rule. From Fig. 6, we can see that the area

) ( ^N

L B

S

p

is also equal to

2 ) ( )

(

1 1

u

p p p

p

D b b

D and the area

)

(

1 (

N

R B

S

nq

is also equal to

2 ) ( )

( 1 1

q n q u q q

n D b b

D ,

where 0 d p d t – 1, t + 1 d q d n – 1 and t =

»¼

»

«¬

« 2 1

n . Then, we

can get

) (

) ( 2

1 1

u

p p

N L p p

B b S

b

^p

D D

and

) (

) ( 2

1 * 1

) 1 (

q n q n

N R q q

B b S

b

ⁿ^q

u

D

D , where 0 d p d t – 1, t + 1 d q d n – 1 and t =

»¼ »

«¬ « 2 1

n . Therefore, the fuzzy interpolative reasoning result B ^N = ( 0 , 1 , ..., , , ..., 2 , 1 )

b b l b r b n b n

b can be

derived as follows [4]:

) , ( 2

1 ¦ 1

t p

k k k

N L l

p

B b S

b

^k

D

D (18)

) , ( 2

1 1

) 1

¦

(

q t

k n k n k

N R r

q

B b S

b

ⁿ^k

D

D (19) where 0 d p d t – 1, t + 1 d q d n – 1 and t =

»¼

»

«¬

« 2 1

n .

Step 6: Based on Eq. (9), get the uncertainty grade functions

A

ij

R ~ ,

A

j

R ~ and

B

i

R ^~ of the interval type-2 fuzzy sets A ~ , _ij A ~ _j

and B ~ _j , respectively, where 1 d i d 2, 1 d j d M, and M denotes

the number of antecedent variables in a fuzzy rule. Get the

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uncertainty grade function R _B ~

of the fuzzy interpolative reasoning result B ~ ^* , shown as follows:

(20)

0, }) (inf{

, if }), (inf{

0, }) (inf{

, if

, }) (inf{

}) (inf{

)

~ ( )

~ ( 2

1

)

~ ( 1

)

~ ( )

~ ( 2

1 1

)

~ (

~

° °

¯

°°

° °

®

u

!

u u

¦

¦ ¦

¦

N A ij N

B i i

i

N A ij M j

N A ij

N B i i

i M

j

N A j

N B

A R ij A

R W

A R ij

A R

A W R

A R

B R

ij i

ij ij i

D D

D D D D

D

(21)

0, }) (sup{

, if }), (sup{

0, }) (sup{

, if

, }) (sup{

}) (sup{

)

~ ( )

~ ( 2

1

)

~ ( 1

)

~ ( )

~ ( 2

1 1

)

~ (

~

° °

¯

°°

° °

®

u

!

u u

¦

¦ ¦

¦

N A ij N

B i i

i

N A ij M j

N A ij

N B i i

i M

j

N A j

N B

A R ij A

R W

A R ij

A R

A W R

A R

B R

ij i

ij ij i

D D

D D D D

D

where R

_P~

(inf{ T

₍

D

₎

}) and R _P ~ (sup{ T ₍ D ₎ }) denote the uncertainty grades of inf{ T _(D ₎ } and sup{ T _(D ₎ } , inf{ T ₍ _D ₎ } and sup{ T ( D ) } denote the minimal element and the maximum element of D -cut T {a} of T, respectively, T { A , _ij ^N A _j ^N , B , _i ^N B ^N }, 0 d D d 1, W i denotes the weight of Rule i, 0 d W i d 1, W 1 + W 2 = 1, 1 d i d 2, 1 d j d M, and M denotes the number of antecedent variables in a fuzzy rule.

Fig. 6. The areas of the polygonal type-1 fuzzy set A [3].

Fig. 7. The areas of the fuzzy interpolative reasoning result B

^N

[3].

Step 7: Get the principal membership function B ^* ^P of the fuzzy interpolative reasoning result ~ ^*

B , shown as follows:

, / )) ( (

) (

³ V

u

y

N B

P y H B y y

B (22)

,

2 1

*^P

1 B

^P

2 B

^P

B W H W H

H u u (23) where B ^P denotes the principle membership function of the fuzzy interpolative reasoning result B ~ ^* , 0 d B ^P ( y ) d 1 , y V, H B

P

1

, H B

P 2

and H _B

denote the maximum membership values of the principle membership functions

B 1 P , B 2 ^P and B ^* ^P , respectively, W i denotes the weight of Rule i, 0 d W i d 1, 1 d i d 2, and W 1 + W 2 = 1.

Step 8: Get the fuzzy interpolative reasoning result ~ ^* B =

< B ~ ^* ^L

, B ~ ^* ^U >, where the lower membership function B ~ ^* ^L and the upper membership function B ~ ^* ^U

of the interval type- 2 fuzzy set B ~ ^* are calculated as follows:

, / 2 )

) ) ( ( max(0, )

~ (

V

³ ~

y

P B

L R y y

y B y

B (24)

, / 2 )

) ) ( ( , 1 ( min )

~ (

V

³ ~

y

P B

U R y y

y B y

B (25)

where B ~ ^* ^L and B ~ ^* ^U

denote the lower membership function B ~ ^* L

and the upper membership function B ~ ^* ^U of the fuzzy interpolative reasoning result ~ _* ,

B respectively, 0 d B ~ ^* ^L (y) d 1, 0 d B ~ ^* ^U

(y) d 1, B ~ ^* ^L (y) d B ~ ^* ^U

(y) and y V.

