Complement of fuzzy k-partitions

Fuzzy Sets and Systems 62 (1994) 175-184 North-Holland

175

Complement of fuzzy k-partitions Slavka Bodjanova Department of Mathematics, Texas A&M University, Kingsville, TX USA Received April 1993 Revised August 1993

Abstract: We define the unary operation of complementation on the set of all fuzzy k-partitions of a finite set of objects. We show that for k = 2 the complement of fuzzy k-partition results in Zadeh's complementation of fuzzy sets. Some further properties of the complement of fuzzy k-partitions are investigated.

Keywords: Fuzzy k-partitions; complement of fuzzy sets; complement of fuzzy k-partitions; sharpness of fuzzy k-partitions.

1. Introduction A fuzzy k-partition of a finite data set X can be considered as a special collection of fuzzy sets on X. The algebraic structure of the set of all fuzzy sets has been studied by numerous authors, see e.g. [1, 7, 10]. However, there are only a few works concerning the structure and mathematical properties of fuzzy partitions [3, 4, 8]. Some of the quantitative characteristics, relations and operations defined on fuzzy sets can be extended to fuzzy partitions. In this paper we present extension of sharpness relation introduced on fuzzy sets by De Luca and Termini [6]. We prove that the set of all fuzzy k-partitions of a set X is partially ordered by this extended relation. Unfortunately, this partial order does not insure the existence of infimum or nontrivial supremum for any two fuzzy k-partitions of X. Therefore we cannot derive from the partial order the binary operations of union and intersection of fuzzy k-partitions. However, using some sharper versions of a fuzzy k-partition U, we can derive a fuzzy partition - U , which has meaningful properties that should be requried for a complement of a fuzzy partition U. The idea of complementation was originally introduced in [5]. In this paper we provide a rigorous mathematical definition of complementation and we study the connection between complementation and the approximation of fuzzy partitions by their sharper versions.

2. K-partitions Let Vk. denote the usual vector space of real k x n matrices. The hard k-partition space Pk associated with a finite set of objects X = {Xl, x2 . . . . , x.} was defined by Bezdek [3] as follows:

Pk = {U~ Vg.;uijE {0, 1}; ~ uij = 1 for all j; ~ui~>Oforalli). i

(1)

j

The fuzzy k-partition space Pc associated with a finite set of objects X = {x~, x2 . . . . . x.} was defined by

Pc = ~(U E Vk.; Uij E [0, 1]; E Uij = 1 for all j; E uij>O for all i}. i

i

Correspondence to: Dr. S. Bodjanova, Department of Mathematics, Texas A&M University, Kingsville TX 78363, USA. 0165-0114/94/$07.00 ~ 1994---Elsevier Science B.V. All rights reserved SSDI 0165-0114(93 )E0236-L

(2)

176

S. Bodjanova / Complement of fuzzy k-partitions

In general, the value uij = ui(x i) specifies the membership of the object xj • X in the ith cluster. The supersets of Pk and P/k obtained by the condition for all i

0 <- ~ uij J

(3)

are denoted by Pk0 and Pp,0, and are called, respectively, the degenerated hard and degenerated fuzzy k-partition space of X. Definition 1. Let U, V • Pik0" We say that V is a sharper version of U, denoted by V < U, if and only if vii ~< uij,

for uij <~ 1/k,

(4)

vii >>-uii,

for u~i >! 1/k.

(5)

and

Note. For k = 2 we get the relation sharpness defined by De Luca and Termini [6]. Theorem 1. P17,o is partially ordered by the relation <. Proof. We need to prove that the relation < satisfies the reflexive, antisymmetric and transitive law. (i) Let U • PJk,,. Then for all i, j we have uij <~ uii for uii <<-1 / k and uii i> uii for uij >1 1/k. Hence U < U and < is reflexive. (ii) If U < V then uii <~vii for vii <~ 1 / k and uij >i vii for vii >t 1/k. If V < U then vii ~< uij for uii <~ 1 / k and vii >i uii for uij >i 1/k. Hence uii = vii for all i, j, therefore U = V and < is antisymmetric. (iii) If U < V and V < W then uij~vii<<-wij for w i i < - l / k and uij>~vij>~wij for wgi>~l/k. Therefore U < W and < is transitive. Theorem 2. Let U E P~o. Let D • Pfko be the partition f o r which uij = 1 / k f o r all i, j. Then U < U. The proof is obvious. The following sharpened versions of U • P~o will be of use for us. Definition 2. Let U, V E Pp,0 be partitions of a set X. We say that V is a linearly sharpened version of U if and only if for each xj • X there exists a constant ci ~> 1 such that vii - ~l = q

(')

uii - ~

for all i.

