Multifactorial fuzzy sets and multifactorial degree of nearness

Fuzzy Sets and Systems 19(1986) 291-297 North-Holland

MULTIFACTORIAL

FUZZY

291

SETS

AND

M U L T I F A C T O R I A L D E G R E E OF N E A R N E S S LI Hongxing Section of Mathematics, Tianjin Institute of Textile Engineering, Tianfin, China Received January 1985 Revised May 1985 Two new concepts, Multifactorial fuzzy sets and multifactorial degree of nearness, are advanced. First. an axiomatic definition for multifactorial functions is given, and multifactorial fuzzy sets and multifactorial degree of nearness are defined by using the multifactorial function. Second, three direct methods of multifactorial pattern recognition and two methods for fuzzy clustering with fuzzy characteristics are given. Last, we advance an interesting open problem.

Keywords: Multifactorial function, Multifactorial fuzzy sets, Degree of nearness, Multifactorial degree of nearness, Pattern recognition, Clustering.

1. Introduction Suppose n standard patterns with m fuzzy characteristics are known, w h e r e the m fuzzy characteristics of the i-th p a t t e r n are A~ (j = 1. . . . . m), which are fuzzy sets on universe Uj (j = 1 , . . . , m), i.e. Air e F(Uj) (j = 1. . . . . m), respectively. Put U* = U1 x U2 x • • • x U,,,. If Uo e U* is an object to be recognized, then we have:

Problem 1. H o w to recognize to which of the n patterns u o relatively belongs? If an object to be recognized has also m fuzzy characteristics (j = 1, . . . , m), then we have:

BieF(Uj)

Problem 2. H o w to recognize to which of the n patterns the pattern to be recognized is relatively closest? L e t X = {xl . . . . . Xn} be a set of objects to be clustered, every object x i has m fuzzy characteristics Aij e F(U~) (j = 1 . . . . . m ) . T h e n we have: P r o b l e m 3. H o w to establish a fuzzy similarity matrix for X ? Put F , , = { / t = ( A , , . . . , A , , , ) I A j e F ( U j ) , j = I . . . . . m}, i.e. F , , , = F ( U , ) x F(U2) x • • • × F ( U , , ) . T h e n any e l e m e n t .,i in F,,,_is a generalized fuzzy vector. If we put ,'i i = ( A i l . . . . . A~,,,) (i = 1. . . . . n), and B = (B~ . . . . . B,,), then: 0165-0114/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

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Problem 1 means how to recognize to which one of A 1 , . . . , fi~,, u0 relatively belongs. Problem 2 means how to recognize to which one of ,4t . . . . . ,4,,/) is relatively closest. Problem 3 means how to establish a fuzzy similarity matrix on X in accordance with the fuzzy characteristics Ai of xi e X (i = 1 , . . . , n)?

2. Multifactorial function In [0, 1]" a partial ordering '~<' is defined by X<~Y

iff x j < ~ y i ( j = l . . . . .

m)

where X = (xl . . . . . x,,), Y = (Yl . . . . . Ym) ~ [0, 1]". It is easy to see that ([0, 1]", ~<) is a complete distributive lattice with the greatest element I = (1 . . . . . 1) and the least element 0 = (0 . . . . . 0).

Definition 2.1. A mapping f : [0, 1]"--* [0, 1] is called a multifactorial function i f f satisfies (m.1) (m.2)

X <~ r implies f ( x ) <-f(Y), minj{xj} <~f(X)<~ maxj{xj}.

Proposition 2.1. Multifactorial functions satisfy (i) (xj = a, j = 1. . . . , m) implies f ( X ) = a, (ii) f(O) = O, f(I) = 1. Examples. The following functions [0, 1]"--* [0, 1] are multifactorial functions:

A:X~ y=! ~ xj, V:x~

~ xj,

j=l it,!

2:x Ecx, j=l

whereaie[O, 1]and ~ a j = l ,

Vl :x~

.Yl" (ajxj)

V2:X~

.="j91( a j ^ x j )

/=1

where aj e [0, 11"' and i=l Q aj = 1, where aj~ [0, II and m, Q aj=l.

Put H - - {f:[0, 1]'--~[0, 1] If satisfies (m.1), (m.2)}. In H a partial ordering '<~' is defined by f~ ~
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Proposition 2.2. (H, <-) is a complete distributive lattice, and /~ is the least element and V is the greatest element in H.

