Representation of uncertain multichannel digital signal spaces and study of pattern recognition based on metrics and difference values on fuzzy n-cell number spaces

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Abstract—In this paper, we discuss the problem of

characterization for uncertain multi-channel digital signal spaces, propose using fuzzy n−cell number space to represent uncertain

−

n channel digital signal space, and put forward a method of constructing such fuzzy n−cell numbers. We introduce two new metrics and concepts of certain types of difference values on fuzzy

−

n cell number space, and study their properties. Further, based on the metrics or difference values appropriately defined we put forward an algorithmic version of pattern recognition in an imprecise or uncertain environment, and we also give practical examples to show the application and rationality of the proposed techniques.

Index Terms—Uncertain multi-channel digital signals, Fuzzy −

n cell numbers, n− dimensional fuzzy vectors, Metrics, Difference values, Pattern recognition

I. INTRODUCTION

T is known that in a precise or certain environment, multi-channel digital signals can be represented by elements of multi-dimensional Euclidean space, i.e., crisp multi-dimensional vectors. If however we wish to study multi channel digital signals in an imprecise or uncertain environment, then the signals themselves are imprecise or have no certain bound, and it becomes unwise to use crisp multidimensional vectors to represent them. In this paper, we recommend using fuzzy n− cell numbers to represent

imprecise or uncertain multi-channel digital signals, and put forward a method of constructing such fuzzy n−cell numbers.

Manuscript received October 15, 2008. This work is supported in partially by Natural Science Foundations of China (No. 60772006) and Natural Science Foundations of Zhejiang Province, China (No. Y7080044), and Engineering and Physical Sciences Research Council, UK (No. EP/F0219195).

G. Wang is with Institute of Operations Research and Cybernetics, Hangzhou Dianzi University, Hangzhou, 310018, China ([email protected]).

P. Shi is with Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd, CF37 1DL, UK ([email protected]). He is also with ILSCM, School of Science and Engineering, Victoria University, Melbourne, 8001, Australia, and School of Mathematics and Statistics, University of South Australia, Adelaide, 5095, Australia.

P. Messenger is with Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd, CF37 1DL, UK ([email protected]).

The concept of general fuzzy numbers was introduced by Chang and Zadeh [2] in 1972 with the consideration of the properties of probability functions. Since then both the numbers and the problems in relation to them (see for example [3, 4, 5, 6, 11, 16, 19, 20, 21]) have been widely studied. With the development of theories and applications of fuzzy numbers, this concept becomes more and more important. In [7] Kaleva ever used a special type of n−dimensional fuzzy number,

whose sets of cuts are all hyper-rectangles. In 2002 we carefully studied the special type of n−dimensional fuzzy

number, and call it fuzzy n−cell number in [14,15]. It has been

demonstrated that fuzzy n−cell number is used much more

conveniently than general n−dimensional fuzzy numbers in

theoretical investigations and some fields of application in [14, 15, 17]. On the other hand, n−dimensional fuzzy vector is also

an important concept, which is the Cartesian product of

n

−

1 _{dimensional fuzzy numbers. In 1985, Kaufmann and Gupta} [8] already studied fuzzy vectors, soon afterwards, Miyakawa and Nakamura et al. [9,10,12] also studied the problems of theories and applications in relating to fuzzy vectors. In 1997, Butnariu [1] studied Methods of solving optimization problems and linear equations in the space of fuzzy vectors. Recently, we [14] showed that fuzzy n−cell numbers and n−dimensional

fuzzy vectors can represent each other, and obtained the representations of the joint membership function and the edge membership functions of a fuzzy n−cell number of each other.

In a previous paper [15], we defined a metric DL on the

fuzzy n−cell number space, and studied its properties. And in

paper [14], we again studied this type of metric in regard to two fuzzy n−cell numbers as the form of n−dimensional fuzzy

vectors. Although metric DL can be more conveniently used in

applications and theoretical investigations, it has some shortcomings. That is, it has a tendency to be rougher, and can not really characterize the degree of difference of two fuzzy

−

n cell numbers in some applications (see Example 3.1 in

Section 3 of this paper). In this paper, in order to discuss the problem of pattern recognition in an imprecise or uncertain environment based on degree of difference, we define two new metrics and some concepts of difference values on fuzzy

−

n cell number space, which may better characterize the

Representation of Uncertain Multi-Channel

Digital Signal Spaces and Study of Pattern

Recognition Based on Metrics and Difference

Values on Fuzzy

n

−

cell Number Spaces

Guixiang Wang, Peng Shi, Senior Member, IEEE, and Paul Messenger

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degree of difference of two fuzzy n−cell numbers in some

applications, and study their properties.

It is well known that pattern recognition is an important field of research. In this aspect many research achievements have been obtained (for example, see [13]). In this paper, as applications of the metrics and difference values (defined by us), we also study the problem of pattern recognition in an imprecise or uncertain environment, put forward an algorithmic version of pattern recognition based on the metrics or difference values (defined by us) of fuzzy n−cell numbers,

and also give examples to show the application and rationality of the method.

The organization of the paper is as follows. In Section 2, we give an example to show how to set up fuzzy n−cell numbers

to represent imprecise or uncertain multi-channel digital signals. In Section 3, we define two new metrics, and study their properties. In Section 4, we introduce concepts of difference values of two fuzzy n− cell numbers, and examine their

properties. In Section 5, an algorithmic version of pattern recognition is given based on the metrics or the difference values defined by us, and examples are also given to show the application and rationality of the method. Finally, in Section 6, we give a brief conclusion of this paper.