(6)

Definition 3. Let U, V e P~o. We say that V is a maximal linearly sharpened (MLS) version of U, denoted by >> U, if and only if V is a linearly sharpened version of U and for all x i E X, v,.i = 0 whenever uri = mini uii < 1 / k and v~i = 1 / k whenever Urj = mini uii = 1/k.

Note. >>/3 =/.J. Theorem 3. Let U ~ P~o. Let urj = mini uii. Then a MLS version o f U is given as follows: For all i, j: 1 >> Uii = -k ~ 1 - kurj

>> uij = uij

uij -

, i f Uri < ~ ,

(7)

otherwise.

(8)

Proofi If is obvious that >> U is a linearly sharpened version of U because, for each j, there exists a constant cj i> 1 such that >> uij - 1 / k = Q(uii - 1/k). We have, for all j: if uri = mini uij < 1 / k then >> uri = 0; if Urj = mini uii = 1 / k then >> uri = 1/k.

S. Bodjanova / Complement of fuzzy k-partitions

177

>> U is a fuzzy k-partition, because (>> uq)/> (>> Urj) >" O, for all i, j, and (>>uq)

~ (~ =

1

( ~)) . Uij .

. q- . 1-ku~i

k k

1 + - -

1-ku,

(~/

uq-

~)

=1

for allj.

Example 1. Let Uj, Vj, Wj, Zj represent the following partitioning of object xj E X into four classes with respect to the partitions U, V, W, Z E P¢~o.

I°3104 .03204 0.2

u, =

/0.175

,

LO.lA

= t

'

L0.025

=

0.401

[03331

0.101

/0.167/

0.45 1' 0.053

Zj = / 0.500 1" LO.OOOA

We can check that Vj, ~ , Zj are sharpened versions of ~ , that Vj, Z j are linearly sharpened versions of Uj and that Z~ is MLS version of ~.

3. Complementation on Pfko Let X = {xl, X 2 , • • • , Xn} be a set of objects, P ( X ) the set of all ordinary subsets of X and F ( X ) the set of all fuzzy subsets of X. Let u E F ( X ) . According to Zadeh [11], the complement of u is the fuzzy set c ( u ) ~ F(X) which assigns a value c ( u ) ( x ) = 1 - u ( x ) to each membership grade u ( x ) . The fuzzy set c ( u ) represents the negation of the concept represented by u. For example, if u is the fuzzy set of young men, its complement c ( u ) is the fuzzy set of men who are not young. Suppose that we would like to classify the elements of a set X (e.g. a set of men) according to their age using the categories young (fuzzy set ul), medium age (fuzzy set u2) and old (fuzzy set u3). Obviously u f f x ) + u2(x) + u3(x) = 1 for each x ~ X. Suppose that xl ~ X is young. Then U l ( X 1 ) = 1, u2(xl) = 0 and u3(xl)= 0. What would be the classification of x2 ~ X, if x2 were not young? That is, what would be the complementary classification ~ uffxl) = ul(x2), ~ u2(x0 = u2(x2), - u3(xl) = u 3 ( x 2 ) ? According to our categories of age, the concept 'not young' means medium age as well as old age. Therefore, we will assign ~ u l ( x l ) = 0, - u2(xl) = 0.5 and ~ u3(xl) = 0.5. Suppose that Ul(Xl) = 0.7, u2(xl) = 0.25 and u3(xl) -- 0.05. What would be the classification of object x2 ~ X, if x2 were not young to the degree 0.7, not medium age to the degree 0.25 and not old to the degree 0.05? Obviously, in the complementary classification - u f f x l ) , ~u2(xl), - u 3 ( x l ) the large coefficients (coefficients t> 1) should be replaced by small coefficients (coefficients ~<~) and vice versa. Also, the rank of coefficients should be reversed. Hence - u 3 ( x l ) > ~ U 2 ( X 1 ) > - U3(XI); 4. /XI(X1) ~ l , while ~ u2(xl) I> ~ and ~ u3(x0 t> ~. Of course, ~ u l ( x l ) + ~ u2(xl) + ~ u3(xl) -- 1 for all x ~ X. How can one find the unique representation of ~ u f f x O , ~ u2(xl), - u 3 ( x 0 ? Or, more generally, how can one find the complement of a fuzzy partition U ~ P~,o? Let W = (w~, w2) where wl, w2 ~ P ( X ) denote a hard partition of X into two disjoint clusters. It is obvious that the complement of W is the hard partition ~ W = ( ~ wl, ~ w2) = ( X - Wl, X - w2). Using the characteristic function q~:P(X)~{0, 1} we have for all x ~ X: q~(- wi(x)) = 1 - qffw,(x))

