GENERALIZED PROBABILISTIC ROUGH SET MODELS

Y.Y. YAO Depar tment of Computer Science, Lakehead University

Thunder B a y , Ontario, Canada P7B 5E1 e-mail: yyao@flash.lakeheadu.ca

S.K.M. Wong Depar tment of Computer Science, University of Regina

Regina, Saskatchewan, Canada SqS OA2

This paper presents a probabilistic version of generalized rough set models. It gen- eralizes the standard algebraic and probabilistic rough set models in two aspects. An arbitrary binary relation is used instead of an equivalence relation. A proba- bility function on the universe is used instead of computing probabilities from the cardinality of sets. Fundamental issues related to probabilistic rough set models are examined.

1 Introduction

Since the introduction of algebraic rough sets: there have been many proposals for incorporating probabilistic i n f o r m a t i ~ n ! > ~ > ~ > ~ ~ They enlarge the application domain of rough sets, and show its connections to other theories, such as belief functions;'>'' fuzzy sets:2 probabilistic modal logic:4 and Bayesian decision theory.16 Most studies on probabilistic rough sets are based on the notion of equivalence relations either explicitly or implicitly. The probability function is often calculated from the cardinality of sets. The main objective of this paper is to generalize further the notion of probabilistic rough sets by considering these two issues, based on generalized rough set models proposed by Yao and Lir~. '~ More specifically, we will use arbitrary binary relations without further restrictions, and probability functions on some cr-algebra of the power set of the universe without referring to the cardinality of sets. The probabilistic in- formation is used for two purposes. From the algebraic approximations of sets, we examine the approximations of the probabilities of these sets!~6~10~11 We de- fine the notion of rough membership functions using probabilistic information! They can in turn be used to derive probabilistic approximations of set^!^,'^

2 Rough Set Models

Let U denote a finite and non-empty set called the universe, and R C U x U a binary relation on the universe. For 5 , y E U , we say that y is R-related to

158 0-7803-3687-9/96(i31996 IEEE

x if xRy. A binary relation may be represented by a mapping r : U -+ 2u.':

r(x) = {y E U I x%y}. (1)

That is, r(x) consists of all %-related elements of z. It may be viewed as a neighborhood of x defined by the binary relation R??l4 If R is an equivalence relation, i.e., R is reflexive, symmetric, and transitive, r(x) = [.]a is the equivalence class containing x.

The pair apr = (U, 92) is called an approximation space. In an approxima- tion space, the relation R describes the connections between elements of the universe. The interpretation of this relation may vary significantly in different applications. In general, it may be viewed as some knowledge or available in- formation about the elements of the universe? For a subset A of the universe, we define a pair of lower and upper approximations of A , _. apr(A) and m ( A ) :

(2)

The lower approximation apr(A) consists of those elements whose R-related elements are all in A, and%e upper approximation W ( A ) consists of those elements such that at least one of whose R-related elements is in A. This interpretation is consistent with the interpretation of the necessity and possi- bility operators in the study of modal 10gic.1~ The pair of approximations may be interpreted as two additional unary set-theoretic operators. 3,14 We call the pair apr, ?if% : 2u --+ 2' rough set approximation operators, or approximation operators for short, and the system (2u, n, U, -,=,?if%) a rough set algebra.13

Although no restriction is imposed on the binary relation, the approxima- tion operators have the following properties: for subsets A, B 5 U ,

- =(A) = {. I .(.) c A } , apr(A) = {x 1 r(x) n A # 0).

__ apr(A) = ( W ( A C ) ) ' ,

- apr(A n B ) = _. apr(A) n - upr(B), __ upr(A U B ) 2 apr(A) U - a p ~ ( B ) ,

__ apr(U) = U ,

W ( A ) = &(Ac))', _. upr(0) = 0, - upr(A U B ) = m ( A ) U ii@(B), upr(A n B ) G - (A) n -(I?), -

where A" = U - A denotes the set complement of A. Properties (LO) and (UO) state that two approximation operators are dual to each other. Properties with the same number may be regarded as dual properties.