II. REPRESENTATIONS OF UNCERTAIN MULTI-CHANNEL DIGITAL SIGNALS

A fuzzy set of the Euclidean space n

R is a function

] 1 , 0 [ : n→

R

u . For fuzzy set u, we denote [ ]r { n:

R x

u = ∈

} ) (x r

u ≥ for r∈[0,1] , and [u]0 ={x∈Rn: u(x)>0} _(the

closure of {x∈Rn: u(x)>0}_{). If}

u is a normal and fuzzy

convex fuzzy set of n

R , u(x) is upper semi-continuous, and 0

]

[u is compact, then we call u a n−dimensional fuzzy

numbers, and denote the n−dimensional fuzzy number space by n

E . If u∈E , and for each r∈[0,1], r

u]

[ is a hyper rectangle, i.e., there exist ui(r), ui(r)∈R with ui(r)≤ui(r),

(i=1,2,_L,n) such that =

∏

n₌

i i i

r _u _r _u _r

u

1[ ( ), ( )]

]

[ , then we call

u is a fuzzy n−cell number, and denote the fuzzy n−cell

number space by ( n)

E

L . A n−dimensional fuzzy vector is an

ordered class (u1,u2,L,un) , where ui∈E (i.e.,

1

E ),

n

i=1,2,_L, . In [14], We have shown that fuzzy n−cell

numbers and n−dimensional fuzzy vectors can represent each other, and as the representation is unique, _L(_En)_{and the}

−

n dimensional fuzzy vector space (i.e., the Cartesian product

4 4 8 4 4 7 6

L n

E E

E× × × ) may be regarded as identical.

When exploring and discussing some quantity, properties or laws of movement of phenomena/objects in the physical world, it is essential for us to establish the description space of them. For instance, when the quantity in question is only the one with a single factor, we can take it as a dot in real number field R,

that is, the space of quantities corresponding to single factor

can be described by 1− dimensional Euclidean space R .

Similarly, we can describe the quantities with n factors, using

−

n dimensional Euclidean spaceRn. However, in the physical

world, many phenomena are imprecise or uncertain (such as, have no certain bound). When the quantity discussed by us possesses some imprecise or uncertain attributes, it is unsuitable that we use still _Rn_{to represent the space of the}

quantities (see Remark 2.1). It is our opinion that using the fuzzy n−cell number space discussed in [14, 15] to describe the quantities with some uncertain factors and discuss these quantities in this n−dimensional fuzzy vector space is a more suitable method to reveal the objective laws of things in physical world (see Remark 2.1).

In the following example, we demonstrate how we construct a fuzzy n−cell number to represent a quantity that possesses some uncertain attributes based on statistical data. About the algorithmic version of such fuzzy n−cell numbers, we can see the first or second step of the algorithmic version in Section 5.

Example 2.1. It is well known that different kinds of terrain

or landcover possess the different reflections of the electromagnetic spectrum. Based on this principle, one can set up a method to recognize the category of landcover, a challenging remote sensing classification problem, using spectral and terrain features for vegetation classification in some zone. In remote sensing classification, the colligation of all species covering a zone of 4500 m2_{can be boiled down to an element of remote}

sensing space. We use “Korean Pine accounts for the main part” to denote forest that mainly contains Korean Pines. Because in different “Korean Pine accounts for the main part” areas, there are many different factors such as the difference of the density of Korean Pines, of the species and quantity of other plants, of the physiognomy and so on, the values of reflections of the electromagnetic spectrum are also different. Therefore “Korean Pine accounts for the main part” should not be a certain crisp value but a fuzzy set without certain bound. So, using a fuzzy number to represent the spectral sensitivity level of the “Korean Pine accounts for the main part” is more suitable than using a crisp number. Suppose that we use 4 wave bands: MSS-4, MSS-5, MSS-6, MSS-7. We take 10 samples, and acquire the following data for some zone of “Korean Pine accounts for the main part”:

45 . 17 08 . 34 48 . 12 38 . 15 10 Sample

02 . 18 10 . 36 58 . 12 50 . 15 9 Sample

62 . 18 64 . 37 67 . 13 82 . 16 8 Sample

29 . 14 87 . 30 98 . 10 90 . 15 7 Sample

54 . 15 10 . 32 94 . 11 80 . 13 6 Sample

75 . 20 10 . 42 80 . 13 10 . 16 5 Sample

75 . 14 50 . 35 70 . 11 90 . 14 4 Sample

16 . 18 70 . 37 79 . 12 82 . 15 3 Sample

35 . 16 81 . 38 56 . 12 60 . 15 2 Sample

37 . 19 50 . 40 30 . 13 01 . 15 1 Sample

7 MSS 6 MSS 5 MSS 4

MSS − − − −

We can directly work out the following means μi

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MSS-4 MSS-5 MSS-6 MSS-7

i

μ : μ1=15.46 μ2=12.58 μ3=36.54 μ4=17.33

i

σ : σ1=1.22 σ2 =0.88 σ3=3.55 σ4=2.08

From the means and the standard deviations, with

⎪ ⎪ ⎪

⎩ ⎪ ⎪ ⎪

⎨ ⎧

+∞ +

− ∉

+∞ +

∈ −

+

+∞ −

∈ −

−

=

) , 0 ( ] 2 , 2 [ if 0

) , 0 ( ] 2 , ( if 2

) 2 (

) , 0 ( ] , 2 [ if 2

) 2 (

) (

I I I

i i i i i

i i i

i i i i i

i i i

i i

x x x

x x

x u

σ μ σ μ

σ μ μ σ

σ μ

μ σ μ σ

σ μ

(i=1,2,3,4)

we can define 4 triangular model one-dimensional fuzzy numbers u₁, u₂, u₃ and u₄ that respectively correspond with

MSS-4, MSS-5, MSS-6 and MSS-7:

⎪ ⎪ ⎪

⎩ ⎪⎪ ⎪

⎨ ⎧

∉ ∈ −

∈ −

=

] 9 . 17 , 02 . 13 [ if

0

] 9 . 17 , 46 . 15 ( if

44 . 2

9 . 17

] 46 . 15 , 02 . 13 [ if 44 . 2

02 . 13 )

(

1 1 1

1 1

x x x

x x

x u

⎪ ⎪ ⎪

⎩ ⎪⎪ ⎪

⎨ ⎧

∉ ∈ −

∈ −

=

] 4.34 1 , 82 . 10 [ if

0

] 4.34 1 , 58 . 12 ( if

76 . 1 34 . 14

] 2.58 1 , 82 . 10 [ if

76 . 1

82 . 10 )