IV. E XAMPLES

Example 4.1: Let us consider the fuzzy rule interpolation scheme, shown as follows:

Rule 1: If X is ~ ₁

A ^{Then Y is} B ~ ₁ Rule 2: If X is ~ A ₂ Then Y is ~ ₂

B Observation: X is A ~ Conclusion: Y is B ~ where ~ ₁

A , ~ A ₂ , B ~ ₁ , B ~ ₂ , A ~ and B ~

are interval type-2 fuzzy sets, ~ ₁

A = <(1.5, 2, 2, 2.5; 0.7), (1, 2, 2, 3; 1)>, ~ ₂

A = <(12.5, 13, 13, 13.5; 0.7), (12, 13, 13, 14; 1), >, B ~ ₁ = <(1, 2, 2, 3; 0.7), (0, 2, 2, 4; 1)>, B ~ ₂ = <(12, 13, 13, 14; 0.7), (11, 13, 13, 15;

1)> and A ~

= <(5.5, 6, 6, 7.5; 0.7), (5, 6, 6, 8; 1)>, as shown in Fig. 8. The fuzzy interpolative reasoning result B ~

of the proposed method is also shown in Fig. 8, where B ~

= <(5, 6, 6,

9; 0.7), (4, 6, 6, 10; 1)>. Because the larger the area of the

footprint of uncertainty of an interval type-2 fuzzy set, the more

the uncertainty of the interval type-2 fuzzy set. From Fig. 8, we

can see that the fuzzy interpolative reasoning result of the

proposed method holds the consistence of uncertainty between

the interval type-2 fuzzy sets appearing in the antecedents and

the consequents of the fuzzy rules.

(6)

Fig. 8. Fuzzy interpolative reasoning scheme of Example 4.1.

Example 4.2: Let us consider the fuzzy rule interpolation scheme, shown as follows:

Rule 1: If X is ~ ₁₁ A and ~ ₁₂

A Then Y is B ~ ₁ Rule 2: If X is A ~ ₂₁ ^and 22

A ~ Then Y is B ~ ₂ Observation: X is ~ ₁

A and A ~ ₂ Conclusion: Y is ~ ^*

B where ~ ₁₁

A , ~ ₁₂

A , ~ A ₂₁ , ~ ₂₂ A , ~ ₁

B , ~ ₂ B , ~ ₁

A , A ~ ₂ and ~ ^* B are interval type-2 fuzzy sets, ~ ₁₁

A = <(1.5, 2, 2, 2.5; 0.8), (1, 2, 2, 3; 1)>, A ~ ₂₁ = <(12.5, 13, 13, 13.5; 0.8), (12, 13, 13, 14; 1)>,

12 ~ A = <(1.5, 2, 2, 2.5; 0.7), (1, 2, 2, 3; 1)>, ~ ₂₂

A = <(12.5, 13, 13, 13.5; 0.7), (12, 13, 13, 14; 1)>, ~ ₁

B = <(1.5, 2, 2, 2.5; 0.6), (1, 2, 2, 3; 1)>, B ~ ₂ = <(12.5, 13, 13, 13.5; 0.6), (12, 13, 13, 14;

1)>, ~ ₁

A = <(5.5, 6, 6, 6.5; 0.8), (5, 6, 6, 7; 1)> and A ~ ₂ = <(7.5, 8, 8, 8.5; 0.7), (7, 8, 8, 9; 1)>, as shown in Fig. 9. The fuzzy interpolative reasoning result B ~ ^* of the proposed method is also shown in Fig. 9, where B ~ ^* = <(6.5, 7, 7, 7.5; 0.6), (6, 7, 7, 8; 1)>. From Fig. 9, we can see that the membership function curve of ~ ₁

A is similar to the membership function curves of

11 ~ A and 21

A ~ , and the membership function curve of A ~ ₂ is similar to the membership function curves of ~ ₁₂

A and ~ ₂₂ A . We also can see that the membership function curve of the fuzzy interpolative reasoning result B ~ ^* of the proposed method is also similar to the membership function curves of ~ ₁

B and ~ ₂ B .

Fig. 9. Fuzzy interpolative reasoning scheme of Example 4.2.

V. C ONCLUSIOINS

In this paper, we have presented a new method for fuzzy rule interpolation in sparse fuzzy rule-based systems based on interval type-2 fuzzy sets. The proposed method can handle fuzzy rule interpolation with polygonal interval type-2 fuzzy sets. It also can handle fuzzy rule interpolation with multiple antecedent variables and can generate reasonable fuzzy interpolative reasoning results in sparse fuzzy rule-based systems based on interval type-2 fuzzy sets.

A CKNOWLEDGMENT

This work was supported in part by the National Science Council, Republic of China, under Grant NSC 95-2221-E-011- 116-MY2.

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A new fuzzy interpolative reasoning method based on interval type-2 fuzzy sets