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

M U L T I F A C T O R I A L F U Z Z Y S E T S A N D 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 Hongx ing 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, Multifac- torial degree of nearness, Pattern recognition, Clustering.

1. Introduction

Suppose n s tandard pa t te rns with m fuzzy characterist ics are known, where the m fuzzy characteris t ics of the i-th pa t t e rn 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 objec t to be recognized, then we have:

Problem 1. H o w to recognize to which of the n pa t te rns u o relatively belongs?

I f an objec t to be recognized has also m fuzzy characterist ics B i e F ( U j ) (j = 1, . . . , m) , then we have:

Problem 2. H o w to recognize to which of the n pa t te rns the pa t te rn to be recognized is relat ively closest?

Le t X = {xl . . . . . Xn} be a set of objec ts to be clustered, every objec t x i has m fuzzy characteris t ics Aij e F(U~) ( j = 1 . . . . . m ) . Then we have:

P r o b l e m 3. H o w to establish a fuzzy similarity matr ix 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 emen t .,i in F,,,_ is a general ized fuzzy vector . I f 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)

292 Li Hongxhlg

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) X <~ r implies f ( x ) <-f(Y), (m.2) 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~ ~ xj, y=!

V : x ~ ~ xj, j=l

i t , !

2:x Ecx, wherea ie [O , 1 ]and ~ a j = l , j = l /=1

V l : x ~ .Yl" (ajxj) where aj e [0, 11"' and i=l Q aj = 1,

V 2 : X ~ .="j91 (a j ^x j ) where aj~ [0, II and m, Q a j = l .

Put H - - {f:[0, 1]'--~[0, 1] I f satisfies (m.1), (m.2)}. In H a partial ordering '<~' is defined by

f~ ~<fz iff (VX)(f~(X) ~fz(X)) .

It is easy to prove:

Mttlttfactorial fuzzy sets 293

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 equali ty relation ' = ' are defined as follows:

f i . c / ) iff A j c B j O ' = I . . . . . m),

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

If ' = ' is regarded as a partially o rdered relat ion, 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 = ( U 1 . . . . . U,,,) is the greatest element in 17,,.

Definit ion 3.1. For any fi, e F,,,, a fuzzy set on U* is constructed by ,,i and deno ted by

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

A is called a multifactorial fuzzy set if

tAA(u) = f(#A,(Ul) . . . . . /AA,,,(Um))

where f e H and u = (u~ . . . . . u,,,) e U*. The set of all multifactorial fuzzy sets is denoted by F(U*, f) .

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) , t J, N ) is a d&tributive lattice with greatest elment U*, and least element O.

Proof . The distributivity is inheri ted 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 __4 A 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)), ]AA(U) ~ ~'B(U), ( IZcl (Ul) ' ' ' " IZc"(u"')) = I.(/~B~(ul) , , ~tB,,,(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. []

The theo rem describes an algebraic structure for multifactorial fuzzy sets.

294 Li Hongxing

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) (C) = (.4) n ' (B)

It is easy to prove:

iff ( 0 ) = (A) U (/}) ,

iff ( 0 ) = (A) f) (/~).

Proposition 3.3. i somorphism.

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 = ( A 1 U B 1 . . . . . A m U B , . ) ,

Then (1) A U B c C , A N B ~ D ; (2) A;% ~,a,4u~) -< ~,.:(u) ~ V?=~ ~,A;(.,).

Proof. (1) By using (re. l) we have

=f(/~a,(U,) . . . . . ~A.(U,.)) V f(l~B,(U,) . . . . . UB.(U,.))

<~f(l~a~(Ua) V I~BI(Ul) . . . . . t~A~(U,. ) V ~B.(Um))

= f(I . tA,UB,(Ul) . . . . . , t t . 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) . . . . , #A.(U, .))

>- 1 UA,(U/) A ( 1 - - ~.£A/(Uj)) ~ ~.LA~(Uj) j=l j=l 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

(F'~, U', A ' ) and (F(U*, f ) , U, A) are two algebraic systems o f

D = ( A I N B , . . . . . A , , ,NB, , , ) .

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~).

295

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 . . . . . n} such that

ItA,(U0) = max{/ ta l ( /2o) . . . . . /~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 . . . . . 1) = 1. It can be proved that N*(1, e) = 0 in the same way.

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

=f(N(B~, A,) . . . . . N(Bm, Am)) = N*(B, .4). ( i i i ) . z i c / ~ c C 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).

296 Li Hongxing

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 me thod 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 me thod 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:

Method 1: Let N be a degree of nearness, take A / = (,4;) (i = 1 . . . . . n). The similarity coefficient of A; and Aj may be calculated by

rii = N(Ai , Ai) , i, j = l, . . . , n.

Based on multifactorial degree of nearness we have

Method 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?

M,dtifactorial fuzzy sets 297

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.