On rule acquisition in decision formal contexts - Springer Link
Int. J. Mach. Learn. & Cyber. (2013) 4:721–731 DOI 10.1007/s13042-013-0150-z
ORIGINAL ARTICLE
On rule acquisition in decision formal contexts Jinhai Li • Changlin Mei • Cherukuri Aswani Kumar Xiao Zhang
•
Received: 6 August 2012 / Accepted: 16 January 2013 / Published online: 6 February 2013 Ó Springer-Verlag Berlin Heidelberg 2013
Abstract Rule acquisition is one of the main purposes in the analysis of decision formal contexts. Up to now, there have existed several types of rules (e.g., the decision rules and the granular rules) in decision formal contexts. This study firstly proposes a new algorithm with less time complexity for deriving the non-redundant decision rules from a decision formal context. Then, we invesigate decision rules and the granular rules in the consistent decision formal contexts and make a contrast between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. Finally, some experiments are conducted to assess the efficiency of the proposed rule acquisition algorithm. Keywords Formal concept analysis Formal context Decision formal context Concept lattice Rule acquisition
1 Introduction Formal concept analysis (FCA), proposed by Wille [1], is oriented towards the discovery of formal concepts and the J. Li (&) C. Mei X. Zhang School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, People’s Republic of China e-mail: [email protected] C. Mei e-mail: [email protected] X. Zhang e-mail: [email protected] C. A. Kumar School of Information Technology and Engineering, VIT University, Vellore, India e-mail: [email protected]
construction of concept hierarchies. Nowadays, this theory has shown a trend of multidisciplinary intersection and fusion and hence has become an effective tool for conceptual data analysis and knowledge processing. FCA starts with the notion of a formal context (U, A, I) which consists of the object set U, the attribute set A and the binary relation I U A indicating that each object of U has what attributes in A. In FCA, the basic tool used to analyze databases is the concept lattice which is constituted by all the formal concepts of a formal context together with the subconcept–superconcept-relation. According to Wille’s definition [1], a formal concept is an ordered pair (X, B) in which X U contains exactly those objects shared by all the attributes in B and B A contains exactly those attributes that all the objects in X have in common. In recent years, many studies [2–9] have been devoted to the issue of reducing the size of the concept lattice of a formal context in order to improve the understandability of the resulting concept lattice. In FCA, a useful way of characterizing attribute dependencies in a formal context is via attribute implication rules or association rules. How to derive these rules efficiently has drawn much attention in the literature [10–18]. Decision formal contexts [19], a useful extension of the formal contexts, were proposed to implement decision analysis using the concept lattice. Rule acquisition is one of the main purposes in the analysis of decision formal contexts. A few studies [20–25] have recently been devoted to the rule acquisition in decision formal contexts. To the best of our knowledge, there have existed several types of rules (e.g., the decision rules [23–25] and the granular rules [22]) in decision formal contexts. Although a rule acquisition method for deriving all the non-redundant decision rules from a decision formal context has been proposed in [24], this method depends heavily on both the conditional
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concept lattice and the decision concept lattice. Since the construction of the concept lattice for a large database is very time-consuming, the method proposed in [24] may not be efficient enough in handling a large database. If some efficient methods for extracting the decision rules from a decision formal context are developed, it is of interest to investigate the relationship between the decision rules and the granular rules. Furthermore, knowledge reduction is always one of the important issues in FCA. Therefore, it is also essential to clarify the relation and the difference between the decision rule oriented knowledge reduction [24] and the granular rule oriented knowledge reduction [22]. In this paper, we first propose an efficient algorithm to derive the non-redundant decision rules from a decision formal context. Then we investigate the relationship between the non-redundant decision rules and the granular rules in the consistent decision formal contexts and further make a contrast between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. Finally, we conduct some experiments to assess the efficiency of the proposed rule acquisition algorithm.
2 Preliminaries In this section, we briefly review some basic notions related to FCA in order to make the paper self-contained. Definition 1 ([1]) A formal context is a triple (U, A, I) consisting of the object set U (called the universe of discourse), the attribute set A and the binary relation I U A in which (x, a) [ I indicates that the object x has the attribute a and ðx; aÞ 62 I means the opposite. A formal context (U, A, I) is said to be regular [19] if for any (x, a) [ U 9 A, the following conditions hold: there exist a1, a2 [ A such that (x, a1) [ I and ðx; a2 Þ 62 I; (ii) there exist x1, x2 [ U such that (x1, a) [ I and ðx2 ; aÞ 62 I:
(i)
It should be noted that an irregular formal context (U, A, I) can be regularized by removing the rows with their objects having all the attributes or having no attribute in A and the columns with their attributes being shared by all the objects in U or not being shared by any object of U. Such way of the regularization causes no effect on the analysis results of the formal context. Thus, without loss of generality, the formal contexts discussed hereinafter are all assumed to be regular. Wille [1] introduced a pair of concept forming operators on a formal context (U, A, I):
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X " ¼ fa 2 A j 8x 2 X; ðx; aÞ 2 IgðX UÞ; B# ¼ fx 2 U j 8a 2 B; ðx; aÞ 2 IgðB AÞ:
ð1Þ
That is, X " is the maximal set of the attributes that all the objects in X have in common, and B; is the maximal set of the objects shared by all the attributes in B. Definition 2 ([1]) Let (U, A, I) be a formal context. The ordered pair (X, B) with X U and B A is called a formal concept of (U, A, I) if X " ¼ B and B; = X. Here, the sets X and B are called the extent and the intent of the formal concept (X, B), respectively. For two formal concepts (X1, B1) and (X2, B2), if X1 X2 (or B2 B1 ), then (X1, B1) is called a subconcept of (X2, B2), or (X2, B2) is called a superconcept of (X1, B1). The subconcept–superconcept-relation between the formal concepts is denoted by B. Then the set of all the formal concepts of a formal context (U, A, I) together with the partial order relation B forms a complete lattice which is denoted by BðU; A; IÞ and is called the concept lattice of the formal context (U, A, I). In BðU; A; IÞ; the infimum and the supremum of {(X1, B1), (X2, B2)} are respectively defined by ðX1 ; B1 Þ ^ ðX2 ; B2 Þ ¼ ðX1 \ X2 ; ðB1 [ B2 Þ#" Þ; ðX1 ; B1 Þ _ ðX2 ; B2 Þ ¼ ððX1 [ X2 Þ"# ; B1 \ B2 Þ:
ð2Þ
In [26], the authors discussed the axiomatic characterization of the concept lattice. It should be pointed out that except Wille’s concept lattice, there have existed several other kinds of concept lattices in the classical FCA, e.g., the rough concept lattice [27], the objectoriented concept lattice [28] and the property-oriented concept lattice [29]. The relationship among Wille’s, object-oriented and property-oriented concept lattices was discussed in [5]. In this paper, we only focus on Wille’s concept lattice to discuss the rule acquisition in decision formal contexts. Definition 3 ([22]) Let (U, A, I) be a formal context and E A: The restriction of I on U 9 E, denoted by IE, is defined as IE(x, a) = I(x, a) for any (x, a) [ U 9 E. The formal context (U, E, IE) is called a subcontext of (U, A, I). Similar to the formal context (U, A, I), a pair of concept forming operators can also be defined on the subcontext (U, E, IE) as follows: X "E ¼ fa 2 E j 8x 2 X; ðx; aÞ 2 IE gðX UÞ; B#E ¼ fx 2 U j 8a 2 B; ðx; aÞ 2 IE gðB EÞ:
ð3Þ
In fact, the operators "E and ;E are the restrictions of the concept forming operators " and ; on the subcontext (U, E, IE). Similar to the discussion in Definition 2, we can give the notion of a formal concept with its extent and
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intent in (U, E, IE). Also, the set of all the formal concepts of the subcontext (U, E, IE) together with the subconcept– superconcept-relation B forms a complete lattice denoted by BðU; E; IE Þ; in line with the notation of the concept lattice of (U, A, I). We denote by BU ðU; E; IE Þ the set of the extents of all the formal concepts of (U, E, IE). Proposition 1 ([1]) Let (U, A, I) be a formal context and E A: For X; X1 ; X2 U and B; B1 ; B2 E; the following properties hold: (i)
"
"
# B2 E
#
X1 X2 ) X2 E X1 E ;
(ii) B1 B2 ) B1E ; "E #E (iii) X X ; B B# E " E ; "E #E "E ; X Þ; ðB#E ; B#E "E Þ 2 BðU; E; IE Þ: (iv) ðX It should be noted that ðfxg"E #E ; fxg"E Þðx 2 UÞ and ðfag#E ; fag#E "E Þða 2 AÞ are referred to as the object concepts and the attribute concepts [10], respectively. For brevity, in the rest of the paper, we write them as ðx"E #E ; x"E Þ and ða#E ; a#E "E Þ; respectively. Proposition 2 ([1]) Let (U, A, I) be a formal context, E A and T be an index set. For Xt U; Bt E (t [ T), we have !"E !#E [ \ [ \ "E Xt ¼ Xt and Bt ¼ B t #E : t2T
t2T
t2T
t2T
3 An efficient rule acquisition algorithm for decision formal contexts Definition 4 ([19]) A decision formal context is a quintuple (U, A, I, D, J), where (U, A, I) and (U, D, J) with A \ D ¼ ; are two formal contexts. Here, A and D are called the conditional attribute set and the decision attribute set of (U, A, I, D, J), respectively. Like the formal context, a decision formal context P ¼ ðU; A; I; D; JÞ is also said to be regular [24] if both (U, A, I) and (U, D, J) are regular. The decision formal context PE ¼ ðU; E; IE ; D; JÞ is called a subcontext of P if (U, E, IE) is a subcontext of (U, A, I). Without loss of generality, the decision formal contexts discussed hereinafter are all assumed to be regular. The concept lattice of (U, D, J) is denoted by BðU; D; JÞ and the set of the extents of all the formal concepts of (U, D, J) is denoted by BU ðU; D; JÞ: Definition 5 ([24]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For any ðX; BÞ 2 BðU; E; IE Þ
and ðY; CÞ 2 BðU; D; JÞ; if X, B, Y and C are all nonempty and X Y; then the expression B ! C is called a decision rule generated between the formal concepts (X, B) and (Y, C). Here, B and C are called the premise and the conclusion of the decision rule B ! C; respectively. The set of all the decision rules generated between the formal concepts in BðU; E; IE Þ and those in BðU; D; JÞ is denoted by RðPE Þ; where PE ¼ ðU; E; IE ; D; JÞ: Thus, for any B ! C 2 RðPE Þ with ðX; BÞ 2 B ðU; E; IE Þ and ðY; CÞ 2 BðU; D; JÞ; we have that each x 2 U having all the attributes in B also has all the attributes in C. So, the decision rule is in fact an ‘‘If–then’’ conjunctive rule. That is, B ! C means ‘‘If ^B, then ^C’’. Moreover, it is easy to verify that B ! C is supported by and only by the objects in X. It should be pointed out that the decision rules have something to do with both the attribute implication rules and the association rules (see, e.g. [10, 11, 20] for the detailed discussion of the attribute implication rules and e.g. [16, 17] for the association rules). Concretely, a decision rule is a special attribute implication rule. However, an attribute implication rule may not be a decision rule since the premise or the conclusion of an attribute implication rule is not required to be the intent of a formal concept but an attribute set only. Also, a decision rule is a special association rule. But an association rule may not be a decision rule because the antecedent of an association rule may not be the intent of a formal concept and the confidence is often less than one. Definition 6 ([24]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For B1 ! C1 ; B2 ! C2 2 R ðPE Þ; if B1 B2 and C2 C1 ; we say that B2 ! C2 can be implied by B1 ! C1 : We denote this implication relationship by B1 ! C1 ) B2 ! C2 : For any B ! C 2 RðPE Þ; if there exists B0 ! C0 2 RðPE ÞnfB ! Cg such that B0 ! C0 ) B ! C; then B ! C is said to be redundant in RðPE Þ; otherwise, B ! C is said to be non-redundant in RðPE Þ: We denote by R ðPE Þ the set of all the nonredundant decision rules in RðPE Þ: It can be known from Definition 6 that for a given decision formal context, it is more appealing to extract the non-redundant decision rules since its redundant decision rules can be implied by the non-redundant ones. Let P ¼ ðU; A; I; D; JÞ be a decision formal context. For any ðX; YÞ 2 BU ðU; A; IÞ BU ðU; D; JÞ; define 8 < 1; if X Y and there does not exist X0 aðX; YÞ ¼ 2 BU ðU; A; IÞ such that X X0 Y; : 0; otherwise; ð4Þ
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bðX; YÞ ¼
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8 < 1; :
0;
if X Yand there does not exist Y0 2 BU ðU; D; JÞ such that X Y0 Y; otherwise. ð5Þ
In [24], the authors put forward a method to derive the non-redundant decision rules from a decision formal context. The method can briefly be described as follows:
Proof
It is immediate from Definitions 6, 7 and 8.
