Home Page. Title Page. Page 1 of 35. Go Back. Full Screen. Close. Quit

Transcription

1 JJ II J I Page 1 of 35

2 General Attribute Reduction of Formal Contexts Tong-Jun Li Zhejiang Ocean University, China September, 2011,University of Milano-Bicocca Page 2 of 35

3 Objective of the paper The paper proposes a kind of attribute reduction of formal context, which can be regarded as the generalization of some existing attribute reduction of formal context. Subsequently, judgement theorems of consistent attribute sets and attribute reducts are obtained. On the basis of rough set theory, all attribute of formal context are classified into three types by means of all the attribute reducts, and the characteristics of these types of attributes are also analyzed. Finally,by using discernibility attribute sets, a method of distinguishing attribute reducts is presented. Page 3 of 35

4 Outline Introduction Basic notions and knowledge Attribute reduction of formal contexts Characterization of attributes Conclusions Page 4 of 35

5 1. Introduction Formal concept analysis (FCA) [3] and rough set theory (RST) [8] are two important mathematical tools for knowledge representation and knowledge discovery. They are related but distinct and complementary with each other [1,2,4,14,15]. Attribute reduction is an important issue in knowledge discovery and data mining. In recent years, much attention has been paid to the attribute reduction of formal context [6,7,12,13,16]. Page 5 of 35

6 1. Introduction Some researchers, such as, Zhang et al. [16], Mi et al. [7], Wu et al. [13], Liu et al. [6], and Wang and Zhang [12], proposed some concepts of attribute reductions by searching a minimal attribute subset such that some granular structures unchanged in the new concept lattice, and presented some approaches to attribute reduction. Page 6 of 35

7 1. Introduction However, in many practical applications, the formal concepts related to the decision classes may be a part of all formal concepts, and the related formal concepts may not form a kind of algebraic structure. Accordingly, in this paper, we propose a new definition of attribute reduction of formal context, and discuss some related issues. Page 7 of 35

8 2. Basic Notions and Knowledge Page 8 of 35

9 Formal Concepts and Concept Lattice A formal context is a triple (U, A, I), where U is nonempty set of objects, A a nonempty set of attributes, and I a relation from U to A. (x, a) I means that object x has attribute a. Example 1 Table 1 A formal context T = (U, A, I) U a b c d e Page 9 of 35

10 Formal Concepts and Concept Lattice Formal concepts are constructed based on the following logical operators: X U, B A, X A = {a A x X, (x, a) I}, B A = {x U a B, (x, a) I}. For any x U and a A, {x} A and {a} A are simply denoted by x A and a A, respectively. For X U, B A, (X, B) is called a formal concept of (U, A, I) if X A = B and B A = X, and X, B are called the extent and the intent of (X, B), respectively. Page 10 of 35

11 Formal Concepts and Concept Lattice In (U, A, I), all of formal concepts forms a complete lattice, called formal concept lattice, denoted as L(U, A, I), under the following partial order relation: (X 1, B 1 ) (X 2, B 2 ) X 1 X 2 (iff B 1 B 2 ). If (X 1, B 1 ) (X 2, B 2 ), then (X 1, B 1 ) is called a sub-concept of (X 2, B 2 ), and (X 2, B 2 ) is called a super-concept of (X 1, B 1 ). If (X 1, B 1 ) < (X 2, B 2 ), and there is no concept (X, B) with (X 1, B 1 ) < (X, B) < (X 2, B 2 ), then (X 1, B 1 ) is called a lower neighbor of (X 2, B 2 ), and (X 2, B 2 ) is called a upper neighbor of (X 1, B 1 ). Denote the set of all extents of formal concept of (U, A, I) as L U (U, A, I). Page 11 of 35

12 Formal Concepts and Concept Lattice Fig. 1. shows the Hasse diagram of L(U, A, I) for the formal context (U, A, I) in Example 1. Page 12 of 35 Figure 1. The formal concept lattice L(U, A, I)

13 Closure systems and closure operators A set system S on U is a family of subsets of U. A closure system C on the universe U is a set system on U which satisfies that (1) U C; (2) D C( X C). X D For any set system S on U, the below closure system C(S) is the least closure system containing S: C(S) = { X D S}. X D Page 13 of 35

14 Closure systems and closure operators For (U, A, I), denote S A = {a A a A}, then the below conclusion can be made. Proposition 1. L U (U, A, I) = C(S A ). Page 14 of 35