for i = 1, 2.

(9)

Let U = (Ul, u2), where ul,/~2 • F ( X ) denote a fuzzy partition of X into two fuzzy clusters. The complement of U is the fuzzy partition - U consisting of the fuzzy clusters ~ ul, ~ u2 ~ F ( X ) , where, according to Zadeh's definition of complementation, for all x E X u,(x) = 1 - u , ( x )

for i = 1, 2.

(10)

S. Bodjanova / Complementof fuzzy k-partitions

178

If k > 2 then the complementation (9), resp. (10) cannot be used as the definition of the complement of a hard or fuzzy k-partition. Example 2. Let U ~ ek3o be a partitioning of X = {Xl, x2, x3, U=

1 1 0 0

X4} given by the matrix

.

Intuitively, we cannot find a complement of U, i.e. the partition - U , which would be a hard partition. For example, the object Xl should belong to the cluster - u 2 and ~ u3, the object x2 to the clusters ~ Ul and - u3, etc. Therefore, the complement of U should be the fuzzy partition - U given by the matrix 0 0.5 0.5

U=

0.5 0 0.5

0.5 0 0.5

0.5) 0.5 . 0

Let us consider the fuzzy partition V e Ps~ogiven by the matrix 0.9 0.3 0.1 0.2 0.0 0.5

V=

0 0.4) 1 0.2 . 0 0.4

If we suppose that u o is high if uij > i / k , and u o is low if uij < 1/k, then a complement - U should satisfy the following conditions: 1

~v,j < ;

1

1

1

iff vii > ; ,

1

~viJ=k

(11) (12)

1

iffviJ

(13)

k"

Of course, conditions (11)-(13) do not define ~ V uniquely. Definition 4. Let V, U ~ Prk0" We say that V is a complementary sharpened version of U, denoted by

V
1 if uij>~-~,

(14)

and 1

vi;>~-~

1

ifuo<~ ~.

(15)

It is obvious that we can define the complementary linearly sharpened version of U e PIk0 as follows. Definition 5. Let U, V E P~0 be partitions of a set X. We say that V is a complementary linearly

sharpened version of U if and only if for each xj ~ X there exists constant dj < 0 such that

,

(')

vii - -~ = dj uij - -k

for all i.

(16)

S. Bodjanova / Complement of fuzzy k-partitions

179

Definition 6. Let U, V E P~,o be partitions of a set X. We say that V is a complementary MLS version of U, denoted by << U, if and only if V is a complementary linearly sharpened version of U and for all xj E X , Vsj = 0 whenever u~j = maxi u 0 > 1 / k and v~j = 1 / k whenever u~ = maxi u 0 = 1/k. Note. << 0

= lJ.

Theorem 4. L e t U ~ Pp, o. L e t Usj = maxi u 0. T h e n the c o m p l e m e n t a r y f o l l o w s : F o r all i, j: 1

<< u0 =

__1

~ + 1 -

(

_ ~)

MLS v e r s i o n o f U is given as

1

kusj uij

<< u o = u o

if Us, >-k,

(17)

otherwise.