Theorem 2 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For ðX; BÞ 2 BðU; E; IE Þ with X 6¼ ; T and B 6¼ ;; if C ¼ x2X x"D 6¼ ;; then B ! C is a feedforward non-redundant decision rule in RðPE Þ; where "D denotes the operator " on (U, D, J). Proof Since the ordered pairs ðx"D #D ; x"D Þðx 2 XÞ are object concepts of (U, D, J) where ;D denotes the operator ; on (U, D, J), the supremum of fðx"D #D ; x"D Þjx 2 Xg; denoted by (Y, C), satisfies ðY; CÞ ¼
W
ðx
"D #D
"D
Then [ X x"D #D
where |LA| and |LD| denote the cardinalities of the concept lattices BðU; A; IÞ and BðU; D; JÞ; respectively. In order to enhance the efficiency of the rule acquisition method in decision formal contexts, we shall propose in the following a new algorithm of deriving the non-redundant decision rules which is of less time complexity than Algorithm 1. Definition 7 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For any B ! C 2 RðPE Þ; if there exists B0 ! C0 2 RðPE Þ such that B0 B and C C0 ; we say that B ! C is a feedforward redundant decision rule in RðPE Þ; otherwise, we say that B ! C is a feedforward non-redundant decision rule in RðPE Þ: Definition 8 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For any B ! C 2 RðPE Þ; if there exists B0 ! C0 2 RðPE Þ such that B0 B and C C0 ; we say that B ! C is a feedback redundant decision rule in RðPE Þ; otherwise, we say that B ! C is a feedback nonredundant decision rule in RðPE Þ: Theorem 1 Let P ¼ ðU; A; I; D; JÞ be a decision formal context, E A and B ! C 2 RðPE Þ: Then the following statements hold: B ! C is redundant in RðPE Þ iff B ! C is a feedforward or feedback redundant decision rule in RðPE Þ: (ii) B ! C is non-redundant in RðPE Þ iff B ! C is both a feedforward and a feedback non-redundant decision rule in RðPE Þ: (i)
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x2X
S
;x Þ ¼
x2X
It is easy to prove that the time complexity of Algorithm 1 is O ðjUj þ jAjÞjAjjLA j þ jUjjLA j2 jLD j þ jUjjLA jjLD j2 ;
h
x
"D #D
" D # D ;
x2X
[
T
! x
"D
2 BðU; D; JÞ:
x2X
!"D #D x
"D #D
¼ Y:
x2X
Thus, B ! C 2 RðPE Þ according to the assumption. Furthermore, we can prove that B ! C is a feedforward non-redundant decision rule in RðPE Þ: In fact, if B ! C is a feedforward redundant decision rule in RðPE Þ; there exists B0 ! C0 2 RðPE Þ such that B0 B and C C0 : Suppose ðY0 ; C0 Þ 2 BðU; D; JÞ: Then we obtain Y0 Y: However, !"D #D !"D #D [ [ "D #D "D #D Y¼ x x ¼ Y0 ; x2X
x2Y0
which is in contradiction with Y0 Y:
h
Theorem 3 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For ðX; BÞ 2 BðU; E; IE Þ with X 6¼ T ;; B 6¼ ; and C ¼ x2X x"D 6¼ ;; then B ? C is nonredundant in RðPE Þ iff there does not exist ðX0 ; B0 Þ 2 BðU; E; IE Þ with X0 6¼ ; and B0 6¼ ; such that B0 ! C 2 RðPE Þ and B0 , B. Proof Necessity. If there exists ðX0 ; B0 Þ 2 BðU; E; IE Þ with X0 6¼ ; and B0 6¼ ; such that B0 ! C 2 RðPE Þ and B0 , B, it follows from Definition 8 that B ? C is a feedback redundant decision rule in RðPE Þ: According to Theorem 1, B ? C is redundant in RðPE Þ; which is in contradiction with the assumption that B ? C is nonredundant in RðPE Þ: Sufficiency. If B ? C is redundant in RðPE Þ; by Theorem 1 we have that B ? C is a feedforward or feedback redundant decision rule in RðPE Þ: (i)
If B ? C is a feedforward redundant decision rule in RðPE Þ; then there exists B0 ! C0 2 RðPE Þ such that B0 B and C , C0. Suppose ðY0 ; C0 Þ 2 BðU; D; JÞ and let Y ¼ C #D : Then we obtain Y0 , Y. However, it follows from Propositions 1 and 2 that
Int. J. Mach. Learn. & Cyber. (2013) 4:721–731
Y ¼C
#D
¼
\
!#D x
"D
\
¼
x2X
[
¼
x
x
x2X
!"D #D
"D #D
! #D "D #D "D
x2X
[
x
!"D #D
"D #D
¼ Y0 ;
x2Y0
which is in contradiction with Y0 Y: (ii) If B ! C is a feedback redundant decision rule in RðPE Þ; there exists B0 ! C0 2 RðPE Þ such that B0 B and C C0 : Since C C0 does not hold, we obtain C = C0. Thus, B0 ! C 2 RðPE Þ and B0 B; which is in contradiction with the assumption. h Definition 9 Let P ¼ ðU; A; I; D; JÞ be a decision formal context, E A and X RðPE Þ:B ! C 2 X is called a premise-minimal decision rule in X if B0 6 B for any B0 ! C 2 X:
725
feedforward non-redundant decision rule in RðPE Þ: By Definition 10, B ! C 2 MðPE Þ follows. Furthermore, we can prove B ! C 2 M ðPE Þ: In fact, if there exists B0 ! C 2 MðPE Þ such that B0 B; then by Definition 8 B ! C is a feedback redundant decision rule in RðPE Þ; which is in contradiction with B ! C 2 R ðPE Þ: Thus, based on Definition 9, we obtain that B ! C is a premise-minimal decision rule in MðPE Þ; i.e. B ! C 2 M ðPE Þ: By Theorem 4, we can obtain all the non-redundant decision rules from a decision formal context P ¼ ðU; A; I; D; JÞ via M ðPÞ: By Definitions 9 and 10, it is natural to first compute MðPÞ and then obtain M ðPÞ: However, here we directly compute M ðPÞ without generating MðPÞ in advance in order to reduce the computational complexity, which is described as follows:
Definition 10 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: Define MðPE Þ ¼ fB ! C j T ðX; BÞ 2 BðU; E; IE Þ; X 6¼ ;; B 6¼ ;; C ¼ x2X x"D 6¼ ;g and denote by M ðPE Þ the set of all the premise-minimal decision rules in MðPE Þ: Theorem 4 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: Then R ðPE Þ ¼ M ðPE Þ: Proof On one hand, for any B ! C 2 M ðPE Þ; we have by Definitions 9 and 10 that B ! C 2 RðPE Þ and B0 6 B for any B0 ! C 2 MðPE Þ: It can be known from Theorem 2 that B ? C is a feedforward non-redundant decision rule in RðPE Þ: If B ? C is a feedback redundant decision rule in RðPE Þ; there exists B0 ! C0 2 RðPE Þ such that B0 , B and C C0 : Since C , C0 is not true, we obtain C = C0. Thus, B0 ! C 2 MðPE Þ and B0 , B, which is in contradiction with B0 6 B for any B0 ! C 2 MðPE Þ: As a result, B ? C is a feedback non-redundant decision rule in RðPE Þ: By Theorem 1, B ! C 2 R ðPE Þ is at hand. On the other hand, for any B ! C 2 R ðPE Þ; there exist X 2 BU ðU; E; IE Þ and Y 2 BU ðU; D; JÞ such that ðX; BÞ 2 BðU; E; IE Þ and ðY; CÞ 2 BðU; D; JÞ: Noting that 0 1 !"D #D [ \ @ x"D #D ; x"D A 2 BðU; D; JÞ x2X
x2X
and X
[ x2X
!"D #D x
"D #D
[
!"D #D "D #D
x
¼ Y;
x2Y
T T we obtain B ! x2X x"D 2 RðPE Þ and C x2X x"D : T Then, we can conclude C ¼ x2X x"D since B ! C is a
It is easy to prove that the time complexity of Algorithm 2 is O ðjUj þ jAjÞjAjjLA j þ jDjjLA j2 ; where |LA| denotes the cardinality of BðU; A; IÞ: Note that |U||LD| |D| generally holds for a given decision formal context. So, Algorithm 2 is of less time complexity than Algorithm 1. We have the following example to illustrate Algorithm 2. h Example 1 Table 1 shows a decision formal context P ¼ ðU; A; I; D; JÞ; where U = {1, 2, 3, 4, 5}, A = {a, b, c, d, e, f} and D = {d1, d2, d3}. In the table, the number 1 in the ith row and jth column represents that the object in the ith row has the attribute in the jth column, and the number 0 in the ith row and jth column means the opposite. Figure 1 depicts the Hasse diagram of the concept lattice of the formal context (U, A, I). By Theorem 2, all the feedforward non-redundant decision rules generated by the formal concepts of (U, A, I) are as follows: r 1:
a ! d1 ; which is generated by ({1, 3, 5}, {a}) due to 1"D \ 3"D \ 5"D ¼ fd1 g and is supported by objects 1, 3, 5;
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Table 1 A decision formal context P ¼ ðU; A; I; D; JÞ U
a
b
c
d
e
f
d1
d2
d3
1
1
0
0
0
0
0
1
0
0
2
0
1
0
1
0
0
0
1
1
3
1
0
1
0
0
1
1
1
0
4
0
1
0
0
1
0
1
1
0
5
1
1
1
0
1
0
1
1
0
(U,∅)
4 Decision rules and granular rules in the consistent decision formal contexts In Sect. 3, we have proposed an efficient algorithm to extract all the non-redundant decision rules from a decision formal context. Since the decision rules and the granular rules are two useful ways of discovering knowledge from a decision formal context, it is of interest to investigate the relationship between the non-redundant decision rules and the granular rules. Furthermore, considering that knowledge reduction is one of the key issues in FCA, it is also essential to clarify the relation and the difference between the decision rule oriented knowledge reduction [24] and the granular rule oriented knowledge reduction [22].