(

2 2 2

2 2

x x x

x x

x u

⎪ ⎪ ⎪

⎩ ⎪⎪ ⎪

⎨ ⎧

∉ ∈ −

∈ −

=

] 3.64 4 , 44 . 29 [ if 0

] 3.64 4 , 54 . 36 ( if 1 . 7 64 . 43

] 6.54 3 , 44 . 29 [ if 1 . 7

44 . 29 )

(

3 3 3

3 3

x x x

x x

x u

⎪ ⎪ ⎪

⎩ ⎪⎪ ⎪

⎨ ⎧

∉ ∈ −

∈ −

=

] 1.49 2 , 17 . 13 [ if 0

] 1.49 2 , 33 . 17 ( if

16 . 4 49 . 21

] 7.33 1 , 17 . 13 [ if

16 . 4

17 . 13 )

(

4 4 4

4 4

x x x

x x

x u

By Theorem 3.1 and 3.2 in [14], we know that u1, u2, u3 and 4

u determine a fuzzy 4−cell numberu=(u₁,u₂_,u₃,u₄), and

the membership function of u is

)} ( ), ( ), ( ), ( min{ ) , , ,

(x1 x2 x3 x4 u1 x1 u2 x2 u3 x3 u4 x4

u =

4 4 3 2

1, , , )

(x x x x ∈R

Then u can be used to represent “Korean Pine accounts for the

main part”.

Likewise, from the means and the standard deviations, according to

⎪⎩ ⎪ ⎨ ⎧

= +∞ ∉

+∞ ∈ −

− =

4 , 3 , 2 , 1 ) , 0 ( if 0

) , 0 ( if ) 2

) ( exp( )

( ₂

2

i x

x x

x v

i i i

i

σ μ

we can also define 4 Gaussian model one-dimensional fuzzy numbers v1, v2, v3 and v4 that respectively correspond with

MSS-4, MSS-5, MSS-6 and MSS-7:

⎪⎩ ⎪ ⎨ ⎧

+∞ ∉

+∞ ∈ −

− =

) , 0 ( if 0

) , 0 ( if ) 98 . 2

) 46 . 15 ( exp( ) (

1 1 2

1 1

1

x x x

x v

⎪⎩ ⎪ ⎨ ⎧

+∞ ∉

+∞ ∈ −

− =

) , 0 ( if 0

) , 0 ( if ) 56 . 1

) 58 . 12 ( exp( ) (

2 2 2

2 2

2

x x x

x v

⎪⎩ ⎪ ⎨ ⎧

+∞ ∉

+∞ ∈ −

− =

) , 0 ( if 0

) , 0 ( if ) 21 . 25

) 54 . 36 ( exp( ) (

3 3 2

3 3

3

x x x

x v

⎪⎩ ⎪ ⎨ ⎧

+∞ ∉

+∞ ∈ −

− =

) , 0 ( if 0

) , 0 ( if ) 65 . 8

) 33 . 17 ( exp( ) (

4 4 2

4 4

4

x x x

x v

and obtain the membership function of the fuzzy 4−cell number v=(v1,v2,v3,v4) determined by v1, v2, v3 and v4 as

)} ( ), ( ), ( ), ( min{ ) , , ,

(x1 x2 x3 x4 v1 x1 v2 x2 v3 x3 v4 x4

v = ,

4 4 3 2

1, , , )

(x x x x ∈R . Then the fuzzy 4−cell number v can also

be used to represent the “Korean Pine accounts for the main part”.

Remark 2.1. Of course, if the quantity to describe is precise

and certain, we should use a crisp multi-dimensional vector to represent it. However, if the quantity to describe is imprecise and uncertain, such as “Korean Pine accounts for the main part”, then using a fuzzy n−cell number to represent it is better than using a crisp n−dimensional vector. If we narrowly use a crisp multi-dimensional vector, such as (15.46 ,12.58 ,36.54 ,17.33)

(i.e., the mean vector), to represent “Korean Pine accounts for the main part”, then it can not clearly tell us the relationship of “Korean Pine accounts for the main part” and the zone whose value of reflection of electromagnetic spectrum is

) 79 . 16 , 50 . 37 , 80 . 12 , 16 . 15

( since (15.16 ,12.80 ,37.50 ,16.79)

) 33 . 17 , 54 . 36 , 58 . 12 , 46 . 15 (

≠ . If we use fuzzy n− cell number

) , , (v1 v2,v3 v4

v= to represent it, then we can almost affirm that the

zone whose value is (15.16 ,12.80 ,37.50 ,16.79) is part of “Korean Pine accounts for the main part” since

) 93 . 0 , 93 . 0 , 94 . 0 , 94 . 0 min( ) 79 . 16 , 50 . 37 , 80 . 12 , 16 . 15

( =

v =0.93, i.e.,

the degree of the zone which is “Korean Pine accounts for the main part” is 0.93.

III. METRICS ON FUZZY n−CELL NUMBER SPACE In [3], the authors studied the metric d_p (⋅ ,⋅) (note that in

this paper we rewrite d_p (⋅ ,⋅) as D_p (⋅ ,⋅) ) on general

−

n dimensional fuzzy number space _En_{, which is defined by}

p p r r

p u v d u v r

D 1 1/

0[ ([ ] ,[ ] )] d )

( ) ,

( =

∫

for any n

E v

u, ∈ , and

point out that the metric Dp is complete.

In [15], we studied the metrics D and D_L on ( n)

E

L , but

the two metrics seem to be ‘rough’ in certain applications (see Example 3.1). In the following, other metrics are defined on

) ( n

E

L , which better reveal the difference between two

(4)

We denote LC R A ai bi i n

n

, , 2 , 1 , exist there : { )

( = ≤ = L

} ] , [ such that =

∏

n₌₁

i ai bi

A , where [ , ]

1

∏

= n

i ai bi is the Cartesian

product [a1,b1]×[a2,b2]×L×[an,bn].