15 Closure systems and closure operators For (U, A, I), X A A (X U) is a closure operator on U, that is, it satisfies (1) X Y X A A Y A A, (2) X X A A, (3) X A A = X A A A A. Proposition 2. X A A = {Y L U (U, A, I) X Y }. Page 15 of 35

16 3. Attribute Reduction of Formal Contexts Page 16 of 35

17 Sub-contexts Definition 1. Let T = (U, A, I) be a formal context. For B A we call the formal context (U, B, I B ) a sub-context of T, where I B = I (U B). Let (U, B, I B ) be a sub-context and X U. Then X B = X A B. Page 17 of 35

18 Attribute Reduction Definition 2. Let (U, A, I) be a formal context and D L U (U, A, I) with D and U D. B A is referred to as a D-consistent set of (U, A, I) if D L U (U, B, I B ). B is referred to as a D-attribute reduct of (U, A, I) if B is a D- consistent set of (U, A, I), and no proper subset of B is a D-consistent set of (U, A, I). Page 18 of 35

19 Attribute Reduction Remark. Therefore, a D-consistent set of (U, A, I) and the original attribute set A produce the same extent subset D, and a D-attribute reduct of (U, A, I) is a minimal consistent set. Specially, if D = L U (U, A, I) then the notion of D- attribute reduction in Definition 2 coincides with the notion of attribute reduction proposed by Zhang et al. in [16], and if D = {x A A x U} then the notion of D- attribute reduction in Definition 2 coincides with the notion of attribute reduction proposed by Wu et al. in [13]. Page 19 of 35

20 Judgement of Attribute Reduction In the following, we assume that the set system D satisfies D = and U D, and U, A of (U, A, I) is finite set. Theorem 1. The D-attribute reduct of (U, A, I) exist. Theorem 2. Let (U, A, I) be a formal context, D L U (U, A, I) and C A, C. Then C is a D-consistent set of (U, A, I) iff D C(S C ). Theorem 3. Let (U, A, I) be a formal context, D L U (U, A, I) and D A, D. Then D is a D-attribute reduct of (U, A, I) iff D C(S C ) and for any a D, there is X D such that X C(S D {a} ). Theorem 4. Let (U, A, I) be a formal context, D L U (U, A, I) and C A, C. Then C is a D-consistent set of (U, A, I) iff (X A C) A (X A (A C)) A for any X D. Page 20 of 35

21 Properties of Attribute Reduction Proposition 3. Let (U, A, I) be a formal context and D 1, D 2 Then L U (U, A, I). (1) if D 1 D 2 then the D 2 -consistent set must be D 1 -consistent set. (2) if C 1, C 2 A are D 1 -consistent set and D 2 -consistent set, then C 1 C 2 is a D 1 D 2 -consistent set. Page 21 of 35

22 Method of Attribute Reduction Definition 3. Let T = (U, A, I) be a formal context and D L U (U, A, I). For any (X i, B i ), (X j, B j ) L U (U, A, I), denote { Bi B j, if X i D, (X i, B i ) (X j, B j ), D((X i, B i ), (X j, B j )) =, otherwise. Then D((X i, B i ), (X j, B j )) is referred to as the D-discernibility attribute set of (X i, B i ) and (X j, B j ), and DA = (D((X i, B i ), (X j, B j )), (X i, B i ), (X j, B j ) L U (U, A, I)) Page 22 of 35 is referred to as the D-discernibility matrix of L(U, A, I).

23 Method of Attribute Reduction Theorem 5. Let (U, A, I) be a formal context, D L U (U, A, I) and C A. Then C is a D-consistent set of (U, A, I) iff for any D((X i, B i ), (X j, B j )), if D((X i, B i ), (X j, B j )) then C D((X i, B i ), (X j, B j )). Page 23 of 35

24 Method of Attribute Reduction Definition 4. Assume DA is the D-discernibility attribute matrix of L(U, A, I). Suppose M D = { {a a D((X i, B i ), (X j, B j ))} D((X i, B i ), (X j, B j )) }, then M D is called the D-discernibility function of L(U, A, I). Theorem 6. Let (U, A, I) be a formal context, D L U (U, A, I). The minimal disjunctive normal form of the D-discernibility function M D is k M D = ( l i i=1 j=1 a ij ), let D i = {a ij j = 1,, l i }(i = 1,, k), then {D i i = 1,, k} is the set of all D-attribute reducts of (U, A, I). Page 24 of 35