({2,4,5},{b})
({1,3,5},{a})
4.1 The relationship between the non-redundant decision rules and the granular rules ({3,5},{a,c})
({4,5},{b,e})
({3},{a,c,f})
({5},{a,b,c,e})
Definition 11 ([22]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context. If x"# x"D #D for any x 2 U; then P is said to be consistent; otherwise, P is said to be inconsistent. ({2},{b,d})
(∅,A)
Fig. 1 Hasse diagram of the concept lattice of (U, A, I)
r 2:
r 3: r 4:
r 5:
r 6:
r 7:
ac ! d1 d2 ; which is generated by ({3, 5}, {a, c}) due to 3"D \ 5"D ¼ fd1 ; d2 g and is supported by objects 3, 5; acf ! d1 d2 ; which is generated by ({3}, {a, c, f}) due to 3"D ¼ fd1 ; d2 g and is supported by object 3; b ! d2 ; which is generated by ({2, 4, 5}, {b}) due to 2"D \ 4"D \ 5"D ¼ fd2 g and is supported by objects 2, 4, 5; be ! d1 d2 ; which is generated by ({4, 5}, {b, e}) due to 4"D \ 5"D ¼ fd1 ; d2 g and is supported by objects 4, 5; abce ! d1 d2 ; which is generated by ({5}, {a, b, c, e}) due to 5"D ¼ fd1 ; d2 g and is supported by object 5; bd ! d2 d3 ; which is generated by ({2}, {b, d}) due to 2"D ¼ fd2 ; d3 g and is supported by object 2.
Thus, by applying Algorithm 2 to the decision formal context P; the rules r1, r2, r4, r5 and r7 are added into M one by one. That is, P has five non-redundant decision rules: r1, r2, r4, r5 and r7.
123
In FCA, there are several other kinds of consistent decision formal contexts (e.g., those in [25, 30–32]) except the one introduced above. However, for our purpose, we here only focus on the kind of consistent decision formal contexts in [22]. That is to say, the consistent decision formal contexts discussed in the following represents the ones whose consistency is defined by Definition 11. Proposition 3 Let P ¼ ðU; A; I; D; JÞ be a decision formal context. Then P is consistent iff x" ! x"D 2 RðPÞ for any x 2 U: Proof Necessity. Since P is consistent, it follows from Definition 11 that x"# x"D #D for any x 2 U: Noting that ðx"# ; x" Þ 2 BðU; A; IÞ; ðx"D #D ; x"D Þ 2 BðU; D; JÞ and P is regular, we know that x"# ; x" ; x"D #D and x"D are all nonempty, which leads to x" ! x"D 2 RðPÞ: Sufficiency. It is immediate from Definitions 5 and 11. h Definition 12 ([22]) Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. For any x 2 U; x" ! x"D is called a granular rule of P: It can be known from Definition 12 that a granular rule is a special decision rule. However, a decision rule may not be a granular rule because the premise or the conclusion of a decision rule is not required to be the intent of an object concept but that of a formal concept only. Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. For any nonempty set X 2 BU ðU; A; IÞ with T T " "D 6¼ ;; the mergence of the granx2X x 6¼ ; and x2X x " "D ular rules x ! x ðx 2 XÞ is defined as
Int. J. Mach. Learn. & Cyber. (2013) 4:721–731
^
727
\ \ " x" ! x"D ¼ x ! x"D :
x2X
x2X
ð6Þ
x2X
It is easy to verify that the mergence of x" ! x"D (x 2 X) is also a decision rule. Theorem 5 Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context and B ! C 2 RðPÞ: Then the decision rule B ! C is non-redundant in RðPÞ iff B ! V C ¼ x2B# x" ! x"D and ðx0 " \ BÞ# 6 C#D for any x0 2 C #D nB# : Proof Since B ! C 2 RðPÞ; there exist X 2 BU ðU; A; I) and Y 2 BU ðU; D; JÞ such that ðX; BÞ 2 BðU; A; IÞ and ðY; CÞ 2 BðU; D; JÞ: If B ? C is non-redundant in RðPÞ; it follows from Theorem 1 that B ? C is both a feedforward and a feedback non-redundant decision rule in RðPÞ: So, B ! T T "D 2 RðPÞ; which leads to C ¼ x2X x"D because x2X x T B ! x2X x"D can imply B ? C. Furthermore, noting that
ðX; BÞ ¼
_ x2X
0 ðx"# ; x" Þ ¼ @
[ x2X
!"# x"#
;
\
1 x " A;
x2X
V we have B!C ¼ x2B# x" !x"D : To prove ðx0 " \BÞ# 6 C #D for any x0 2 C #D nB# ; it is sufficient to show ðx0 " \ BÞ# 6 Y for any x0 2 YnX: If there exists x0 2 YnX such that ðx0 " \ BÞ# Y; then ðx0 " \ BÞ# ¼ ðx0 " \ X " Þ# ¼ ðfx0 g [ XÞ"# Y yielding ðfx0 g [ XÞ" 6¼ ;: Since ððfx0 g [XÞ"# ; ðfx0 g [ XÞ" Þ 2 BðU; A; IÞ; we obtain ðfx0 g[XÞ" ! C 2 RðPÞ; i.e. x0 " \ B ! C 2 RðPÞ: Since B ! C is a feedback non-redundant decision rule in RðPÞ; we conclude x0: \ B = B, which is in contradiction with x0 2 YnX: V If B ! C ¼ x2B# x" ! x"D and ðx0 " \ BÞ# 6 C #D for any x0 2 C#D nB# ; then by Theorem 2, B ! C is a feedforward non-redundant decision rule in RðPÞ: Thus, to prove that B ! C is non-redundant in RðPÞ; it is sufficient to show that B ! C is a feedback non-redundant decision rule in RðPÞ: In fact, if B ! C is a feedback redundant decision rule in RðPÞ; there exists B0 ! C0 2 RðPÞ such that B0 B and C C0 : Suppose ðX0 ; B0 Þ 2 BðU; A; IÞ and ðY0 ; C0 Þ 2 BðU; D; JÞ: Then X X0 Y0 Y: Therefore, there exists x 2 X0 nX such that ðx" \ BÞ# ¼ ðfxg [ XÞ"# X0"# ¼ X0 Y ¼ C#D ; which is in contradiction with ðx0 " \ BÞ# 6 C #D for any x0 2 C #D nB# : Theorem 5 clarifies the relationship between the nonredundant decision rules and the granular rules in a consistent decision formal context. h
4.2 A comparison of the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction In general, both the decision rules and the granular rules derived directly from a consistent decision formal context are not concise or compact. In order to derive more compact decision rules and/or granular rules, the issue of rule acquisition oriented knowledge reduction was discussed in [22, 24]. Since knowledge reduction is one of the important issues in FCA, it is of interest to clarify the relation and the difference between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. This issue will be discussed in the following. Definition 13 ([24]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For X RðPÞ and X0 RðPE Þ; if each decision rule of X can be implied by a decision rule of X0 ; we say that X can be implied by X0 : We denote this implication relationship by X0 ) X: Definition 14 ([24]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context. E A is called a consistent set of P if RðPE Þ ) RðPÞ: Furthermore, if E is a consistent set of P and any F E is not a consistent set of P; then E is called a reduct of P: It can easily be observed from Definitions 13 and 14 that this kind of knowledge reduction can preserve the decision rule information of a decision formal context and allows us to obtain more compact decision rules from a decision formal context. Proposition 4 Let P ¼ ðU; A; I; D; JÞ be a decision formal context. Then E A is a consistent set of P iff R ðPE Þ ) R ðPÞ: Proof Necessity. Since E A is a consistent set of P; it follows from Definition 14 that RðPE Þ ) RðPÞ: According to Definition 13, it is easy to prove R ðPE Þ ) RðPE Þ and RðPÞ ) R ðPÞ: As a result, R ðPE Þ ) RðPE Þ ) RðPÞ ) R ðPÞ: Sufficiency. Based on Definition 13, it is also easy to prove RðPE Þ ) R ðPE Þ and R ðPÞ ) RðPÞ: Thus, according to the assumption R ðPE Þ ) R ðPÞ; we can obtain RðPE Þ ) R ðPE Þ ) R ðPÞ ) RðPÞ: By Definition 14, E is a consistent set of P: h Definition 15 ([22]) Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. E A is called a granular consistent set of P if x"E #E x"D #D for any x 2 U: Furthermore, if E is a granular consistent set of P and any F E is not a granular consistent set of P; then E is called a granular reduct of P:
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Proposition 5 Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. Then E A is a granular consistent set of P iff x"E ! x"D 2 RðPE Þ for any x 2 U:
and x"D C )C #D x"D #D )Y x"D #D : Therefore, x"E #E X Y x"D #D : By Definition 15, we conclude that E is a granular consistent set of P:
Proof Necessity. If E A is a granular consistent set of P; then by Definition 15, x"E #E x"D #D holds for any x 2 U: Since P is regular, we have x"D 6¼ ;: Furthermore, we can prove x"E 6¼ ;: In fact, if x"E ¼ ;; then x"E #E ¼ U: Thus, x"D #D ¼ U yielding x"D ¼ ;; which is in contradiction with x"D 6¼ ;: Consequently, x"E ! x"D 2 RðPE Þ:
It can be known from Theorem 7 that the decision rule oriented knowledge reduction of a consistent decision formal context can preserve the granular rule information. However, the granular rule oriented knowledge reduction may not preserve the decision rule information. That is to say, it may happen that a granular consistent set is not a consistent set. In what follows, we use a counterexample to confirm this assertion. h
Sufficiency. It is immediate from Definitions 5 and 15. Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context and E A be a granular consistent set of P: We denote by R# ðPE Þ the set of all the granular rules of the subcontext PE ¼ ðU; E; IE ; D; JÞ: h Theorem 6 Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. If E A is a granular consistent set of P; then R# ðPE Þ ) R# ðPÞ: Proof It is immediate from Definition 13 and Propositions 3 and 5. It can easily be seen from Definition 15 and Theorem 6 that the granular rule oriented knowledge reduction can preserve the granular rule information of a consistent decision formal context and allows us to derive more compact granular rules from a consistent decision formal context. The following theorem illustrates the relation between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. h Theorem 7 Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. If E A is a consistent set of P; then E is a granular consistent set of P: Proof Since P is consistent, it follows from Definition 11 that x"# x"D #D for any x 2 U: Noting that ðx"# ; x" Þ 2 BðU; A; IÞ; ðx"D #D ; x"D Þ 2 BðU; D; JÞ and P is regular, we know that x"# ; x" ; x"D #D and x"D are all nonempty, which leads to x" ! x"D 2 RðPÞ: If E A is a consistent set of P; then there exists B ! C 2 RðPE Þ such that B x" and x"D C: Since there exist X 2 BU ðU; E; IE Þ and Y 2 BU ðU; D; JÞ such that ðX; BÞ 2 BðU; E; IE Þ and ðY; CÞ 2 BðU; D; JÞ; we have by Proposition 1 that B x " ) x 2 B# E ) x"# X
Example 2 Let P ¼ ðU; A; I; D; JÞ be the decision formal context in Example 1. Then the following statements hold: 1"# ¼ f1; 3; 5g f1; 3; 4; 5g ¼ 1"D #D ; 2"# ¼ f2g ¼ 2"D #D ; 3"# ¼ f3g f3; 4; 5g ¼ 3"D #D ; 4"# ¼ f4; 5g f3; 4; 5g ¼ 4"D #D ; 5"# ¼ f5g f3; 4; 5g ¼ 5"D #D : Thus, it follows from Definition 11 that the decision formal context P is consistent. According to the results in Example 1 and the discussion in Sect. 4.1, we obtain the following seven feedforward non-redundant decision rules via the mergence of granular rules: r 1:
r 2:
r 3:
r 4:
r 5:
r 6:
r 7:
a ! d1 ; which is generated by the extent {1, 3, 5} due to 1: \ 3: \ 5: = {a} and 1"D \ 3"D \ 5"D ¼ fd1 g; is supported by objects 1, 3, 5; ac ! d1 d2 ; which is generated by the extent {3,5} due to 3: \ 5: = {a,c} and 3"D \ 5"D ¼ fd1 ; d2 g; is supported by objects 3, 5; acf ! d1 d2 ; which is generated by the extent {3} due to 3" ¼ fa; c; f g and 3"D ¼ fd1 ; d2 g; is supported by object 3; b ! d2 ; which is generated by the extent {2, 4, 5} due to 2" \ 4" \ 5" ¼ fbg and 2"D \ 4"D \ 5"D ¼ fd2 g; is supported by objects 2, 4, 5; be ! d1 d2 ; which is generated by the extent {4,5} due to 4" \ 5" ¼ fb; eg and 4"D \ 5"D ¼ fd1 ; d2 g; is supported by objects 4, 5; abce ! d1 d2 ; which is generated by the extent {5} due to 5: = {a, b, c, e} and 5"D ¼ fd1 ; d2 g; is supported by object 5; bd ! d2 d3 ; which is generated by the extent {2} due to 2: = {b,d} and 2"D ¼ fd2 ; d3 g; is supported by object 2.