Theorem 3.1. We define mappings

) , 0 [ ) ( ) ( :

ˆ n _× n _→ _+∞

R LC R LC

d_α

and

) , 0 [ ) ( ) ( :

~ n _× n _→ _+∞

R LC R LC

d_α

by

∑

= ⋅ − −

= n

i i ai bi ai bi

B A d

1 max{| |, | |}

) , (

ˆ _α

α

and

∑

=

− + − = n

i

i i i i i

b a b a B

A

d ₁

2 | | | | )

, (

~ _α

α

for any n₁[ , ] ( n)

i ai ai LCR

A=

∏

₌ ∈ and n₁[ , ] ( n)

i bi bi LCR

B=

∏

₌ ∈ ,

where α=(α1,α2,L,αn) satisfies

∑

₌₁ =1

n

i αi and αi ≥0 ,

n

i=1,2,L, . Then for any =

∏

₌₁[ , ,] =

∏

n₌₁[ , ,]

i i i

n

i ai ai B b b

A

∏

=

= n

i ci ci

C ₁[ , ] in LC(Rn) and each k∈R , dˆ and α dα

~ satisfies

(1) dˆα(A,B)=dˆα(B,A) and ( , ) ~ ) , ( ~

A B d B A

dα = α ;

(2) dˆα(A,B)≥0 and ( , ) 0

~ _≥

B A

dα ;

(3) dˆα(A,B)=0 ⇔ A=B ⇔ ( , ) 0

~ ₌

B A

dα ;

(4) dˆ_α(A,B)≤dˆ_α(A,C)+dˆ_α(C,B) and d~_α(A,B)≤ d~_α(A,C) )

, ( ~

B C dα

+ ;

(5) dˆ_α(A+C,B+C)= dˆ_α(A,B) and d~_α(A+C,B+C) )

, ( ~

B A dα

= ;

(6) dˆ_α(kA,kB)=|k|dˆ_α(A,B) and d~_α(kA,kB)=|k|d~_α(A,B).

Proof. We only show Proofs (4), (5) and (6) (the other

proofs are easy). From

) , ( ˆ ) , ( ˆ

|}) | |, max{| |} | |, (max{|

|} | | | |, | | max{|

|} | |, max{| )

, ( ˆ

1 1 1

B C d C A d

b c b c c

a c a

b c c a b c c a

b a b a B

A d

i i i i n

i i i i i i

n

i i i i i i i i i i

n

i i i i i i

α α

α

α α α

+ =

− − +

− −

⋅ ≤

− + − − + − ⋅

≤

− − ⋅

=

∑

= = =

) , ( ˆ

} | |

|, max{|

} | |

|, max{|

) , ( ˆ

1 1

B A d

c b c a c b c a

c b c a c b c a C

B C A d

i i i i i i n

i i i i

i i i i i i n

i i i i

α α

=

− − + −

− + ⋅

=

+ − + +

− + ⋅

= + +

∑

= =

) , ( ˆ | |

|} |

|, max{|

|} |

|, max{|

) , ( ˆ

1 1

B A d k

b k a k b k a k

kb ka kb ka kB

kA d

i i i n

i i i

i i i n

i i i

α α

=

− −

⋅ =

− −

⋅ =

∑

= =

we see that (4), (5) and (6) of the theorem hold for dˆ . For α dα

~ , we can similarly prove that (4), (5) and (6) of the theorem also hold. 

Theorem 3.2. We define mappings

) , 0 [ ) ( ) ( : ˆ

, × → +∞

n n

p L E L E

Dα

and

) , 0 [ ) ( ) ( : ~

, × → +∞

n n

p L E L E

Dα

by

p p r r

p u v r d u v r

D 1 1/

0

, ( , ) ( [ ˆ ([ ],[ ] )] d )

ˆ ₌

_∫

_⋅

α α

and

∫

⋅ = 1

0

/ 1

, ([ ],[ ] )] d )

~ [ ( ) , (

~ r r p p

p uv r d u v r

Dα α

i.e.,

p p n

i i i i i i p

r r v r u r v r u r

v u D

/ 1 1

0 1

,

) d ] |} ) ( ) ( | |, ) ( ) ( max{| [

(

) , ( ˆ

∫ ∑

= ⋅ − −

= α

α

and

p n

i

p i i i

i i p

r r v r u r v r u r

v u D

/ 1 1

0 1

,

) d |)] ) ( ) ( | | ) ( ) ( (| [

( 2 1

) , ( ~

∫ ∑

= ⋅ − + −

= α

α

for any ( , ) ( n) ( n)

E L E L v

u ∈ × , where p≥1, and α=(α1,α2,L, )

n

α satisfies

∑

n₌₁ =1

i αi and αi >0, i=1,2,L,n. Then for

any , , ( n)

E L w v

u ∈ and each k∈R, Dˆα,p and D,p

~

α satisfy

(1) Dˆα,p(u,v)=Dˆα,p(v,u) and ( , ) ~ ) , ( ~

,

, uv D vu

Dαp = αp ;

(2) Dˆα,p(u,v)≥0 and ( , ) 0 ~

, u v ≥

Dαp ;

(3) Dˆα,p(u,v)=0 ⇔ u= ⇔v ( , ) 0 ~

, u v =

Dαp ;

(4) Dˆα,p(u,v)≤Dˆα,p(u,w)+Dˆα,p(w,v) and ( , )

~

, u v

D_α _p

) , ( ~ ) , ( ~

,

, uw D wv

D_α_p + _α_p

≤ ;

(5) Dˆα,p(u+w,v+w)=Dˆα,p(u,v) and ( , )

~

, u wv w

Dα _p + +

) , ( ~

, uv

Dα_p

= ;

(6) Dˆα,p(ku,kv)=|k|Dˆα,p(u,v) and ( , )

~

, ku kv

Dα p

) , ( ~ |

|k Dα,p uv

= .

Proof. It is obvious that (1) and (2) of the theorem hold.