3. Multifactorial fuzzy sets In F,,,, the inclusion relation ' c ' and the equality relation ' = ' are defined as follows: fi.c/)

iff A j c B j O ' = I

.....

fI=B

iff , ~ c B a n d B = f i ~ .

m),

If ' = ' is regarded as a partially o r d e r e d relation, then it is easy to prove:

Proposition 3.1. (F,,,, c ) is a complete distributive lattice, and 0 = (~. . . . . O) is the least element and i = ( U1. . . . .

U,,,) is the greatest element in 17,,.

Definition 3.1. For any fi, e F,,,, a fuzzy set on U* is constructed by ,,i and d e n o t e d by

A ~ ( A ) ~- (A, . . . . .

A.,).

A is called a multifactorial fuzzy set if

tAA(u) = f(#A,(Ul) . . . . . where f e H and u = (u~ . . . . . denoted by F(U*, f).

/AA,,,(Um)) u,,,) e U*. T h e set of all multifactorial fuzzy sets is

Proposition 3.2. (i) O, U* ~ F(U*, f) = F(U*). (ii) J : F,,,~ F ( U*, f), 7t ~ ( ft ) , is an ordered-preserving mapping from ( F,,, c ) to (F(U*,f), =).

Theorem 3.1. (F(U*, f), tJ, N ) is a d&tributive lattice with greatest elment U*, and least element O. Proof. T h e distributivity is inherited to F(U*, f ) by F(U*). Now we prove that F ( U * , f ) is a lattice. VA, B e F ( U * , f ) , 3.4, B, such that A = (,4), B = ( B ) . Write C __4A t3 B, we only need to prove 3 C e F,,, such that C = ((~). In fact, we take (~ = (C1 . . . . . Cm) e F,,,, which satisfies

[(~AI(Ul) . . . . . #am(Urn)), (IZcl(Ul)'''" IZc"(u"')) = I.(/~B~(ul) , , ~tB,,,(u,,,)),

]AA(U)~ ~'B(U), otherwise.

It is easy to check C = ((~), thus C e F(U*,f). That A N B e F ( U * , f ) can be proved in the same way, so F(U*, f) is a lattice. [] T h e t h e o r e m describes an algebraic structure for multifactorial fuzzy sets.

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An equivalence relation '~" on F,. may be determined by the mapping J: ~/}

iff J(.4) = J(/1)

Hence we obtain a quotient set of F,.:

F'~-F~/~ &{(A) IAeFm) where (A) is an equivalence class for .4. In F~, the operations U ' and N ' are defined by

(C) = (.4) u '(B) iff ( 0 ) = (A) U (/}), (C) = (.4) n'(B) iff ( 0 ) = (A) f) (/~). It is easy to prove:

Proposition 3.3. (F'~, U', A ' ) and (F(U*, f), U, A) are two algebraic systems of isomorphism. The proposition shows that a multifactorial fuzzy set is regarded as an equivalence class of generalized fuzzy vectors. Proposition 3.4. Let A = (,4 ), B = { B ),

C=(A1UB Then (1) A U B c C ,

1. . . . .

AmUB,.),

D= (AINB ,.....

A,,,NB,,,).

ANB~D;

(2) A;% ~,a,4u~) -< ~,.:(u) ~ V?=~ ~,A;(.,). Proof. (1) By using (re.l) we have

~A.(U,.)) V f(l~B,(U,) . . . . .

=f(/~a,(U,) . . . . .

<~f(l~a~(Ua) V I~BI(Ul) . . . . . = f(I.tA,UB,(Ul) . . . . .

UB.(U,.)) t~A~(U,. ) V ~B.(Um))

,tt.a~we.(Um)) = ~ c ( U )-

That I~AnB(U ) >1 #D(U) can be proved in the same way. (2) By using (m.2) we have /L4,(u) = 1 - # A ( U ) = 1 -- f(#A,(Ul) . . . .

>- 1

j=l

UA,(U/)

A j=l

(1 --

, #A.(U,.))

~.£A/(Uj)) ~ ~.LA~(Uj) j=l

That/~,~,(u) ~< V~'=I IL~7(ui) can be proved in the same way. Note. If f = A , then A N B = D and

~,Ao(u)= ~/ ~,,,:(u~). i=1

[]

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Mult~fuctorial .fuzz), sets

If f = V , t h e n A U B = C a n d m

If f = ~, then A ~= (A~ . . . . . A~). 4. Method for solving Problem 1 Let .2,1. . . . . fi'n • F , , be n known patterns, Uo• U* be an object to be recognized. Since the patterns differ from the object on levels of concepts (the former are generalized fuzzy vectors, but latter is an element in U*), this problem of pattern recognition is called multifactorial pattern recognition of distinct levels. Based on the principle of highest g a d e of membership [6] we have:

Direct method for multifactorial pattern recognition on distinct levels: Take A i = (fi, i) (i = 1. . . . .

n). If 3i • {1 . . . . .