25 An example of Attribute Reduction Example 2. For the formal context of Example 1, select D = {{1}, {2}}, then Table 2 shows the D-discernibility matrix. Table 2. The D-discernibility matrix of L(U, A, I) (23, ab) (24, be) (12, de) (2, abde) de ad ab (14, ce) (12, de) (1, cde) d c Thus the D-discernibility function is M D = (d e) (a d) (a b) d c = c d (a b) = (a c d) (b c d) therefore, we can find that there are two attribute reducts, that is, {a, c, d} and {b, c, d}. Page 25 of 35

26 4. Characterization of Attributes Page 26 of 35

27 Characterization of Attributes With reference to rough set theory [34], all attributes of (U, A, I) can be classified into three types: the absolutely necessary attributes, the relatively necessary attributes, and the unnecessary attributes. Definition 5. Let (U, A, I) be a formal context and D L U (U, A, I). D- Reduct(U, A, I) denotes the set system of all D-attribute reducts of (U, A, I), then three attribute subsets can be defined: (1) D absolute necessary attribute set: C D = E. E D Reduct(U,A,I) (2) D Relative necessary attribute set: K D = E E. E D Reduct(U,A,I) E D Reduct(U,A,I) (3) D Unnecessary attribute set: N D = A E. E D Reduct(U,A,I) Page 27 of 35

28 Characterization of Attributes Remark. An absolutely necessary attribute is an attribute which is in all D- attribute reducts so that it cannot be deleted in the construction of the formal concepts with the extent subset D. A relatively necessary attribute is an attribute which is in some but not all D-attribute reducts, deleting it may not change the structure of the formal concepts with the extent subset D. An unnecessary attribute is an attribute which is not in any D-attribute reduct, and it is superfluous for constructing the formal concepts with the extent subset D. Page 28 of 35

29 An example of Attribute Reduction Example 3. For the formal context, (U, A, I), of Example 2, according to Example 2, there are two attribute reducts: {a, c, d} and {b, c, d}. Hence, by Definition 5, three types of attributes of (U, A, I) are C D = {c, d}, K D = {a, b}, N D = {e}. Page 29 of 35

30 Characterization of Attributes Theorem 7. Let (U, A, I) be a formal context, D L U (U, A, I) and a A. Then a belongs to C D iff there exists(x i, B i ), (X j, B j ) L(U, A, I) such that D((X i, B i ), (X j, B j )) = {a}. Theorem 8. Let (U, A, I) be a formal context, D L U (U, A, I) and a A. Then a belongs to C D iff D L U (U, A {a}, I A {a} ). Page 30 of 35

31 Characterization of Attributes For (U, A, I), we can define a equivalence relation, E A, on A as follows: E A = {(a, b) a A = b A }. For any a A, the equivalence class containing attribute a is denoted by [a], that is, [a] = {b A (a, b) E A }. Denote I(a) = {b A a A b A }. Page 31 of 35

32 Characterization of Attributes Proposition 4. Let (U, A, I) be a formal context and D L U (U, A, I). (1) If a C D then [a] = 1 (where [a] is the cardinality of the set [a]). (2) If b I(a) [a] then b C D. (3) C D = C {X}. X D Proposition 5. Let (U, A, I) be a formal context, D L U (U, A, I) and a A. If a C D then [a] K D or [a] N D. Page 32 of 35

33 Characterization of Attributes Theorem 9. Let (U, A, I) be a formal context, D L U (U, A, I) and a A. Then (1) a C D X D(a X A, (X A (A {a})) A a A ). (2) a N D X D(X(a) X), where X(a) = {(X A (C [a])) A C is D consistent set}. (3) a K D X D((X A (A {a})) A = X, ), X D(X(a) X). Page 33 of 35

34 5. Conclusions Formal contexts are a common framework of formal concept analysis and the theory of rough sets by which two theories are related closely. In this paper, a general attribute reduction of formal context is proposed, this kind of attribute reduction is defined as minimal attribute subsets which can preserve the extents of some formal concepts. Thus, some existing attribute reductions of formal context are its special cases. some judgment theorems of the attribute reducts are obtained, all attribute reducts can be computed by means of the discernibility attribute sets between formal concepts. Based on the significance of attributes in the attribute reducts, all attributes of a formal context are classified into three types, and the characteristics of different types of attributes have been analyzed. Page 34 of 35

35 Thanks! Page 35 of 35