) X "E x"#"E ) B x"E ) x"E #E B#E ¼ X
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For r3 : acf ! d1 d2 ; there is 5 2 fd1 ; d2 g#D nfa; c; f g# such that
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729
ð5" \ fa; c; f gÞ# ¼ f3; 5g f3; 4; 5g ¼ fd1 ; d2 g#D ; for r6 : abce ! d1 d2 ; there is 4 2 fd1 ; d2 g#D nfa; b; c; eg# such that ð4" \fa;b;c;egÞ# ¼f4;5gf3;4;5g¼fd1 ;d2 g#D : Thus, by Theorem 5, we know that the rules r3 and r6 are redundant in RðPÞ: Similarly, it can be verified that r1, r2, r4, r5 and r7 are all non-redundant in RðPÞ: Let E = {a, d, e, f}. Then we can obtain a subcontext PE ¼ ðU; E; IE ; D; JÞ which satisfies 1
"E #E
2
"E #E
¼ f1; 3; 5g f1; 3; 4; 5g ¼ 1 ¼ f2g ¼ 2
"D #D
"D #D
;
;
3"E #E ¼ f3g f3; 4; 5g ¼ 3"D #D ; 4"E #E ¼ f4; 5g f3; 4; 5g ¼ 4"D #D ; and 5"E #E ¼ f5g f3; 4; 5g ¼ 5"D #D : Thus, E is a granular consistent set of P: Figure 2 depicts the Hasse diagram of the concept lattice of the formal context (U, E, IE). According to Theorem 5, the nonredundant decision rules derived from PE are as follows: r10 : a ! d1 ; r20 : af ! d1 d2 ; r30 : e ! d1 d2 ; r40 : d ! d2 d3 : It is easy to check that the rules r2 and r4 derived from P cannot be implied by the non-redundant decision rules derived from the subcontext PE : By Proposition 4, E is not a consistent set of P although it is a granular consistent set of P: That is to say, to remove the attributes b and c from the decision formal context P will lose the decision rule information although it can preserve the granular rule information. Example 2 illustrates the difference between the decision rule oriented knowledge reduction and the granular (U,∅)
rule oriented knowledge reduction. Furthermore, by combining the difference with the relation between these two kinds of knowledge reduction, we understand that the decision rule oriented knowledge reduction can preserve more decision rule information than the granular rule oriented knowledge reduction.
5 Experiments In this section, we conduct some experiments to compare the proposed algorithm with the existing one in [24] in terms of the efficiency of extracting the non-redundant decision rules from a decision formal context. In the experiments, five reallife databases including Bacteria [33], Zoo [34], Balance Scale [34], Wine [34], and Car Evaluation [34] are analyzed to achieve the task of comparing the efficiency. The detailed information on the five real-life databases is shown in Table 2. For each of the chosen databases, we took the classification attribute as the decision attribute and the other attributes (variables) as the conditional attributes. Then, using the scaling approach [10] to convert the discrete (but not Boolean) attributes of Bacteria, Zoo and Balance Scale into Boolean ones, we obtained three decision formal contexts which are denoted by Date sets 1, 2 and 3, respectively. In the Wine database, there are 178 instances (each of them denotes a wine) characterized by 13 variables (each of them denotes a constituent found in each of the wines) whose values are all continuous. Here, we classified, from small to large, the values of each variable into three pairwise disjoint intervals with their length being the same. For example, for the seventh variable (Flavanoids in the Wine database), since its minimum value is 0.34 and its maximum value is 5.08, we obtained three pairwise disjoint intervals: [0.34, 1.92), [1.92, 3.50), [3.50, 5.08] according to the above classification approach. Then, by using the scaling approach, a decision formal context was obtained and is denoted by Data set 4. In the Car Evaluation database, its 1728 instances Table 2 The main characters of the five chosen real-life databases Database
({1,3,5},{a})
Instances
Classes
({4,5},{e})
Input attributes excluding the classification attribute Boolean
Discrete but not Boolean
({2},{d})
({3},{a,f})
({5},{a,e})
(∅,E)
Fig. 2 Hasse diagram of the concept lattice of (U, E, IE)
Continuous
Bacteria
17
6
16
0
0
Zoo Balance Scale
101 625
7 3
15 0
1 4
0 0
Wine
178
3
0
0
13
1,728
4
0
6
0
Car Evaluation
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Table 3 A contrast between Algorithm 1 and Algorithm 2 in terms of the running time Data set
|U|
|A|
|D|
Number of NR-rules
Running time (s) Algorithm 1
Algorithm 2
Data set 1
17
16
6
6
0.036
0.018
Data set 2
101
21
7
9
0.799
0.428
Data set 3
625
20
3
303
109.347
78.011
Data set 4
178
39
3
256
2,826.048
680.908
Data set 5
1,728
19
4
141
4,121.354
2,810.434
are characterized by six attributes which are Buying price, Price of the maintenance, Number of doors, Capacity in terms of persons to carry, The size of luggage boot, Estimated safety of the car. For our purpose, the values of the third attribute are divided into two hierarchies: I (2 or 3) and II (4 or 5). Similarly, using the scaling approach, we obtained another decision formal context denoted by Data set 5. Then Algorithms 1 and 2 were applied to Data sets 1, 2, 3, 4 and 5. The corresponding running time is reported in Table 3, in which |U|, |A| and |D| denote the cardinalities of the object set, the conditional attribute set and the decision attribute set of the concerned decision formal context, respectively, and NR is the abbreviation of the term ‘nonredundant’. It can be seen from Table 3 that by the running time of extracting the non-redundant decision rules, Algorithm 2 is much more efficient than Algorithm 1 for each of the chosen data sets.
6 Conclusion Rule acquisition is one of the main purposes in the analysis of decision formal contexts. In this paper, we have proposed a new algorithm of deriving the non-redundant decision rules from a decision formal context, proved that the proposed rule acquisition algorithm is of less time complexity than the existing one in [24], and conducted some experiments to compare their efficiency. Furthermore, the relationship between the non-redundant decision rules and the granular rules has been investigated in the consistent decision formal contexts. Also, the relation and the difference between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction have been clarified. From the point of view of applications, the results obtained in this paper need to be further extended to the case of fuzzy decision formal contexts [35], incomplete
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decision formal contexts [36] or even real decision formal contexts [37–39] since in the real world the relationship between some objects and attributes of a decision formal context may be fuzzy-valued, interval-valued or even realvalued. This issue will be discussed in our future work. Acknowledgments The authors would like to thank the anonymous reviewers for their valuable comments and helpful suggestions which lead to a significant improvement on the manuscript. This work was supported by the National Natural Science Foundation of China (Nos. 10971161, 61005042, 11071281 and 61202018).