ItA,(U0) = max{/tal(/2o) . . . . .

n} such that

/~A~(U0)}

then it is decided that u o belongs to Ai. 5. Multifactorial degree of nearness For solving Problem 2, first we solve the problem of nearness for two generalized fuzzy vectors. Definition 5.1. Let N be degree of nearness which is defined by axiomatization, f • H. The mapping

N*: F,~ x F.,---~[O, 11 (A, [~),--~N*(A, B)=f(N(A~, BI) . . . . . N(Am, B,,)) is called multifactorial degree of nearness. Theorem 5.1. N* satisfies the axioms of degree of nearness [6], i.e. (i) N*(A, A) = 1, N*(L 0) = 0; (ii) N*(A, B) = N*(/}, A); (iii) .A ~ / ~ ~ C implies N*(A, C) < g*(.4, B) A N*(S, C). Proof. (i) N*(A.A_) =f(N(A~, A1) . . . . . N(Am, A.,)) = f ( 1 . . . . . proved that N*(1, e) = 0 in the same way. (ii)

1) = 1. It can be

N*(A, ~) =f(m(A. B,) . . . . . N(Am, B.,)) =f(N(B~, A,) . . . . . N(Bm, Am)) = N*(B, .4).

(iii).zic/~cC implies A j c B j ~ Q ( ] = 1 . . . . . m) implies N(A L,G) <~ N(A v Bj) A N(Bj, Q) (j = 1. . . . . m) implies N*(./i, C) ~ N*(A, B) ^ N*(B, C).

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6. Methods for solving Problem 2 Let A~. . . . , ~,l, e F,,, be n known patterns and /~ e F,, be a pattern to be recognized. Since the known pattern and the pattern to recognized are the same level in the concept (they are all generalized fuzzy vectors), this problem of pattern recognition is called multifactorial pattern recognition of the same level. Based on the principle for choosing degree of nearness [6] we have: Direct m e t h o d 1 o f multifactorial pattern recognition o f the same level: Let N be a degree of nearness, take A i = (,'i,i) (i = 1. . . . . n), B = (/3). If 3i e {1 . . . . . n} such that N(A~, B) = max{N(Al, B1) . . . . .

N ( A , , 13,)}

then it is decided that/} is closest to .4~. Direct m e t h o d 2 o f multifactorial pattern recognition o f the same level: If 3 i e {1 . . . . . n} such that N*(TI,, B) = max{N*(A,,/~) . . . . .

N*(.A,,, B)}

then it is decided that/} is closest to ,4 i.

7. Method for solving Problem 3 Based on multifactorial fuzzy sets we have: M e t h o d 1: Let N be a degree of nearness, take A / = (,4;) (i = 1. . . . . similarity coefficient of A; and Aj may be calculated by rii = N(Ai, Ai),

n). The

i, j = l, . . . , n.

Based on multifactorial degree of nearness we have M e t h o d 2: The similarity coefficient of A,. and Aj may be calculated by rii= N*(]i,, ]ii),

i, j = l . . . . .

n.

8. An interesting open problem In Section 6 we have given two direct methods for multifactorial pattern recognition of the same level: Method 1 and Method 2. Thus we have:

Open problem. Is Method 1 equivalent to Method 2?

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References [1] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms (Plenum Press, New York, 1981). [2] D. Dubois and H. Prade, Fuzzy Sets and Systems (Academic Press, New York, 1980). [3] A. Kandel, Fuzzy Techniques in Pattern Recognition (John Wiley & Sons, New York, 1982). [4] Li Hongxing, Stability of solutions of fuzzy relation equations, BUSEFAL 20 (1984)106-114. [5] Li Hongxing, Inverse problem of fuzzy multifactorial decision, Sino-U.S. Symposium on Fuzzy Methodology and Modern Decision with Application to Electric Power System, Vol. 7 (1984) (in Chinese). [6] Wang Peizhuang, Fuzzy Sets Theory and its Applications (Shanghai Scientific and Technical Publishers, 1983) 91-94 (in Chinese). [7] You Zhaoyong, Methods for constructing triangular norm, Fuzzy Mathematics 1 (1983) (in Chinese) 71-78. [8] Zhang Wenxiu and Le Huffing, Fuzzy truth possibility degree, Fuzzy Mathematics 1 (1984) (in Chinese) 7-13.