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The proof is analogous to the proof of Theorem 3.

[o3/ w'=/o,2o/'

Example 3. Let Uj, Wj, Zj represent the following partitioning of object xj ~ X into four classes with respect to the partitions U, W , Z E Pf4o.

U,=

0.2 0.4'

LO.IJ

/0.30/

Lo,3oJ

o.167] r/0.333/ z, = / 0 000 / k0.500J

It is obvious that Wj, Zj are complementary sharpened versions of Uj, and that Z~ is the complementary MLS version of Uj. From Theorem 3 and Theorem 4 it follows that for every fuzzy k-partition U ¢ / 9 we can define the nontrivial MLS version (>> U) and the nontrivial complementary MLS version (<> Uh << U j ) d ( ~ , - Uj) d(>> ~ , Uj) d(U# ~ ) '

(19)

where d is Euclidean distance.

Definition 7. Let U E Pe~o" The complement of U is a fuzzy partition - U E Pf~o defined by uij - A J k ~ uij =

(20)

1 - Aj

where

Aj =

k(maxi u 0 - mini uo) 1 - k mini u o 0

1 if min; uij < ~, 1 if mini u o = ~.

(21)

It is routine to check the following properties: (1) - U e Pp,o, becasue - uij >I 0 and •i ( ~ uo) = 1 for all i, j. (2) - U is a complementary linearly sharpened version of U, because for each xj ~ X there exists a constant dj < 0 such that - uij - 1 / k = dj(u o - i / k ) .

S. Bodjanova / Complement of fuzzy k-partitions

180

(3) Let d be Euclidean distance. Then for all xj E X, d(>> UI., << Uj)_ d ( U h ~ g ) d(>> Uh ~.)

d ( U h Uj)

k(maxi uij - mini u~j)= Aj > 1. 1 - k min~ uij

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(4) ~ / ] = / J , therefore U is an equilibrium of ~.

Example 4. Refer to the partitions from Example 2. Partition complement of U, can be obtained from (20) as follows: u~-3/3

1-u~

1- 3

2

u~ -

-U,

obtained as an intuitive

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The complement of fuzzy partition V obtained by (20) is given by the following matrix: 0.00 0.47 \0.53

-V=

0.36 0.44 0.20

0.50 0.00 0.50

0.20~ 0.60 / . / 0.20/

Theorem 5. Let

U E P~o, where complementation o r f u z z y sets.

k =2.

Then

~uij = 1-uij,

which

is

Zadeh's

definition

of

Proof. (i) If mini uij = 1, then - uij = uij = ½= 1 - uq for all i. (ii) If mini uij < ½, then hj = 2 and uij - x h j _ uij - 1

-uiJ=

1-hi

- 1-2

=l-ui#

Theorem 6. Let uo, u~j be two partitionings o f the object xj E X into two clusters ui, Ur with respect to a f u z z y partition U E Pf~o. We have that 1

1

1

1

i f uij <~ u~j < -k then ~ uij >- ~ u~j > ~

(24)

where

(25)

if uij >- urj > -~ then ~ uij <~ ~ uri < -~.

Proof. If u o <~urj < 1 / k , then

At

Aj
uij---£<~u'J- k

A~=I(I_At).

-k- k

Because )tj > 1, we get uij- hj/k 1 -

aj

u r j - h j / k > ( 1 - Aj)/k 1 - Aj

Therefore, - uij >1 ~ urj > 1/k.

1 - Aj

1 k

S. Bodjanova / Complementof fuzzy k-partitions A n a l o g o u s l y , if uij/> u~j > 1 / k ,

uij-

>/u~j

_Aj>I k k

then

Aj_I k ~c(1-Aft.