159

3 Probabilistic Rough Set Models

For a given approximation space upr = (U, X), we consider an a-algebra S C 2u and a probability function P over S. The set S must contain W({z}) for all z E U . By properties of approximation operators and n-algebra, this implies that S contains - upr(A) and P ( A ) for all A C U . The triple upr = (U, X, (S, P ) ) is called a probabilistic approximation space9 It is used to study approximations of both sets and their probabilities.

3.1 Lower and upper probabilities

For a subset A U , it is approximated by upr(A) and P ( A ) in a probabilistic approximation space upr = (U, 8, (S , P ) ) . Through these two approximations, we define a pair of lower and upper probabilities as follows:

- - P(A) = P(W-(A) ) , - P ( A ) = P ( W ( A ) ) . (3)

By properties (LO)-(L3) and (U0)-(U3), we have:

(LPO) P(A) = 1 - P(A"), (LPl) P ( U ) = 1,

(LP2)

(UP1) P(0) = 0,

P(A U B ) 2 P(A) + P(B) - P ( A U B ) , -

(UPO) P(A) = 1 - I'(A"), -

- (UP2) P ( A n B ) 5 P ( A ) + P ( B ) - F ( A U B) .

Properties (LPO) and (UPO) state that are dual functions from 2u to [0, l]!5 Properties (LP2) and (UP2) can be expressed in a more stronger version: for every positive integer n and every collection A I , . . . , A, C U ,

and

(LP2) P(A1 U . . . U An) 1 P(Aa) - P ( A n Ag) f i i<g

. . . + (-l)n+lP(Al n . . . n A,), -

(up21 P ( A ~ n . . . n A,) 5 CF(A~) - F ( A ~ U A ~ ) f i a<3

. . . + (-1)"+'P(A1 U . . . U A,).

Properties with numbers 1 and 2 are subsets of axioms of belief and plausibility functions.15 They become belief and plausibility functions if they obey the following axioms:

(LP3) P(0) = 0, (UP3) F ( V ) = 1.

160

A binary relation can be interpreted as a multivalued mapping from U to U using T . Our formulation of lower and upper probabilities in the theory of rough sets is related to the framework suggested by Dempster! In defining lower probability, Dempster used the following lower approximation:

4 apr (A) = {. I T(.) # 0,T(.) c_ A). (4) The lower approximation operator apr satisfies axioms (L2) and (L3). It is related to tijF by:

-d

(LDO) (UDO)

%(A) = Z@F(U) - @F(Ac), W ( A ) = W ( U ) - aprd(AC).

By definition, % ( U ) = W ( U ) . The corresponding lower and upper proba- bilies are defined by:

&(A) = P ( a P r , ( A > ) / ~ ( W ( U ) ) , pd(A) = f'(W(A))/P(W(U)). (5) -

The function & satisfies axioms (LPO)-(LP3), and p d satisfies (uPO)-(uP3). They are dual belief and plausibility functions.

If the binary relation 8 is an equivalence relation, lower and upper proba- bilities are the same as rough probabilities introduced by Pawlak6 In this set- ting, its connection to belief functions has been studied by many authors!>lOJ1 Instead of using a probability function, Skowron and Grzymala-Busse com- puted a pair of belief and plausibility functions as follows:

- P(A) = lapr(A)I/IUI, P(A) = lZ@F(A)l/IU/, (6)

where I . I denotes the cardinality of a set. Our formulation may be considered as an extension of these studies. Recently, Lingras and Yao showed that if the binary relation R is a serial relation, i.e., T ( Z ) # 0 for every 2 E U , E and P defined by equation (3) is a pair of belief and plausibility functions!

3.2 Rough membership functions

For an approximation apr = (U, 8) with an equivalence relation 8, the rough membership function of A C_ U is defined by Pawlak and Skowron as:

PA(.) = lAn [Z]~l/l[xl~l. (7)

PA(.) = )A n T ( Z > l / l T ( Z ) l . (8)

Yao and Lin generalized it with respect to an arbitrary binary relation by: l4

161

In this definition, we assume that !R is at least a serial relation. In order to generalize such a notion further, we consider a probabilistic approximation space upr = (U, 8, (2', P)) . For A C U , its rough membership function is defined as:

Similarly, we assume that 3? is a serial relation and furthermore P(r ( z ) ) # 0. Given a number a E [0,1], we define a pair of a-level lower and upper

approximations as:

P A ( Z ) = P ( A I ~ ( 3 ) ) = P ( A n r ( x ) ) / P ( r ( x ) ) . (9)

- %(A) = {x I pA(2) 2 1 - a}, upra(A) Z= {x I PA(%) a}. (10)

We call upr and ZjF, probabilistic rough set approximation operators. They can be explained in the framework of Bayesian decision theory by minimizing certain loss function. For probabilistic approximation operators, we have:

-a

As stated by (PLO) and (PUO), apr and ma are dual operators on 2u. Properties (PL2) and (PU2) are much weaker versions of (L2) and (U2). We call the system (2', n, U, -, upra, Fa), cy E [0,1], a probabilistic rough set algebra. When the rough membership function is computed by equation (8), the algebraic and probabilistic approximations are related to each other by - apr(A) = %(A) and ?@=(A) = F o ( A ) .

-a

4 Conclusion

In this paper, we have generalized the notion of probabilistic rough set mod- els by incorporating arbitrary binary relations and probability functions. The probabilistic information is used for approximating both sets and their proba- bilities. Our discussion focused mainly on the formulation of the basic frame- work, and the explanation of the basic notions. It established the groundwork for the study of probabilistic rough set models. The results of a more system- atic study of this topic will be reported in a forthcoming paper.

162

References

1. A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Annals of Mathematical Statistics, 38, 325-339, 1967.

2. T.Y. Lin, Neighborhood systems: a qualitative theory for fuzzy and rough sets, Proc. 2nd Joint Conf. on Infor. Sci., 255-258, 1995.

3. T.Y. Lin and Q. Liu, Rough approximate operators: axiomatic rough set theory, in: Rough Sets, Fuzzy Sets and Knowledge Discovery, W.P. Ziarko (Ed.) , Springer-Verlag, London, 256-260, 1994.

4. P.J. Lingras and Y.Y. Yao, Belief functions in rough set models, Proc. 2nd Joint Conf. on Infor. Sci., 190-193, 1995.

5. Z. Pawlak, Rough sets. International Journal of Computer and Infor- mation Sciences, 11, 341-356, 1982.

6. Z. Pawlak, Rough probability, Bulletin of Polish Academy of Sciences. Mathematics, 32, 607-615, 1984.

7. Z. Pawlak, Hard and soft sets, in: Rough Sets, Fuzzy Sets and Knowledge Discovery, W.P. Ziarko (Ed.), Springer-Verlag, London, 130-135, 1994.

8. Z. Pawlak and A. Skowron, Rough membership functions, in: Fuzzy Logic for the Management of Uncertainty, L.A. Zadeh and J. Kacprzyk (Eds.), John Wiley & Sons, New York, 251-271, 1994.

9. Z. Pawlak, S.K.M. Wong, and W. Ziarko, Rough sets: probabilistic versus deterministic approach, Int. J. Man-machine Stud., 29, 81-95, 1988.

10. A. Skowron and J. Grzymala-Busse, From rough set theory to evidence theory, in: Advances in the Dempster-Shafer Theory of Evidence, R.R. Yager, M. Fedrizzi, and J. Kacprzyk (Eds.), Wiley, New York, 193-236, 1994.

11. S.K.M. Wong and P.J. Lingras, The compatibility view of Shafer- Dempster theory using the concept of rough set, Methodologies of In- telligent Systems 4 , 33-42, 1989.

12. S.K.M. Wong and W. Ziarko, Comparison of the probabilistic approxi- mate classification and the fuzzy set model, Fuzzy Sets and Systems, 21,

13. Y.Y. Yao, Two views of the theory of rough sets in finite universes,

14. Y.Y. Yao and T.Y. Lin, Generalization of rough sets using modal logic,

15. Y.Y. Yao, S.K.M. Wong, and L.S. Wang, A non-numeric approach to

16. Y.Y. Yao and S.K.M. Wong, A decision theoretic framework for approx-

357-362,1987.

International Journal of Approximate Reasoning, to appear.

Intelligent Automation and Soft Computing, 2, 103-120, 1996.

uncertain reasoning, Int. J. Gen. Syst., 23, 343-359, 1995.

imating concepts, Int. J. Man-machine Stud., 37, 793-809, 1992.

163