By the definition of Dˆα,p , it is obvious thatu = v ⇒

0 ) , ( ˆ

, u v =

Dαp . Otherwise, let Dˆα,p(u,v)=0. Then we have 0 d ] |} ) ( ) ( | |, ) ( ) ( max{| [

1

0 1 ⋅ − − =

∫ ∑

r = u r v r u r v r r

p n

i αi i i i i .

Taking note of αi>0, we see that

∑

₌ ⋅ −

n

i i ui r

r( ₁α max{| ( )

0 |}) ) ( ) ( | |, )

(r u r −v r =

vi i i holds for r almost everywhere on

] 1 , 0

[ . Further, we have that

∑

n₌ ⋅ −

i 1αi max{|ui(r) vi(r)|, 0

|} ) ( ) (

| u_i r −v_i r = holds for r almost everywhere on [0,1], so

we can see that ui(r)=vi(r)and ui(r)=vi(r) holds for r

almost everywhere on [0,1] for i=1,2,_L,n. Therefore, we

obtain that r r

v u] [ ]

[ = holds for r almost everywhere on [0,1],

so we know that u=v holds by Lemma 2.1 in [18]. Thus,

0 ) , ( ˆ

, u v =

(5)

0 ) , ( ~

, u v =

Dαp ⇔ u=v, so (3) of the theorem holds. From

Theorem 3.1, we have

) , ( ˆ ) , ( ˆ

) d ] ) ] [ , ] ([ ˆ [ ( ) )] ] [ , ] ([ ˆ [ (

] d ) )] ] [ , ] ([ ˆ ) ] [ , ] ([ ˆ [ ( [

) d ] |} ) ( ) ( | |, ) ( ) ( max{| [

( ) , ( ˆ

, ,

/ 1 1

0

p /

1 1

0

/ 1 p 1

0

/ 1 1

0 1 ,

v w D w u D

r v w d r dr

w u d r

r v w d w u d r

r r v r u r v r u r

v u D

p p

p r r p

p r r

p r

r r

r

p p n

i i i i i i

p

α α

+ =

⋅ + ⋅

≤

+ ⋅

≤

− −

⋅ =

∫

∫ ∑

=

The proofs of D~α,p(u,v)≤D~α,p(u,w)+D~α,p(w,v), (5) and (6) can be similarly proved. 

Remark 3.1. From Theorems 3.1 and 3.2, we know dα dα

~ , ˆ and D,p D ,p

~ , ˆ

α

α are metrics on LC(Rn) and L(En), respectively,

and satisfy translation invariance, absolute homogeneity. Also, from the factor r of the integrands in the definitions of

) , ( ˆ

, uv

Dαp and ( , )

~

, u v

Dαp , we can see that the bigger the

degrees of the points are, which belong to the fuzzy n−cell numbers u and v, the greater the effects on the metric of u

and v. This is true in reality.

Example 3.1. Let u,v,w be the 2−cell numbers defined

by: u=(u1,u2), v=(v1,v2) and w=(w1,w2), where,

] 2 , 0 [ if 0

] 2 , 1 ( if 2

] 1 , 0 [ if ) (

⎪ ⎩ ⎪ ⎨ ⎧

∉ ∈ −

∈ =

x x x

x x

x

ui ,

] 4 , 2 [ if 0

] 4 , 3 ( if 4

] 3 , 2 [ if 2 ) (

⎪ ⎩ ⎪ ⎨ ⎧

∉ ∈ −

∈ −

=

x x x

x x

x vi

(i=1,2)

] 2 , 0 [ if 0

] 2 , 1 ( if 2

] 1 , 0 [ if )

( 1

⎪ ⎩ ⎪ ⎨ ⎧

∉ ∈ −

∈ =

x x x

x x

x

w ,

] 4 , 2 [ if 0

] 4 , 3 ( if 4

] 3 , 2 [ if 2 )

(

2

⎪ ⎩ ⎪ ⎨ ⎧

∉ ∈ −

∈ −

=

x x x

x x

x w

Then we know that u_i(r)=r , ui(r)=2−r , vi(r)=2+r ,

r r

v_i( )=4− (i=1,2),w₁(r)=r ,w₁(r)=2−r,w₂(r)=2+r

and w₂(r)=4−r for r∈[0,1]. From the definitions of D_L,

we have D_L(u,v)=2=D_L(u,w), i.e., D_L can not tell us the

difference of D_L(u,v) and D_L(u,w), so we say that D_L seems

to be ‘rough’ (similar proof for D). However, as a matter of

fact, D_L(u,v) and D_L(u,w) should have some difference.

Takingα =(1/2, 1/2), from the definitions of D,p ~

α , we can

obtain p _p

p v

u

D , ₁_/

) 1 (

2 ) , ( ~

+ =

α > ~ ( , )

) 1 (

1

, /

1 D u w

p p = αp

+ , this

accord with fact.

If we restrain the metric Dp (i.e. dp (⋅ ,⋅) defined by

Diamond in [3], see paragraph 1 of this section) on general −

n dimensional fuzzy number space n

E into on ( n)

E

L , then

it also becomes a metric on ( n)

E

L . In the following, we give

the relationships of the metrics D,p D,p ~ , ˆ

α

α and Dp.

Theorem 3.3. Metrics D,p D,p ~ , ˆ

α

α and Dp satisfy

(1) D,p≤D,p≤Dˆ ,p≤Dp≤D

~ ˆ 2 1

α α

α , i.e., ( , )

~ ) , ( ˆ 2 1

,

, u v D uv

D_α_p ≤ _α_p

) , ( ) , ( ) , ( ˆ

, uv D uv D uv

D _p ≤ _p ≤

≤ _α for any , ( n)

E L v

u ∈ (D is

discussed in [15]).

(2) D

p D p

D ,_p ₁_/_p _L ₁_/_p

) 1 (

1 )

1 (

1 ˆ

+ ≤ +

≤

α , i.e., Dˆα,p(u,v)

) , ( ) 1 (

1 ) , ( ) 1 (

1

/ 1 /

1 Duv

p v u D

p₊ p L ≤ ₊ p

≤ for any , ( n)

E L v

u ∈ .