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Int. J. Mach. Learn. & Cyber. (2013) 4:721–731 20. Qu KS, Zhai YH, Liang JY et al (2007) Study of decision implications based on formal concept analysis. Int J Gen Syst 36(2):147–156 21. Shao MW (2007) Knowledge acquisition in decision formal contexts. In: Proceedings of the sixth international conference on machine learning and cybernetics, Hong Kong, pp 4050–4054 22. Wu WZ, Leung Y, Mi JS (2009) Granular computing and knowledge reduction in formal contexts. IEEE Trans Knowl Data Eng 21(10):1461–1474 23. Li J, Mei C, Lv Y (2011) A heuristic knowledge-reduction method for decision formal contexts. Comput Math Appl 61(4):1096–1106 24. Li J, Mei C, Lv Y (2011) Knowledge reduction in decision formal contexts. Knowl Based Syst 24(5):709–715 25. Li J, Mei C, Lv Y (2012) Knowledge reduction in formal decision contexts based on an order-preserving mapping. Int J Gen Syst 41(2):143–161 26. Song XX, Wang X, Zhang WX (2012) Independence of axiom sets characterizing formal concepts. Int J Mach Learn Cybern. doi:10.1007/s13042-012-0110-z 27. Kent RE (1994) Rough concept analysis. In: Ziarko WP (ed) Rough sets, Fuzzy sets and knowledge discovery. Springer, London, pp 248–255 28. Yao YY (2004) Concept lattices in rough set theory. In: Proceedings of 2004 annual meeting of the north American fuzzy information processing society, Banff, Canada, pp 796–801 29. Du¨ntsch I, Gediga G (2003) Approximation operators in qualitative data analysis. In: Swart H et al (eds) Theory and applications of relational structures as knowledge instruments, Lecture Notes in Computer Science, vol 2929. Springer, Berlin, pp 214–230
731 30. Wei L, Qi JJ, Zhang WX (2008) Attribute reduction theory of concept lattice based on decision formal contexts. Sci China Ser F 51(7):910–923 31. Pei D, Mi JS (2011) Attribute reduction in decision formal context based on homomorphism. Int J Mach Learn Cyber 2(4):289–293 32. Wang H, Zhang WX (2008) Approaches to knowledge reduction in generalized consistent decision formal contexts. Math Comput Model 48:1677–1684 33. Fielding AH (2007) Clustering and classification techniques for the biosciences. Cambridge University Press, London 34. Frank A, Asuncion A (2010) UCI machine learning repository [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science 35. Pei D, Li MZ, Mi JS (2011) Attribute reduction in fuzzy decision formal contexts. In: International conference on machine learning and cybernetics. IEEE Press, New York, pp 204–208 36. Li J, Mei C, Lv Y (2013) Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction. Int J Approx Reason 54(1):149–165 37. Li J, Mei C, Lv Y (2012) Knowledge reduction in real decision formal contexts. Inform Sci 189:191–207 38. Li J, Mei C, Lv Y, Zhang X (2012c) A heuristic knowledge reduction algorithm for real decision formal contexts. In: Yao JT et al (eds) Proceedings of RSCTC, Lecture Notes in Artificial Intelligence, vol 7413. Springer, Berlin, pp 303–312 39. Yang HZ, Leung Y, Shao MW (2011) Rule acquisition and attribute reduction in real decision formal contexts. Soft Comput 15(6):1115–1128
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ORIGINAL ARTICLE
On rule acquisition in decision formal contexts Jinhai Li • Changlin Mei • Cherukuri Aswani Kumar Xiao Zhang
•
Received: 6 August 2012 / Accepted: 16 January 2013 / Published online: 6 February 2013 Ó Springer-Verlag Berlin Heidelberg 2013
Abstract Rule acquisition is one of the main purposes in the analysis of decision formal contexts. Up to now, there have existed several types of rules (e.g., the decision rules and the granular rules) in decision formal contexts. This study firstly proposes a new algorithm with less time complexity for deriving the non-redundant decision rules from a decision formal context. Then, we invesigate decision rules and the granular rules in the consistent decision formal contexts and make a contrast between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. Finally, some experiments are conducted to assess the efficiency of the proposed rule acquisition algorithm. Keywords Formal concept analysis Formal context Decision formal context Concept lattice Rule acquisition
1 Introduction Formal concept analysis (FCA), proposed by Wille [1], is oriented towards the discovery of formal concepts and the J. Li (&) C. Mei X. Zhang School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, People’s Republic of China e-mail: [email protected] C. Mei e-mail: [email protected] X. Zhang e-mail: [email protected] C. A. Kumar School of Information Technology and Engineering, VIT University, Vellore, India e-mail: [email protected]
construction of concept hierarchies. Nowadays, this theory has shown a trend of multidisciplinary intersection and fusion and hence has become an effective tool for conceptual data analysis and knowledge processing. FCA starts with the notion of a formal context (U, A, I) which consists of the object set U, the attribute set A and the binary relation I U A indicating that each object of U has what attributes in A. In FCA, the basic tool used to analyze databases is the concept lattice which is constituted by all the formal concepts of a formal context together with the subconcept–superconcept-relation. According to Wille’s definition [1], a formal concept is an ordered pair (X, B) in which X U contains exactly those objects shared by all the attributes in B and B A contains exactly those attributes that all the objects in X have in common. In recent years, many studies [2–9] have been devoted to the issue of reducing the size of the concept lattice of a formal context in order to improve the understandability of the resulting concept lattice. In FCA, a useful way of characterizing attribute dependencies in a formal context is via attribute implication rules or association rules. How to derive these rules efficiently has drawn much attention in the literature [10–18]. Decision formal contexts [19], a useful extension of the formal contexts, were proposed to implement decision analysis using the concept lattice. Rule acquisition is one of the main purposes in the analysis of decision formal contexts. A few studies [20–25] have recently been devoted to the rule acquisition in decision formal contexts. To the best of our knowledge, there have existed several types of rules (e.g., the decision rules [23–25] and the granular rules [22]) in decision formal contexts. Although a rule acquisition method for deriving all the non-redundant decision rules from a decision formal context has been proposed in [24], this method depends heavily on both the conditional
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concept lattice and the decision concept lattice. Since the construction of the concept lattice for a large database is very time-consuming, the method proposed in [24] may not be efficient enough in handling a large database. If some efficient methods for extracting the decision rules from a decision formal context are developed, it is of interest to investigate the relationship between the decision rules and the granular rules. Furthermore, knowledge reduction is always one of the important issues in FCA. Therefore, it is also essential to clarify the relation and the difference between the decision rule oriented knowledge reduction [24] and the granular rule oriented knowledge reduction [22]. In this paper, we first propose an efficient algorithm to derive the non-redundant decision rules from a decision formal context. Then we investigate the relationship between the non-redundant decision rules and the granular rules in the consistent decision formal contexts and further make a contrast between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. Finally, we conduct some experiments to assess the efficiency of the proposed rule acquisition algorithm.
2 Preliminaries In this section, we briefly review some basic notions related to FCA in order to make the paper self-contained. Definition 1 ([1]) A formal context is a triple (U, A, I) consisting of the object set U (called the universe of discourse), the attribute set A and the binary relation I U A in which (x, a) [ I indicates that the object x has the attribute a and ðx; aÞ 62 I means the opposite. A formal context (U, A, I) is said to be regular [19] if for any (x, a) [ U 9 A, the following conditions hold: there exist a1, a2 [ A such that (x, a1) [ I and ðx; a2 Þ 62 I; (ii) there exist x1, x2 [ U such that (x1, a) [ I and ðx2 ; aÞ 62 I:
(i)
It should be noted that an irregular formal context (U, A, I) can be regularized by removing the rows with their objects having all the attributes or having no attribute in A and the columns with their attributes being shared by all the objects in U or not being shared by any object of U. Such way of the regularization causes no effect on the analysis results of the formal context. Thus, without loss of generality, the formal contexts discussed hereinafter are all assumed to be regular. Wille [1] introduced a pair of concept forming operators on a formal context (U, A, I):
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X " ¼ fa 2 A j 8x 2 X; ðx; aÞ 2 IgðX UÞ; B# ¼ fx 2 U j 8a 2 B; ðx; aÞ 2 IgðB AÞ:
ð1Þ
That is, X " is the maximal set of the attributes that all the objects in X have in common, and B; is the maximal set of the objects shared by all the attributes in B. Definition 2 ([1]) Let (U, A, I) be a formal context. The ordered pair (X, B) with X U and B A is called a formal concept of (U, A, I) if X " ¼ B and B; = X. Here, the sets X and B are called the extent and the intent of the formal concept (X, B), respectively. For two formal concepts (X1, B1) and (X2, B2), if X1 X2 (or B2 B1 ), then (X1, B1) is called a subconcept of (X2, B2), or (X2, B2) is called a superconcept of (X1, B1). The subconcept–superconcept-relation between the formal concepts is denoted by B. Then the set of all the formal concepts of a formal context (U, A, I) together with the partial order relation B forms a complete lattice which is denoted by BðU; A; IÞ and is called the concept lattice of the formal context (U, A, I). In BðU; A; IÞ; the infimum and the supremum of {(X1, B1), (X2, B2)} are respectively defined by ðX1 ; B1 Þ ^ ðX2 ; B2 Þ ¼ ðX1 \ X2 ; ðB1 [ B2 Þ#" Þ; ðX1 ; B1 Þ _ ðX2 ; B2 Þ ¼ ððX1 [ X2 Þ"# ; B1 \ B2 Þ:
ð2Þ
In [26], the authors discussed the axiomatic characterization of the concept lattice. It should be pointed out that except Wille’s concept lattice, there have existed several other kinds of concept lattices in the classical FCA, e.g., the rough concept lattice [27], the objectoriented concept lattice [28] and the property-oriented concept lattice [29]. The relationship among Wille’s, object-oriented and property-oriented concept lattices was discussed in [5]. In this paper, we only focus on Wille’s concept lattice to discuss the rule acquisition in decision formal contexts. Definition 3 ([22]) Let (U, A, I) be a formal context and E A: The restriction of I on U 9 E, denoted by IE, is defined as IE(x, a) = I(x, a) for any (x, a) [ U 9 E. The formal context (U, E, IE) is called a subcontext of (U, A, I). Similar to the formal context (U, A, I), a pair of concept forming operators can also be defined on the subcontext (U, E, IE) as follows: X "E ¼ fa 2 E j 8x 2 X; ðx; aÞ 2 IE gðX UÞ; B#E ¼ fx 2 U j 8a 2 B; ðx; aÞ 2 IE gðB EÞ:
ð3Þ
In fact, the operators "E and ;E are the restrictions of the concept forming operators " and ; on the subcontext (U, E, IE). Similar to the discussion in Definition 2, we can give the notion of a formal concept with its extent and
Int. J. Mach. Learn. & Cyber. (2013) 4:721–731
723
intent in (U, E, IE). Also, the set of all the formal concepts of the subcontext (U, E, IE) together with the subconcept– superconcept-relation B forms a complete lattice denoted by BðU; E; IE Þ; in line with the notation of the concept lattice of (U, A, I). We denote by BU ðU; E; IE Þ the set of the extents of all the formal concepts of (U, E, IE). Proposition 1 ([1]) Let (U, A, I) be a formal context and E A: For X; X1 ; X2 U and B; B1 ; B2 E; the following properties hold: (i)
"
"
# B2 E
#
X1 X2 ) X2 E X1 E ;
(ii) B1 B2 ) B1E ; "E #E (iii) X X ; B B# E " E ; "E #E "E ; X Þ; ðB#E ; B#E "E Þ 2 BðU; E; IE Þ: (iv) ðX It should be noted that ðfxg"E #E ; fxg"E Þðx 2 UÞ and ðfag#E ; fag#E "E Þða 2 AÞ are referred to as the object concepts and the attribute concepts [10], respectively. For brevity, in the rest of the paper, we write them as ðx"E #E ; x"E Þ and ða#E ; a#E "E Þ; respectively. Proposition 2 ([1]) Let (U, A, I) be a formal context, E A and T be an index set. For Xt U; Bt E (t [ T), we have !"E !#E [ \ [ \ "E Xt ¼ Xt and Bt ¼ B t #E : t2T
t2T
t2T
t2T
3 An efficient rule acquisition algorithm for decision formal contexts Definition 4 ([19]) A decision formal context is a quintuple (U, A, I, D, J), where (U, A, I) and (U, D, J) with A \ D ¼ ; are two formal contexts. Here, A and D are called the conditional attribute set and the decision attribute set of (U, A, I, D, J), respectively. Like the formal context, a decision formal context P ¼ ðU; A; I; D; JÞ is also said to be regular [24] if both (U, A, I) and (U, D, J) are regular. The decision formal context PE ¼ ðU; E; IE ; D; JÞ is called a subcontext of P if (U, E, IE) is a subcontext of (U, A, I). Without loss of generality, the decision formal contexts discussed hereinafter are all assumed to be regular. The concept lattice of (U, D, J) is denoted by BðU; D; JÞ and the set of the extents of all the formal concepts of (U, D, J) is denoted by BU ðU; D; JÞ: Definition 5 ([24]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For any ðX; BÞ 2 BðU; E; IE Þ
and ðY; CÞ 2 BðU; D; JÞ; if X, B, Y and C are all nonempty and X Y; then the expression B ! C is called a decision rule generated between the formal concepts (X, B) and (Y, C). Here, B and C are called the premise and the conclusion of the decision rule B ! C; respectively. The set of all the decision rules generated between the formal concepts in BðU; E; IE Þ and those in BðU; D; JÞ is denoted by RðPE Þ; where PE ¼ ðU; E; IE ; D; JÞ: Thus, for any B ! C 2 RðPE Þ with ðX; BÞ 2 B ðU; E; IE Þ and ðY; CÞ 2 BðU; D; JÞ; we have that each x 2 U having all the attributes in B also has all the attributes in C. So, the decision rule is in fact an ‘‘If–then’’ conjunctive rule. That is, B ! C means ‘‘If ^B, then ^C’’. Moreover, it is easy to verify that B ! C is supported by and only by the objects in X. It should be pointed out that the decision rules have something to do with both the attribute implication rules and the association rules (see, e.g. [10, 11, 20] for the detailed discussion of the attribute implication rules and e.g. [16, 17] for the association rules). Concretely, a decision rule is a special attribute implication rule. However, an attribute implication rule may not be a decision rule since the premise or the conclusion of an attribute implication rule is not required to be the intent of a formal concept but an attribute set only. Also, a decision rule is a special association rule. But an association rule may not be a decision rule because the antecedent of an association rule may not be the intent of a formal concept and the confidence is often less than one. Definition 6 ([24]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For B1 ! C1 ; B2 ! C2 2 R ðPE Þ; if B1 B2 and C2 C1 ; we say that B2 ! C2 can be implied by B1 ! C1 : We denote this implication relationship by B1 ! C1 ) B2 ! C2 : For any B ! C 2 RðPE Þ; if there exists B0 ! C0 2 RðPE ÞnfB ! Cg such that B0 ! C0 ) B ! C; then B ! C is said to be redundant in RðPE Þ; otherwise, B ! C is said to be non-redundant in RðPE Þ: We denote by R ðPE Þ the set of all the nonredundant decision rules in RðPE Þ: It can be known from Definition 6 that for a given decision formal context, it is more appealing to extract the non-redundant decision rules since its redundant decision rules can be implied by the non-redundant ones. Let P ¼ ðU; A; I; D; JÞ be a decision formal context. For any ðX; YÞ 2 BU ðU; A; IÞ BU ðU; D; JÞ; define 8 < 1; if X Y and there does not exist X0 aðX; YÞ ¼ 2 BU ðU; A; IÞ such that X X0 Y; : 0; otherwise; ð4Þ
123
724
bðX; YÞ ¼
Int. J. Mach. Learn. & Cyber. (2013) 4:721–731
8 < 1; :
0;
if X Yand there does not exist Y0 2 BU ðU; D; JÞ such that X Y0 Y; otherwise. ð5Þ
In [24], the authors put forward a method to derive the non-redundant decision rules from a decision formal context. The method can briefly be described as follows:
Proof
It is immediate from Definitions 6, 7 and 8.