Therefore,

uij- AJk 1 -

u~j- Aj/k< (1- Aj)/k

A]

1 - Aj

1 - Aj

1 k

H e n c e ~ uij <~ ~ Urj < 1/k. C o r o l l a r y 1. I f u~j = m i n i u o, then ~ Urj = maxi ~ Ui# I f u,j = maxi uij, then ~ u~j = m i n i - ui# 7. L e t U ~ P~o. T h e n m i n i ~ uq = m i n i ui#

Theorem

ProoL

min - u o =

m a x uq - A j / k = (1 - k m i n i u q ) m a x i uij - maxi uil + m i n i uij 1 - A]

i

1 - k mini

Uij

--

k m a x i uii + k m i n i uij

m i n i uij(1 - k maxi uu) _ m i n i uij. 1 - k maxi u 0 8. L e t U ~ P~o. T h e n ~ ~ U = U.

Theorem

ProoL

u 0 - Aj/k uij

1 - )tj

'

w h e r e Aj -

k ( m a x i u u - m i n i uij) 1 - k m i n i uij

uij A * / k 1 - ~* ' -

~uij= where A* = k(maxi

-

uii - m i n i -

uij) = k(-

1 - k m i n i - uij

uij - Aj/k

m i n i uii - ~ m a x / u i j )

1 - k(- max/uij) k ( m a x i u o - A J k ) ~ -l

max/uij - A,/k](1

= k(mini

(-- 2j

= k(mini

uq-Aj/k-maxi uq+ Aj/k) ( 1 - Aj

)

1 -

Aj - k max/uii

+A

k ( m i n i uij - m a x i uo) 1 - k m a x i u~] Then

1

(uij-aj/k

~*~

kuij-aj-a*(1-aj) k(1 - A*)(1 - Aj)

W e will p r o v e t h a t - A j - A*(I - Aj) = 0 a n d (1 - A*)(1 - Aj) = 1.

181

S. Bodjanova / Complement of fuzzy k-partitions

182

Part I. - A t - )t*(1 - At) = k(maxi uit k(mini uij - maxi uit) (1 k(maxi uit - mini u°)) mini uit) + -k ma----xiu i ; - l 1 - k mini uit / k mini uit - 1 = k(maxi uit

miu'+C"miniu, maxi.')(i m'n,u kmax,.+kmi.,ut

k mini u o - 1

\

k maxi uij - 1

1 - k mini uit

= k(maxi uit - mini ui/) _ k(mini uit - maxi uit) k mini uit - 1

1 - k mini uit

= k(maxi uq - mini uit) + k(mini uit - max/uo) k mini uit - 1

=0.

Part II.

(1- At)0- A*)

= \ l - k(maxi 1 - kuit-mini - mini 1nit uij)/\

k(m'1~ u_iL- max_xiiuo!) 1 - k maxiuit /

1 - k maxi uit 1 - k mini uit = 1. 1 - k mini uit 1 - k maxi uit Therefore - ~ uit = (kuit - 0)/(1 • k) = uit. Let h : F(X)---> [0, 1] be a complementation function on the set of all fuzzy sets. According to Bellman and Giertz (reference from [7]), h should satisfy the following properties: (1) h ( u ( x ) ) depends only on u(x), (2) h(0) = 1 and h(1) = 0, (3) h is continuous and strictly monotonically decreasing, (4) h is involutive, i.e. h ( h ( u ( x ) ) ) = u(x), (5) for all X l , X 2 ~ X : U ( X l ) - u ( x 2 ) = h(u(x1) ) - h(u(x2) ). Property (5) means, that a certain change in the membership value in u should have the same effect on the membership in - u . The complement of a fuzzy k-partition given by Definition 7 satisfies the properties (1)-(5) for k = 2. For k/> 3, conditions (1), (3) and (4) hold. We replace condition (2) by the following weaker condition: (u 0 = 1 ) = 0

and

1/k~-(u~=O)~l.

We replace condition (5) by the following condition: For up, Uq E F ( X ) and for all x t e X:

up(xj) ) -

.q(xj) ) = Cj(up(xj) - uq(xt) )

where Cj is constant which depends on U(xj). That means that a certain change in membership of xj in U(xj) is proportional to a change in membership of xj in - U(xj). It is easy to check that, according to Definition 7, Cj = 1/(1 - Aj). 9. Let U E Pt~o and let >> U be the MLS o f U. Then the MLS o f the complement o f U is identical to the complement o f >> U, which is the complementary MLS version o f U. Hence

Theorem

>> ( - U) = - (>> U) = (<< U).