Proof. For any , ( n)

E L v

u ∈ and r∈[0,1], by the definitions

of dˆα and dα

~

, we have

) ] [ , ] ([ ˆ

2

|} ) ( ) ( | |, ) ( ) ( max{| 2

)) ] [ , ] ([ ~ ( 2

| ) ( ) ( | | ) ( ) ( |

4

| ) ( ) ( | | ) ( ) ( | | ) ( ) ( | | ) ( ) ( |

2

| ) ( ) ( | | ) ( ) ( | | ) ( ) ( | | ) ( ) ( |

2 1

} | ) ( ) ( | |, ) ( ) ( max{| 2

1

) ] [ , ] ([ ˆ 2 1

1 1 1 1 1

r r n i

i i i i i

r r n

i

i i i i i n i

i i i i i i i i i n i

i i i i i i i i i n

i i i i i i

r r

v u d

r v r u r v r u

v u d r v r u r v r u

r v r u r v r u r v r u r v r u

r v r u r v r u v u d

α

α α

α α α α α

=

− −

⋅ ≤

= − + − ⋅ ≤

− + − + − + − ⋅ ≤

− − − + − + − ⋅ =

− −

⋅ =

∑

= = = = =

From this, we can directly obtain ˆ ( , ) ~ ( , ) 2

1

,

, uv D u v

D_α_p ≤ _α _p

) , ( ˆ

, uv

D_α_p

≤ .

By Theorem 4.4 in [15], we know dL ≤d≤ ndL, where, |}

| |, {| max ) ,

( ₁_i_n _i _i _i _i

L A B a b a b

d = ≤≤ − − for any =

∏

=

n i ai ai

A ₁[ , ]

) ( n

R LC

∈ and n₁[ , ] ( n)

i bi bi LC R

B=

∏

∈

= . Therefore, for any

) (

, n

E L v

u ∈ , we have

p p n

i i j n j j j j p p n

i i i i i i p

r r v r u r v r u r

v u D

/ 1 1

0 1 1

/ 1 1

0 1

,

) d ] |} ) ( ) ( | |, ) ( ) ( {| max [

(

) d ] |} ) ( ) ( | |, ) ( ) ( max{| [

(

) , ( ˆ

∫ ∑

= ≤≤

=

− −

⋅ ≤

− −

⋅ =

α α

α

) , (

) d ] ) , ( [ (

) d ] ) ] [ , ] ([ [sup (

)) , ( ( ) d ] ) ] [ , ] ([ [ (

) d ] ) ] [ , ] ([ [ (

/ 1 1

0

/ 1 1

0 [0,1]

/ 1 1

0

/ 1 n

1

i i

1

0

v u D

r v u D

r v u d

v u D r

v u d

r v

u d r

p p

p p r r r

p p p r r

p p r

r L

≤ = ≤

= ≤

⋅ =

∫

∑

∫

∈

=α

Thus, we also obtain Dˆα,p(u,v)≤Dp(u,v)≤D(u,v) and the

proof of (1) of the theorem is complete. From

p p n

i i i i i i p

r r v r u r v r u r

v u D

/ 1 1

0 1

,

) d ] |} ) ( ) ( | |, ) ( ) ( max{| [

(

) , ( ˆ

∫ ∑

= ⋅ − −

= α

(6)

) , ( ) 1 (

1

) d )( , (

) d ] ) , ( [ (

) d ] ) ] [ , ] ([ sup

[ (

) d ] ) ] [ , ] ([ [ (

/ 1

/ 1 1

0

/ 1 1

0

/ 1 1

0 [0,1]

/ 1 1

0

v u D p

r r v u D

r v u rD

r v u d r

p

p p

p p L

p p r r L r

p p r r L

+ = ≤ =

⋅ ≤

∫

∈

and Theorem 4.5 in [15], we can see that (2) of the theorem holds. 

Remark 3.2. From (1) of Theorem 3.3, we see

p p

p D D

D, , ˆ ,

~ ˆ 2 1

α α

α ≤ ≤ , i.e., Dˆα,p and D,p ~

α are equivalent, so we

know that Dˆα,p and D,p ~

α induce equivalent topologies on

) ( n

E

L by the knowledge of topological space.

IV. DIFFERENCE VALUES ON FUZZY n−CELL NUMBER SPACE In Section 3, we discussed metrics on ( n)

E

L . But sometimes

these have some shortcomings demonstrating the difference of two objects. For example, we consider that the degree of difference of 1 and 2 is bigger than the degree of difference of

10

10 and 1010₊1 _{though their metrics (Euclidean metric)}

measure both are 1. A mapping from the Cartesian product

X

X× of a set X into R needs to satisfy stronger conditions

in order that it can become a metric, and this brings limitations in some applications. The measure used to characterize the differences does not need to satisfy all metric conditions, for example, when we set up a method of pattern recognition basing on the principle of minimal difference (i.e. the principle of maximal likelihood), the measure used to characterize the differences does not need to satisfy all metric conditions. To conveniently set up methods of pattern recognition using fuzzy

−

n cell numbers, we introduce the concepts of difference

values on ( n)

E

L , and study the properties.

Let _u_∈_L(_En) _and n

n ∈R

= (α₁,α₂, ,α )

α _L satisfy

∑

= =

n

i1αi 1 and αi ≥0 ( i=1,2,L,n ). We denote

∑

=

∫

+

= n

i i rui r ui r dr

u

M ₁ 1

0 [ ( ) ( )]

)

( α

α , denote M(u)=Mα(u)

as (1,1, ,1)

n n

n L

=

α , and denote M(u)=M1(u) as u∈E.