Theorem 2 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For ðX; BÞ 2 BðU; E; IE Þ with X 6¼ ; T and B 6¼ ;; if C ¼ x2X x"D 6¼ ;; then B ! C is a feedforward non-redundant decision rule in RðPE Þ; where "D denotes the operator " on (U, D, J). Proof Since the ordered pairs ðx"D #D ; x"D Þðx 2 XÞ are object concepts of (U, D, J) where ;D denotes the operator ; on (U, D, J), the supremum of fðx"D #D ; x"D Þjx 2 Xg; denoted by (Y, C), satisfies ðY; CÞ ¼
W
ðx
"D #D
"D
Then [ X x"D #D
where |LA| and |LD| denote the cardinalities of the concept lattices BðU; A; IÞ and BðU; D; JÞ; respectively. In order to enhance the efficiency of the rule acquisition method in decision formal contexts, we shall propose in the following a new algorithm of deriving the non-redundant decision rules which is of less time complexity than Algorithm 1. Definition 7 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For any B ! C 2 RðPE Þ; if there exists B0 ! C0 2 RðPE Þ such that B0 B and C C0 ; we say that B ! C is a feedforward redundant decision rule in RðPE Þ; otherwise, we say that B ! C is a feedforward non-redundant decision rule in RðPE Þ: Definition 8 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For any B ! C 2 RðPE Þ; if there exists B0 ! C0 2 RðPE Þ such that B0 B and C C0 ; we say that B ! C is a feedback redundant decision rule in RðPE Þ; otherwise, we say that B ! C is a feedback nonredundant decision rule in RðPE Þ: Theorem 1 Let P ¼ ðU; A; I; D; JÞ be a decision formal context, E A and B ! C 2 RðPE Þ: Then the following statements hold: B ! C is redundant in RðPE Þ iff B ! C is a feedforward or feedback redundant decision rule in RðPE Þ: (ii) B ! C is non-redundant in RðPE Þ iff B ! C is both a feedforward and a feedback non-redundant decision rule in RðPE Þ: (i)
123
x2X
S
;x Þ ¼
x2X
It is easy to prove that the time complexity of Algorithm 1 is O ðjUj þ jAjÞjAjjLA j þ jUjjLA j2 jLD j þ jUjjLA jjLD j2 ;
h
x
"D #D
" D # D ;
x2X
[
T
! x
"D
2 BðU; D; JÞ:
x2X
!"D #D x
"D #D
¼ Y:
x2X
Thus, B ! C 2 RðPE Þ according to the assumption. Furthermore, we can prove that B ! C is a feedforward non-redundant decision rule in RðPE Þ: In fact, if B ! C is a feedforward redundant decision rule in RðPE Þ; there exists B0 ! C0 2 RðPE Þ such that B0 B and C C0 : Suppose ðY0 ; C0 Þ 2 BðU; D; JÞ: Then we obtain Y0 Y: However, !"D #D !"D #D [ [ "D #D "D #D Y¼ x x ¼ Y0 ; x2X
x2Y0
which is in contradiction with Y0 Y:
h
Theorem 3 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For ðX; BÞ 2 BðU; E; IE Þ with X 6¼ T ;; B 6¼ ; and C ¼ x2X x"D 6¼ ;; then B ? C is nonredundant in RðPE Þ iff there does not exist ðX0 ; B0 Þ 2 BðU; E; IE Þ with X0 6¼ ; and B0 6¼ ; such that B0 ! C 2 RðPE Þ and B0 , B. Proof Necessity. If there exists ðX0 ; B0 Þ 2 BðU; E; IE Þ with X0 6¼ ; and B0 6¼ ; such that B0 ! C 2 RðPE Þ and B0 , B, it follows from Definition 8 that B ? C is a feedback redundant decision rule in RðPE Þ: According to Theorem 1, B ? C is redundant in RðPE Þ; which is in contradiction with the assumption that B ? C is nonredundant in RðPE Þ: Sufficiency. If B ? C is redundant in RðPE Þ; by Theorem 1 we have that B ? C is a feedforward or feedback redundant decision rule in RðPE Þ: (i)
If B ? C is a feedforward redundant decision rule in RðPE Þ; then there exists B0 ! C0 2 RðPE Þ such that B0 B and C , C0. Suppose ðY0 ; C0 Þ 2 BðU; D; JÞ and let Y ¼ C #D : Then we obtain Y0 , Y. However, it follows from Propositions 1 and 2 that
Int. J. Mach. Learn. & Cyber. (2013) 4:721–731
Y ¼C
#D
¼
\
!#D x
"D
\
¼
x2X
[
¼
x
x
x2X
!"D #D
"D #D
! #D "D #D "D
x2X
[
x
!"D #D
"D #D
¼ Y0 ;
x2Y0
which is in contradiction with Y0 Y: (ii) If B ! C is a feedback redundant decision rule in RðPE Þ; there exists B0 ! C0 2 RðPE Þ such that B0 B and C C0 : Since C C0 does not hold, we obtain C = C0. Thus, B0 ! C 2 RðPE Þ and B0 B; which is in contradiction with the assumption. h Definition 9 Let P ¼ ðU; A; I; D; JÞ be a decision formal context, E A and X RðPE Þ:B ! C 2 X is called a premise-minimal decision rule in X if B0 6 B for any B0 ! C 2 X:
725
feedforward non-redundant decision rule in RðPE Þ: By Definition 10, B ! C 2 MðPE Þ follows. Furthermore, we can prove B ! C 2 M ðPE Þ: In fact, if there exists B0 ! C 2 MðPE Þ such that B0 B; then by Definition 8 B ! C is a feedback redundant decision rule in RðPE Þ; which is in contradiction with B ! C 2 R ðPE Þ: Thus, based on Definition 9, we obtain that B ! C is a premise-minimal decision rule in MðPE Þ; i.e. B ! C 2 M ðPE Þ: By Theorem 4, we can obtain all the non-redundant decision rules from a decision formal context P ¼ ðU; A; I; D; JÞ via M ðPÞ: By Definitions 9 and 10, it is natural to first compute MðPÞ and then obtain M ðPÞ: However, here we directly compute M ðPÞ without generating MðPÞ in advance in order to reduce the computational complexity, which is described as follows:
Definition 10 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: Define MðPE Þ ¼ fB ! C j T ðX; BÞ 2 BðU; E; IE Þ; X 6¼ ;; B 6¼ ;; C ¼ x2X x"D 6¼ ;g and denote by M ðPE Þ the set of all the premise-minimal decision rules in MðPE Þ: Theorem 4 Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: Then R ðPE Þ ¼ M ðPE Þ: Proof On one hand, for any B ! C 2 M ðPE Þ; we have by Definitions 9 and 10 that B ! C 2 RðPE Þ and B0 6 B for any B0 ! C 2 MðPE Þ: It can be known from Theorem 2 that B ? C is a feedforward non-redundant decision rule in RðPE Þ: If B ? C is a feedback redundant decision rule in RðPE Þ; there exists B0 ! C0 2 RðPE Þ such that B0 , B and C C0 : Since C , C0 is not true, we obtain C = C0. Thus, B0 ! C 2 MðPE Þ and B0 , B, which is in contradiction with B0 6 B for any B0 ! C 2 MðPE Þ: As a result, B ? C is a feedback non-redundant decision rule in RðPE Þ: By Theorem 1, B ! C 2 R ðPE Þ is at hand. On the other hand, for any B ! C 2 R ðPE Þ; there exist X 2 BU ðU; E; IE Þ and Y 2 BU ðU; D; JÞ such that ðX; BÞ 2 BðU; E; IE Þ and ðY; CÞ 2 BðU; D; JÞ: Noting that 0 1 !"D #D [ \ @ x"D #D ; x"D A 2 BðU; D; JÞ x2X
x2X
and X
[ x2X
!"D #D x
"D #D
[
!"D #D "D #D
x
¼ Y;
x2Y
T T we obtain B ! x2X x"D 2 RðPE Þ and C x2X x"D : T Then, we can conclude C ¼ x2X x"D since B ! C is a
It is easy to prove that the time complexity of Algorithm 2 is O ðjUj þ jAjÞjAjjLA j þ jDjjLA j2 ; where |LA| denotes the cardinality of BðU; A; IÞ: Note that |U||LD| |D| generally holds for a given decision formal context. So, Algorithm 2 is of less time complexity than Algorithm 1. We have the following example to illustrate Algorithm 2. h Example 1 Table 1 shows a decision formal context P ¼ ðU; A; I; D; JÞ; where U = {1, 2, 3, 4, 5}, A = {a, b, c, d, e, f} and D = {d1, d2, d3}. In the table, the number 1 in the ith row and jth column represents that the object in the ith row has the attribute in the jth column, and the number 0 in the ith row and jth column means the opposite. Figure 1 depicts the Hasse diagram of the concept lattice of the formal context (U, A, I). By Theorem 2, all the feedforward non-redundant decision rules generated by the formal concepts of (U, A, I) are as follows: r 1:
a ! d1 ; which is generated by ({1, 3, 5}, {a}) due to 1"D \ 3"D \ 5"D ¼ fd1 g and is supported by objects 1, 3, 5;
123
726
Int. J. Mach. Learn. & Cyber. (2013) 4:721–731
Table 1 A decision formal context P ¼ ðU; A; I; D; JÞ U
a
b
c
d
e
f
d1
d2
d3
1
1
0
0
0
0
0
1
0
0
2
0
1
0
1
0
0
0
1
1
3
1
0
1
0
0
1
1
1
0
4
0
1
0
0
1
0
1
1
0
5
1
1
1
0
1
0
1
1
0
(U,∅)
4 Decision rules and granular rules in the consistent decision formal contexts In Sect. 3, we have proposed an efficient algorithm to extract all the non-redundant decision rules from a decision formal context. Since the decision rules and the granular rules are two useful ways of discovering knowledge from a decision formal context, it is of interest to investigate the relationship between the non-redundant decision rules and the granular rules. Furthermore, considering that knowledge reduction is one of the key issues in FCA, it is also essential to clarify the relation and the difference between the decision rule oriented knowledge reduction [24] and the granular rule oriented knowledge reduction [22].