(26)

S. Bodjanova / Complement o f f u z z y k-partitions

183

Proof. (I). (>>

-

>> uii - A j / k

>> uij - maxi(>> uij)

1 - At

1 - k max/(>> uij) "

Because '

1 max/(>> uij)

k +

1 - k min~ uij

we get _ maxi u o - uij (>> uij) k max/uij - 1" (II).

>> ( - u/j) , = --+

k

, ~ U i j --

1-kmin/~uij

1

1

k

1 - k mini uij

(u/j(1-kminiuo)-maxiuij+miniuo

1)

1 - k max/uij

(1 - k mini u0)(1 - k max/uii) + kuij(1 - k mini uij) - k max/uij + k mini u 0 - (1 - k m a x / u o ) k(1 - k mini uij)(1 - k max/uij) = k ( u 0 - max/uii)(1 - k mini uij) _ uij - maxi u o

k(1 - k mini uij)(1 - k maxi uij)

1 - k maxi uij

m a x / u q - uij k maxi uij - 1"

F r o m (I) and (II) we get max/uij - uii _ maxi uii - uij = k ( u q - maxi uo) (>> uij) = (>> ( - u o ) ) = k maxi uij - 1 - k max/uij - 1 k(1 - k maxi vii) 1 - k maxi uij + kuij - 1 k ( 1 - k max/uij) 1

uo-1/k

+ 1 - k max/uij

1 - k maxi uij

= k(1 - k maxi uij)

+

kuii_-__11

k(1 - k maxi uij)

_ l 1 /I u i j - l~']~ = ( < < u i j ) , k + 1 - k maxi u/j

4. Conclusion

W e have i n t r o d u c e d a simple o p e r a t i o n of c o m p l e m e n t a t i o n on the set of all fuzzy k-partitions, which is a generalization of Z a d e h ' s c o m p l e m e n t a t i o n of fuzzy sets. W e have shown s o m e interesting p r o p e r t i e s of c o m p l e m e n t a t i o n and the a p p r o x i m a t i o n of fuzzy k-partition by its linearly s h a r p e n e d version. F r o m practical point of view, the c o m p l e m e n t of a fuzzy k-partition of a set X can be used for characterization of the counter-objects ( c o u n t e r - e x a m p l e s ) of the objects f r o m X. It might be useful e.g. in decision m a k i n g or in dissimilarity analysis. In our further w o r k we will study s o m e o t h e r relations and o p e r a t i o n s on P/ko b a s e d on sharpness and linear sharpness of fuzzy k-partitions.

184

S. Badjanova / Complement of fuzzy k-partitions

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I.Z. Batyrschin, On fuzzinestic measures of entropy on Kleene algebras, Fuzzy Sets and Systems 34 (1990) 47-60. R.E. Bellman and M. Giertz, On the analytic formalism of the theory of fuzzy sets, Inf. Sci. 5 (1973) 149-157. J. Bezdek, Pattern Recognition with Fuzzy Objective Functions Algorithms (Plenum Press, New York, 1981). J. Bezdek and J. Harris, Convex decomposition of fuzzy partitions, Journal of Math. Anal and Applications 67 (1980) 490-512. S. Bodjanova, Fuzzy sets and fuzzy partitions, Proceedings of the 1992 Meeting of the Gesellschaft fur Klassifikation, Dortmund, Germany. A. De Luca and S. Termini, A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory, Information and Control 20 (1972) 301-312. D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). A. Kandel, Fuzzy Techniques in Pattern Recognition (J. Wiley, New York, 1982). J. Klir and T. Folger, Fuzzy Sets, Uncertainty, and Information (Prentice Hall, Englewood Cliffs, 1988). S. Miyamoto, Fuzzy Sets in Information Retrieval and Cluster Analysis (Kluwer Academic Publishers, Dodrecht-Boston, 1990). L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353.