Definition 4.1. Let u,v∈L(En) with

0 ) ( ),

(u M v ≥

Mα α and Mα(u)+Mα(v)≠0. We denote

a n

i i i i

a

v M u M

dr r v r u r v

u L

)] ( ) ( [

| ) ( ) ( | 2 )

,

( 1

1

0 ,

α α

α

+ −

=

∑ ∫

=

and

a n

i i i i

a

v M u M

dr r v r u r v

u R

)] ( ) ( [

| ) ( ) ( | 2 )

,

( 1

1

0 ,

α α

α

+ −

=

∑ ∫

=

and call L_α_,_a(u,v) and R_α_,_a(u,v) a left difference value and a

right difference value of u and v (with respect to the weight

α and parameter a), respectively. And we denote )] , ( ) , ( [ 2 1 ) ,

( , ,

,a u v Lαa u v Rαa u v

α = +

Δ

i.e.,

a n

i i i i i i

n

i i i i i i

a

dr r v r u r v r u r

dr r v r u r v r u r v

u

⎟ ⎠ ⎞ ⎜

⎝

⎛ ₊ ₊ ₊

− + − =

Δ

∑ ∫

= =

1 1

0 1

1

0 ,

)] ( ) ( ) ( ) ( [

|] ) ( ) ( | | ) ( ) ( ( | )

, (

α α

α

and call Δα,a(u,v) a difference value of

u

and

v

(with

respect to the weight α and parameter a ), where,

n n ∈R

=(α₁,α₂, ,α )

α _L with

∑

n₌ =

i 1αi 1 and αi≥0

( i =1,2,_L,n ) , and a∈(0,+∞) . Specially, we denote

) , ( ) ,

(u v ,a u v

a =Δ_α

Δ as (1,1, ,1)

n n

n L

=

α , and Δa(u,v)

) , ( , 1a u v

Δ

= as u,v∈E.

Remark 4.1. (1) Generally speaking, we consider that the

degree of the difference of two numbers is related not only by the metric of them but also by the sizes of them. As the metrics are the same, the bigger the sizes of the two numbers are, the smaller the degree of their difference is. The denominator

a

v M u

M ( ) ( )]

[ _α + _α in the definition of Δα,a just plays the action

(see Example 4.1), and the exponent a in a

v M u

M ( ) ( )]

[ α + α

can be properly chosen accordingly to the case in question. (2) Taking the note of that

∑

= ⎟⎟_⎠ =

⎞ ⎜⎜

⎝ ⎛

+ +

n

a i i

i

i _M _u _M _v

v M u M

1 ( ) ( ) 1

) ( ) (

α α

α holds as

) 1 , , 1 , 1 (

n n

n L

=

α and a=1, and

∑

= ⎟⎟_⎠ =

⎞ ⎜⎜

⎝ ⎛

+ +

n

a i i i

i M u M v

v M u M

1 ( ) ( ) 1

) ( ) (

α α

α

does not necessarily hold as (1,1, ,1)

n n

n L

≠

α or a≠1, from

∑

∫

∑

∫

∑

∫

∑

∫

= =

Δ ⎟⎟ ⎠ ⎞ ⎜⎜

⎝ ⎛

+ + =

⎟ ⎠ ⎞ ⎜

⎝

⎛ ₊ ₊ ₊

− +

− ⋅

⎟⎟ ⎟ ⎟

⎠ ⎞

⎜⎜ ⎜ ⎜

⎝ ⎛

+ + +

+ + + =

⎟ ⎠ ⎞ ⎜

⎝

⎛ ₊ ₊ ₊

− +

− =

Δ

= = =

n

i i a a i i i

a i i i i

i i i

i n

a n

i i i i i i i i i i i

a n

i i i i i i n

i i i i i i a

i i

v u v

M u M

v M u M

dr r v r u r v r u r v u

1 1

) , ( ) ( ) (

) ( ) (

)] ( ) ( ) ( ) ( [

|] ) ( ) ( | | ) ( ) ( [|

)] ( ) ( ) ( ) ( [

|] ) ( ) ( | | ) ( ) ( [| ) , (

1

0 1

1

0 1

1

0 1

1

0 ,

α α

α

α α

(7)

we can directly see that Δ1(u,v) is a convex combination of )

, (

1 ui vi

Δ , i=1,2,L,n , but Δα,a(u,v) is not necessarily a

convex combination of Δa(ui,vi) , i=1,2,L,n as )

1 , , 1 , 1 (

n n

n L

≠

α or a≠1.

Example 4.1. Let u,v,u′,v′ be the 2− cell numbers

defined by: u=(u1,u2) , v=(v1,v2) , u′=(u1′,u2′) and )

, (v1 v2

v′= ′ ′ , where,

⎪ ⎩ ⎪ ⎨ ⎧

∉ ∈ −

∈ =

] 2 , 0 [ if 0

] 2 , 1 ( if 2

] 1 , 0 [ if )

(

i i i

i i

x x x

x x

x u

⎪ ⎩ ⎪ ⎨ ⎧

∉ ∈ −

∈ +

− =

] 12 , 10 [ if 0

] 12 , 11 ( if 12

] 11 , 10 [ if 10 ) (

i i i

i i

x x x

x x

x v

⎪ ⎩ ⎪ ⎨ ⎧

∉ ∈ −

∈ +

− = ′

] 102 , 100 [ if 0

] 102 , 101 ( if 102

] 101 , 100 [ if 100 )

(

i i i

i i

x x x

x x

x u

⎪ ⎩ ⎪ ⎨ ⎧

∉ ∈ −

∈ +

− = ′

] 12 , 10 [ if 0

] 12 , 11 ( if 112

] 11 , 10 [ if 110 )