({2,4,5},{b})
({1,3,5},{a})
4.1 The relationship between the non-redundant decision rules and the granular rules ({3,5},{a,c})
({4,5},{b,e})
({3},{a,c,f})
({5},{a,b,c,e})
Definition 11 ([22]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context. If x"# x"D #D for any x 2 U; then P is said to be consistent; otherwise, P is said to be inconsistent. ({2},{b,d})
(∅,A)
Fig. 1 Hasse diagram of the concept lattice of (U, A, I)
r 2:
r 3: r 4:
r 5:
r 6:
r 7:
ac ! d1 d2 ; which is generated by ({3, 5}, {a, c}) due to 3"D \ 5"D ¼ fd1 ; d2 g and is supported by objects 3, 5; acf ! d1 d2 ; which is generated by ({3}, {a, c, f}) due to 3"D ¼ fd1 ; d2 g and is supported by object 3; b ! d2 ; which is generated by ({2, 4, 5}, {b}) due to 2"D \ 4"D \ 5"D ¼ fd2 g and is supported by objects 2, 4, 5; be ! d1 d2 ; which is generated by ({4, 5}, {b, e}) due to 4"D \ 5"D ¼ fd1 ; d2 g and is supported by objects 4, 5; abce ! d1 d2 ; which is generated by ({5}, {a, b, c, e}) due to 5"D ¼ fd1 ; d2 g and is supported by object 5; bd ! d2 d3 ; which is generated by ({2}, {b, d}) due to 2"D ¼ fd2 ; d3 g and is supported by object 2.
Thus, by applying Algorithm 2 to the decision formal context P; the rules r1, r2, r4, r5 and r7 are added into M one by one. That is, P has five non-redundant decision rules: r1, r2, r4, r5 and r7.
123
In FCA, there are several other kinds of consistent decision formal contexts (e.g., those in [25, 30–32]) except the one introduced above. However, for our purpose, we here only focus on the kind of consistent decision formal contexts in [22]. That is to say, the consistent decision formal contexts discussed in the following represents the ones whose consistency is defined by Definition 11. Proposition 3 Let P ¼ ðU; A; I; D; JÞ be a decision formal context. Then P is consistent iff x" ! x"D 2 RðPÞ for any x 2 U: Proof Necessity. Since P is consistent, it follows from Definition 11 that x"# x"D #D for any x 2 U: Noting that ðx"# ; x" Þ 2 BðU; A; IÞ; ðx"D #D ; x"D Þ 2 BðU; D; JÞ and P is regular, we know that x"# ; x" ; x"D #D and x"D are all nonempty, which leads to x" ! x"D 2 RðPÞ: Sufficiency. It is immediate from Definitions 5 and 11. h Definition 12 ([22]) Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. For any x 2 U; x" ! x"D is called a granular rule of P: It can be known from Definition 12 that a granular rule is a special decision rule. However, a decision rule may not be a granular rule because the premise or the conclusion of a decision rule is not required to be the intent of an object concept but that of a formal concept only. Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. For any nonempty set X 2 BU ðU; A; IÞ with T T " "D 6¼ ;; the mergence of the granx2X x 6¼ ; and x2X x " "D ular rules x ! x ðx 2 XÞ is defined as
Int. J. Mach. Learn. & Cyber. (2013) 4:721–731
^
727
\ \ " x" ! x"D ¼ x ! x"D :
x2X
x2X
ð6Þ
x2X
It is easy to verify that the mergence of x" ! x"D (x 2 X) is also a decision rule. Theorem 5 Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context and B ! C 2 RðPÞ: Then the decision rule B ! C is non-redundant in RðPÞ iff B ! V C ¼ x2B# x" ! x"D and ðx0 " \ BÞ# 6 C#D for any x0 2 C #D nB# : Proof Since B ! C 2 RðPÞ; there exist X 2 BU ðU; A; I) and Y 2 BU ðU; D; JÞ such that ðX; BÞ 2 BðU; A; IÞ and ðY; CÞ 2 BðU; D; JÞ: If B ? C is non-redundant in RðPÞ; it follows from Theorem 1 that B ? C is both a feedforward and a feedback non-redundant decision rule in RðPÞ: So, B ! T T "D 2 RðPÞ; which leads to C ¼ x2X x"D because x2X x T B ! x2X x"D can imply B ? C. Furthermore, noting that
ðX; BÞ ¼
_ x2X
0 ðx"# ; x" Þ ¼ @
[ x2X
!"# x"#
;
\
1 x " A;
x2X
V we have B!C ¼ x2B# x" !x"D : To prove ðx0 " \BÞ# 6 C #D for any x0 2 C #D nB# ; it is sufficient to show ðx0 " \ BÞ# 6 Y for any x0 2 YnX: If there exists x0 2 YnX such that ðx0 " \ BÞ# Y; then ðx0 " \ BÞ# ¼ ðx0 " \ X " Þ# ¼ ðfx0 g [ XÞ"# Y yielding ðfx0 g [ XÞ" 6¼ ;: Since ððfx0 g [XÞ"# ; ðfx0 g [ XÞ" Þ 2 BðU; A; IÞ; we obtain ðfx0 g[XÞ" ! C 2 RðPÞ; i.e. x0 " \ B ! C 2 RðPÞ: Since B ! C is a feedback non-redundant decision rule in RðPÞ; we conclude x0: \ B = B, which is in contradiction with x0 2 YnX: V If B ! C ¼ x2B# x" ! x"D and ðx0 " \ BÞ# 6 C #D for any x0 2 C#D nB# ; then by Theorem 2, B ! C is a feedforward non-redundant decision rule in RðPÞ: Thus, to prove that B ! C is non-redundant in RðPÞ; it is sufficient to show that B ! C is a feedback non-redundant decision rule in RðPÞ: In fact, if B ! C is a feedback redundant decision rule in RðPÞ; there exists B0 ! C0 2 RðPÞ such that B0 B and C C0 : Suppose ðX0 ; B0 Þ 2 BðU; A; IÞ and ðY0 ; C0 Þ 2 BðU; D; JÞ: Then X X0 Y0 Y: Therefore, there exists x 2 X0 nX such that ðx" \ BÞ# ¼ ðfxg [ XÞ"# X0"# ¼ X0 Y ¼ C#D ; which is in contradiction with ðx0 " \ BÞ# 6 C #D for any x0 2 C #D nB# : Theorem 5 clarifies the relationship between the nonredundant decision rules and the granular rules in a consistent decision formal context. h
4.2 A comparison of the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction In general, both the decision rules and the granular rules derived directly from a consistent decision formal context are not concise or compact. In order to derive more compact decision rules and/or granular rules, the issue of rule acquisition oriented knowledge reduction was discussed in [22, 24]. Since knowledge reduction is one of the important issues in FCA, it is of interest to clarify the relation and the difference between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. This issue will be discussed in the following. Definition 13 ([24]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context and E A: For X RðPÞ and X0 RðPE Þ; if each decision rule of X can be implied by a decision rule of X0 ; we say that X can be implied by X0 : We denote this implication relationship by X0 ) X: Definition 14 ([24]) Let P ¼ ðU; A; I; D; JÞ be a decision formal context. E A is called a consistent set of P if RðPE Þ ) RðPÞ: Furthermore, if E is a consistent set of P and any F E is not a consistent set of P; then E is called a reduct of P: It can easily be observed from Definitions 13 and 14 that this kind of knowledge reduction can preserve the decision rule information of a decision formal context and allows us to obtain more compact decision rules from a decision formal context. Proposition 4 Let P ¼ ðU; A; I; D; JÞ be a decision formal context. Then E A is a consistent set of P iff R ðPE Þ ) R ðPÞ: Proof Necessity. Since E A is a consistent set of P; it follows from Definition 14 that RðPE Þ ) RðPÞ: According to Definition 13, it is easy to prove R ðPE Þ ) RðPE Þ and RðPÞ ) R ðPÞ: As a result, R ðPE Þ ) RðPE Þ ) RðPÞ ) R ðPÞ: Sufficiency. Based on Definition 13, it is also easy to prove RðPE Þ ) R ðPE Þ and R ðPÞ ) RðPÞ: Thus, according to the assumption R ðPE Þ ) R ðPÞ; we can obtain RðPE Þ ) R ðPE Þ ) R ðPÞ ) RðPÞ: By Definition 14, E is a consistent set of P: h Definition 15 ([22]) Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. E A is called a granular consistent set of P if x"E #E x"D #D for any x 2 U: Furthermore, if E is a granular consistent set of P and any F E is not a granular consistent set of P; then E is called a granular reduct of P:
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Proposition 5 Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. Then E A is a granular consistent set of P iff x"E ! x"D 2 RðPE Þ for any x 2 U:
and x"D C )C #D x"D #D )Y x"D #D : Therefore, x"E #E X Y x"D #D : By Definition 15, we conclude that E is a granular consistent set of P:
Proof Necessity. If E A is a granular consistent set of P; then by Definition 15, x"E #E x"D #D holds for any x 2 U: Since P is regular, we have x"D 6¼ ;: Furthermore, we can prove x"E 6¼ ;: In fact, if x"E ¼ ;; then x"E #E ¼ U: Thus, x"D #D ¼ U yielding x"D ¼ ;; which is in contradiction with x"D 6¼ ;: Consequently, x"E ! x"D 2 RðPE Þ:
It can be known from Theorem 7 that the decision rule oriented knowledge reduction of a consistent decision formal context can preserve the granular rule information. However, the granular rule oriented knowledge reduction may not preserve the decision rule information. That is to say, it may happen that a granular consistent set is not a consistent set. In what follows, we use a counterexample to confirm this assertion. h
Sufficiency. It is immediate from Definitions 5 and 15. Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context and E A be a granular consistent set of P: We denote by R# ðPE Þ the set of all the granular rules of the subcontext PE ¼ ðU; E; IE ; D; JÞ: h Theorem 6 Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. If E A is a granular consistent set of P; then R# ðPE Þ ) R# ðPÞ: Proof It is immediate from Definition 13 and Propositions 3 and 5. It can easily be seen from Definition 15 and Theorem 6 that the granular rule oriented knowledge reduction can preserve the granular rule information of a consistent decision formal context and allows us to derive more compact granular rules from a consistent decision formal context. The following theorem illustrates the relation between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction. h Theorem 7 Let P ¼ ðU; A; I; D; JÞ be a consistent decision formal context. If E A is a consistent set of P; then E is a granular consistent set of P: Proof Since P is consistent, it follows from Definition 11 that x"# x"D #D for any x 2 U: Noting that ðx"# ; x" Þ 2 BðU; A; IÞ; ðx"D #D ; x"D Þ 2 BðU; D; JÞ and P is regular, we know that x"# ; x" ; x"D #D and x"D are all nonempty, which leads to x" ! x"D 2 RðPÞ: If E A is a consistent set of P; then there exists B ! C 2 RðPE Þ such that B x" and x"D C: Since there exist X 2 BU ðU; E; IE Þ and Y 2 BU ðU; D; JÞ such that ðX; BÞ 2 BðU; E; IE Þ and ðY; CÞ 2 BðU; D; JÞ; we have by Proposition 1 that B x " ) x 2 B# E ) x"# X
Example 2 Let P ¼ ðU; A; I; D; JÞ be the decision formal context in Example 1. Then the following statements hold: 1"# ¼ f1; 3; 5g f1; 3; 4; 5g ¼ 1"D #D ; 2"# ¼ f2g ¼ 2"D #D ; 3"# ¼ f3g f3; 4; 5g ¼ 3"D #D ; 4"# ¼ f4; 5g f3; 4; 5g ¼ 4"D #D ; 5"# ¼ f5g f3; 4; 5g ¼ 5"D #D : Thus, it follows from Definition 11 that the decision formal context P is consistent. According to the results in Example 1 and the discussion in Sect. 4.1, we obtain the following seven feedforward non-redundant decision rules via the mergence of granular rules: r 1:
r 2:
r 3:
r 4:
r 5:
r 6:
r 7:
a ! d1 ; which is generated by the extent {1, 3, 5} due to 1: \ 3: \ 5: = {a} and 1"D \ 3"D \ 5"D ¼ fd1 g; is supported by objects 1, 3, 5; ac ! d1 d2 ; which is generated by the extent {3,5} due to 3: \ 5: = {a,c} and 3"D \ 5"D ¼ fd1 ; d2 g; is supported by objects 3, 5; acf ! d1 d2 ; which is generated by the extent {3} due to 3" ¼ fa; c; f g and 3"D ¼ fd1 ; d2 g; is supported by object 3; b ! d2 ; which is generated by the extent {2, 4, 5} due to 2" \ 4" \ 5" ¼ fbg and 2"D \ 4"D \ 5"D ¼ fd2 g; is supported by objects 2, 4, 5; be ! d1 d2 ; which is generated by the extent {4,5} due to 4" \ 5" ¼ fb; eg and 4"D \ 5"D ¼ fd1 ; d2 g; is supported by objects 4, 5; abce ! d1 d2 ; which is generated by the extent {5} due to 5: = {a, b, c, e} and 5"D ¼ fd1 ; d2 g; is supported by object 5; bd ! d2 d3 ; which is generated by the extent {2} due to 2: = {b,d} and 2"D ¼ fd2 ; d3 g; is supported by object 2.
) X "E x"#"E ) B x"E ) x"E #E B#E ¼ X
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For r3 : acf ! d1 d2 ; there is 5 2 fd1 ; d2 g#D nfa; c; f g# such that
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729
ð5" \ fa; c; f gÞ# ¼ f3; 5g f3; 4; 5g ¼ fd1 ; d2 g#D ; for r6 : abce ! d1 d2 ; there is 4 2 fd1 ; d2 g#D nfa; b; c; eg# such that ð4" \fa;b;c;egÞ# ¼f4;5gf3;4;5g¼fd1 ;d2 g#D : Thus, by Theorem 5, we know that the rules r3 and r6 are redundant in RðPÞ: Similarly, it can be verified that r1, r2, r4, r5 and r7 are all non-redundant in RðPÞ: Let E = {a, d, e, f}. Then we can obtain a subcontext PE ¼ ðU; E; IE ; D; JÞ which satisfies 1
"E #E
2
"E #E
¼ f1; 3; 5g f1; 3; 4; 5g ¼ 1 ¼ f2g ¼ 2
"D #D
"D #D
;
;
3"E #E ¼ f3g f3; 4; 5g ¼ 3"D #D ; 4"E #E ¼ f4; 5g f3; 4; 5g ¼ 4"D #D ; and 5"E #E ¼ f5g f3; 4; 5g ¼ 5"D #D : Thus, E is a granular consistent set of P: Figure 2 depicts the Hasse diagram of the concept lattice of the formal context (U, E, IE). According to Theorem 5, the nonredundant decision rules derived from PE are as follows: r10 : a ! d1 ; r20 : af ! d1 d2 ; r30 : e ! d1 d2 ; r40 : d ! d2 d3 : It is easy to check that the rules r2 and r4 derived from P cannot be implied by the non-redundant decision rules derived from the subcontext PE : By Proposition 4, E is not a consistent set of P although it is a granular consistent set of P: That is to say, to remove the attributes b and c from the decision formal context P will lose the decision rule information although it can preserve the granular rule information. Example 2 illustrates the difference between the decision rule oriented knowledge reduction and the granular (U,∅)
rule oriented knowledge reduction. Furthermore, by combining the difference with the relation between these two kinds of knowledge reduction, we understand that the decision rule oriented knowledge reduction can preserve more decision rule information than the granular rule oriented knowledge reduction.
5 Experiments In this section, we conduct some experiments to compare the proposed algorithm with the existing one in [24] in terms of the efficiency of extracting the non-redundant decision rules from a decision formal context. In the experiments, five reallife databases including Bacteria [33], Zoo [34], Balance Scale [34], Wine [34], and Car Evaluation [34] are analyzed to achieve the task of comparing the efficiency. The detailed information on the five real-life databases is shown in Table 2. For each of the chosen databases, we took the classification attribute as the decision attribute and the other attributes (variables) as the conditional attributes. Then, using the scaling approach [10] to convert the discrete (but not Boolean) attributes of Bacteria, Zoo and Balance Scale into Boolean ones, we obtained three decision formal contexts which are denoted by Date sets 1, 2 and 3, respectively. In the Wine database, there are 178 instances (each of them denotes a wine) characterized by 13 variables (each of them denotes a constituent found in each of the wines) whose values are all continuous. Here, we classified, from small to large, the values of each variable into three pairwise disjoint intervals with their length being the same. For example, for the seventh variable (Flavanoids in the Wine database), since its minimum value is 0.34 and its maximum value is 5.08, we obtained three pairwise disjoint intervals: [0.34, 1.92), [1.92, 3.50), [3.50, 5.08] according to the above classification approach. Then, by using the scaling approach, a decision formal context was obtained and is denoted by Data set 4. In the Car Evaluation database, its 1728 instances Table 2 The main characters of the five chosen real-life databases Database
({1,3,5},{a})
Instances
Classes
({4,5},{e})
Input attributes excluding the classification attribute Boolean
Discrete but not Boolean
({2},{d})
({3},{a,f})
({5},{a,e})
(∅,E)
Fig. 2 Hasse diagram of the concept lattice of (U, E, IE)
Continuous
Bacteria
17
6
16
0
0
Zoo Balance Scale
101 625
7 3
15 0
1 4
0 0
Wine
178
3
0
0
13
1,728
4
0
6
0
Car Evaluation
123
730
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Table 3 A contrast between Algorithm 1 and Algorithm 2 in terms of the running time Data set
|U|
|A|
|D|
Number of NR-rules
Running time (s) Algorithm 1
Algorithm 2
Data set 1
17
16
6
6
0.036
0.018
Data set 2
101
21
7
9
0.799
0.428
Data set 3
625
20
3
303
109.347
78.011
Data set 4
178
39
3
256
2,826.048
680.908
Data set 5
1,728
19
4
141
4,121.354
2,810.434
are characterized by six attributes which are Buying price, Price of the maintenance, Number of doors, Capacity in terms of persons to carry, The size of luggage boot, Estimated safety of the car. For our purpose, the values of the third attribute are divided into two hierarchies: I (2 or 3) and II (4 or 5). Similarly, using the scaling approach, we obtained another decision formal context denoted by Data set 5. Then Algorithms 1 and 2 were applied to Data sets 1, 2, 3, 4 and 5. The corresponding running time is reported in Table 3, in which |U|, |A| and |D| denote the cardinalities of the object set, the conditional attribute set and the decision attribute set of the concerned decision formal context, respectively, and NR is the abbreviation of the term ‘nonredundant’. It can be seen from Table 3 that by the running time of extracting the non-redundant decision rules, Algorithm 2 is much more efficient than Algorithm 1 for each of the chosen data sets.
6 Conclusion Rule acquisition is one of the main purposes in the analysis of decision formal contexts. In this paper, we have proposed a new algorithm of deriving the non-redundant decision rules from a decision formal context, proved that the proposed rule acquisition algorithm is of less time complexity than the existing one in [24], and conducted some experiments to compare their efficiency. Furthermore, the relationship between the non-redundant decision rules and the granular rules has been investigated in the consistent decision formal contexts. Also, the relation and the difference between the decision rule oriented knowledge reduction and the granular rule oriented knowledge reduction have been clarified. From the point of view of applications, the results obtained in this paper need to be further extended to the case of fuzzy decision formal contexts [35], incomplete
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decision formal contexts [36] or even real decision formal contexts [37–39] since in the real world the relationship between some objects and attributes of a decision formal context may be fuzzy-valued, interval-valued or even realvalued. This issue will be discussed in our future work. Acknowledgments The authors would like to thank the anonymous reviewers for their valuable comments and helpful suggestions which lead to a significant improvement on the manuscript. This work was supported by the National Natural Science Foundation of China (Nos. 10971161, 61005042, 11071281 and 61202018).
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