(

i i i

i i

x x x

x x

x v

and i=1,2 . Then we know that u_i(r)=r , u_i(r)=2−r , r

r

vi( )=10+ , vi(r)=12−r, ui′(r)=100+r, ui′(r)=102−r,

r r

v′_i( )=110+ , v′_i(r)=112−r (r∈[0,1], i=1,2). From the

definitions of D,p ~

α and Δα,a, we can obtain

]

p

p p p

p

dr r r r r

r r r

r r

v u D

/ 1

/ 1 2

1

0 1

,

) 1 (

20 2 1

|))] 12 2 | | 10 (|

|) 12 2 | | 10 (| ( [ 2 1

) , ( ~

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

+ =

+ − − + − − + ⎢⎣

⎡ ₋ ₋ ₊ ₋ ₋ ₊

=

_∫

α α α

]

p p

p p p

p

dr r r

r r

r v u D

/ 1

/ 1 2

1

0 1

,

) 1 (

20 2 1

|))] 112 102

| | 110 100

(|

|) 112 102

| | 110 100

(| ( [ 2 1

) , ( ~

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

+ =

+ − − + − − + +

⎢⎣

⎡ ₊ ₋ ₋ ₊ ₋ ₋ ₊

=

′ ′

∫

α α

α

a

a a

dr r r r r r dr r r r r r

v u

12 10

) 12 2 10 ( )

12 2 10 (

|) 12 2 | | 10 (| |)

12 2 | | 10 (|

) , (

1 0 2 1

0 1

1 0 2 1

0 1

,

=

⎟ ⎠ ⎞ ⎜

⎝

⎛ ₊ ₊ ₊ ₋ ₊ ₋ ₊ ₊ ₊ ₊ ₋ ₊ ₋

+ − − + − − +

+ − − + − − =

Δ

∫

α α

α

a

a a

dr r r r r r dr r r r r r

v u

212 10

) 112 102 110 100 ( ) 112 102 110 100 (

|) 112 102 | | 110 100 (| |) 112 102 | | 110 100 (|

) , (

1 0 2 1

0 1

1 0 2 1

0 1

,

=

⎟ ⎠ ⎞ ⎜

⎝

⎛ ₊ ₊ ₊ ₊ ₋ ₊ ₋ ₊ ₊ ₊ ₊ ₊ ₋ ₊ ₋

+ − − + − − + +

+ − − + − − + =

′ ′ Δ

∫ ∫

α α

α

so we have D~α,p(u,v)=D~α,p(u′,v′) and Δα,a(u,v)>Δα,a(u′,v′).

Property 4.1. Let u,v∈L(En) with Mα(u), Mα(v)≥0 and

0 ) ( )

(u +M v ≠

Mα α , α=(α1,α2,L,αn)∈Rn with

∑

= =

n i1αi 1

and α_i>0 (i=1,2,_L,n), and a∈(0,+∞). Then

(1) Δ_α_,_a(u,v)≥0;

(2) Δ_α_,_a(u,v)=0 if and only if u=v;

(3) Δ_α_,_a(u,v)=Δ_α_,_a(v,u);

(4) Δα,a(u,w)≥Δα,a(v,w) for any ( )

n

E L

w∈ and u≤v≤w;

(5) Δα,a(u+w,v+w)≤Δα,a(u,v) for any ( )

n

E L

w∈ and

) 0ˆ , , 0ˆ , 0ˆ

( L

≥

w ;

(6) ( , ) , ( , )

1

, ku kv k a u v

a

a α

α = Δ

Δ − _{for any} _>₀

k .

Proof. It is obvious that the conclusions (1) and (3) hold.

The proof of conclusion (2) can also be completed by imitating the proof of (3) of Theorem 3.2 by using Lemma 2.1 in [18].

From u≤v≤w, we know that u_i(r)≤v_i(r)≤w_i(r) and

) ( ) ( )

(r v r w r

ui ≤ i ≤ i ( i=1,2,L,n ), so |ui(r)−wi(r)|

| ) ( ) ( |vi r −wi r

≥ and|u_i(r)−w_i(r)|≥|v_i(r)−w_i(r)|. Therefore,

we have

∑ ∫

= =

− + − ≥

− + −

n

i i i i i i n

i i i i i i

dr r w r v r w r v r

dr r w r u r w r u r

1 1

0 1

1

0

|] ) ( ) ( | | ) ( ) ( [|

α α

On the other hand, from ui(r)≤vi(r)≤wi(r) and

) ( ) ( )

(r v r w r

ui ≤ i ≤ i (i=1,2,L,n), we can also see that Mα(u)

) ( )]

( ) ( [ )]

( ) (

[ ₁ 1

0 1

1

0ru r u r dr r v r v r dr M v

n

i i i i n

i αi i + i ≤ α + = α

=

∑ ∫

₌

∑ ∫

₌ ,

so we can obtain 0≤Mα(u)≤Mα(v)≤Mα(w). Thus, we have

(

)

a

n

i i i i i i a

w M u M

dr r w r u r w r u r w

u

) ( ) (

|] ) ( ) ( | | ) ( ) ( [| )

,

( 1

1

0 ,

α α

α

+

− + − =

Δ

∑ ∫

=

(

)

) , (

) ( ) (

|] ) ( ) ( | | ) ( ) ( [|

, 1

1

0

w v

w M v M

dr r w r v r w r v r

a

a n

i i i i i i

α

α α

α

Δ =

+

− + −

≥

∑ ∫

=

so conclusion (4) holds.

From w≥0ˆ, we know M_α(w)≥0, so we have

( )a

n

i i i i i i

a n

i i i i i i i i i i

n

i i i i i i i i i i

a n

i i i i i i

n

i i i i i i

a

w M v M u M

dr r w r v r w r u r w r v r w r u r

dr r w v r w u r w v r w u r

w v w u

) ( 2 ) ( ) (

|] ) ( ) ( | | ) ( ) ( [|

)] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( [

|] ) ( ) ( ) ( ) ( | | ) ( ) ( ) ( ) ( [|

)] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( [

|] ) ( ) ( ) ( ) ( | | ) ( ) ( ) ( ) ( [|

) , (

1 1 0 1

1 0 1

1 0 ,

α α α α

α α α

α α

+ +

− + − =

⎟ ⎠ ⎞ ⎜

⎝

⎛ ₊ ₊ ₊ ₊ ₊ ₊ ₊

− − + + − − + =

⎟ ⎠ ⎞ ⎜

⎝

⎛ ₊ ₊ ₊ ₊ ₊ ₊ ₊

+ − + + + − + =

+ + Δ

∑

∫

∑

∫

∑

∫

∑

∫

∑

∫

= = =