International Journal of Approximate Reasoning

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1 International Journal of Approximate Reasoning 53 (22) Contents lists available at SciVerse ScienceDirect International Journal of Approximate Reasoning journal homepage: Rough sets based matrix approaches with dynamic attribute variation in set-valued information systems Junbo Zhang a, Tianrui Li a,,daruan b,c,dunliu d, a School of Information Science and Technology, Southwest Jiaotong University, Chengdu 63, China b Belgian Nuclear Research Centre (SCK CEN), Boeretang 2, 24 Mol, Belgium c Department of Applied Mathematics & Computer Science, Ghent University, 9 Gent, Belgium d School of Economics and Management, Southwest Jiaotong University, Chengdu 63, China ARTICLE INFO Article history: Received 22 June 2 Received in revised form 9 October 2 Accepted 3 January 22 Available online 9 January 22 Keywords: Rough sets Knowledge discovery Matrix Set-valued information systems ABSTRACT Set-valued information systems are generalized models of single-valued information systems. The attribute set in the set-valued information system may evolve over time when new information arrives. Approximations of a concept by rough set theory need updating for knowledge discovery or other related tasks. Based on a matrix representation of rough set approximations, a basic vector H(X) is induced from the relation matrix. Four cut matrices of H(X), denotedbyh [μ,ν] (X), H (μ,ν] (X), H [μ,ν) (X) and H (μ,ν) (X), are derived for the approximations, positive, boundary and negative regions intuitively. The variation of the relation matrix is discussed while the system varies over time. The incremental approaches for updating the relation matrix are proposed to update rough set approximations. The algorithms corresponding to the incremental approaches are presented. Extensive experiments on different data sets from UCI and user-defined data sets show that the proposed incremental approaches effectively reduce the computational time in comparison with the non-incremental approach. 22 Elsevier Inc. All rights reserved.. Introduction Rough set theory (RST) was proposed by Pawlak [ 4] in the early 98s. It is a powerful mathematical tool to describe the dependencies among attributes, evaluate the significance of attributes, and derive decision rules. Nowadays, many rough sets based-approaches have been successfully applied in machine learning and data mining. They have been found to be particularly useful for rule induction [5 7] and feature subset selection [8 2]. An information system is a quadruple IS = (U, A, V, f ), where U is a non-empty finite set of objects; A is a non-empty finite set of attributes; V = a A V a and V a is a domain of attribute a; f : U A V is an information function such that f (x, a) V a for every x U, a A. In many practical issues, it happens that some of the attribute values for an object are set-valued, which are always used to characterize uncertain information and missing information in the information system [3]. For an information system IS = (U, A, V, f ), if each attribute has a unique attribute value, then IS with V = a A V a is called a single-valued information system; if an information system is not a single-valued information system, it is called a set-valued (multi-valued) information system [4,5]. Due to uncertainty and incompleteness of information, there exist a lot of missing data in the information system, called incomplete information system [6,7]. Such information systems can also be regarded as a special case of set-valued information systems, in which all missing values can be represented by the set of all possible values of each attribute [5]. Corresponding author. addresses: JunboZhang86@63.com, jbzhang86@gmail.com (J. Zhang), trli@swjtu.edu.cn (T. Li), druan@sckcen.be, da.ruan@ugent.be (D. Ruan), newton83@63.com (D. Liu) X/$ - see front matter 22 Elsevier Inc. All rights reserved.

2 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) Due to the dynamic characteristics of data collection, an information system varies with time. It retrains the system from scratch when the information system varies, which is known as a non-incremental approach [8]. However, the nonincremental approach is often very costly or even intractable. Alternatively, it can apply an incremental updating scheme, which is an effective method to maintain knowledge dynamically, to avoid unnecessary computations by utilizing the previous data structures or results. It has successfully been used to data analysis in the real-time applications and the applications with limited memory or computation ability. Since an information system is consisted of the attributes (features), the objects (instances), and the domain of attributes values, some incremental updating approaches with respect to rough sets have been proposed in such three aspects. Incremental updating approaches under the variation of the attribute set. Chan proposed an incremental method for updating the approximations of a concept based on the lower and upper boundary sets [9]. Li et al. presented an incremental method for updating approximations of a concept in an incomplete information system through the characteristic relation [2]. Qian et al. introduced a theoretic framework based on RST, called positive approximation, which could be used to accelerate a heuristic process of attribute reduction [9]. Cheng presented two incremental methods for fast computing the rough fuzzy approximations based on boundary sets and cut sets, respectively [2]. Incremental updating approaches under the variation of the object set. Shan and Ziarko presented a discernibility-matrix based incremental methodology to find all maximally generalized rules [5]. Liu et al. proposed an incremental model and approach as well as its algorithm for inducing interesting knowledge when the object set varies over time [6]. Furthermore, in business intelligent information systems, Liu et al. presented an optimization incremental approach as well as its algorithm for inducing interesting knowledge [7]. Incremental updating approaches under the variation of the attribute values. Chen et al. proposed an incremental algorithm for updating the approximations of a concept under coarsening or refining of attributes values [22]. Generally, the computation of approximations is a necessary step in knowledge representation and reduction based on rough sets. Approximations may further be applied to data mining and machine learning related work. To our best knowledge, most incremental algorithms mainly focus on the single-valued information systems, but not consider the situation of the set-valued information system. In order to compute and update approximations effectively in the set-valued information systems, it is interesting and desirable to investigate incremental approaches in this kind of information systems. To incrementally update the approximations of a set, the boundary set of a set was taken as a springboard [9,2]. Besides, Cheng took the boundary set and cut set as springboards [2]. By using the inner product method and matrix method, Liu proposed a unified axiomatic system to characterize the upper approximation operations [23]. In this paper, we define a basic vector generated by the relation matrix to drive the lower and upper approximations directly in the set-valued information system. Then we present approaches for updating the lower and upper approximations incrementally by use of the variation of the relation matrix. The remainder of this paper is organized as follows. Section 2 introduces the basic concepts of RST in the set-valued information system. Section 3 proposes the matrix characterizations of the lower and upper approximations. Section 4 presents two approaches for updating the relation matrix and gives some illustrative examples. Section 5 proposes the nonincremental and incremental algorithms. In the Section 6, the performances of non-incremental and incremental methods are evaluated on UCI and user-defined data. The paper ends with conclusions in Section Preliminaries In this section, we outline basic concepts, notations and results of set-valued information systems [,4,5,24]. Let IS = (U, A, V, f ) be a set-valued information system, where U is a non-empty finite set of objects; A is a non-empty finite set of attributes; V is the set of attributes values; f is a mapping from U A to V, where f : U A 2 V is a set-valued mapping. Table illustrates a set-valued information system, where U ={x, x 2, x 3, x 4, x 5, x 6 }, A ={a, a 2, a 3, a 4 } and V ={,, 2}. A set-valued decision information system (U, C {d}, V, f ) is a special case of a set-valued information system, where U is a non-empty finite set of objects; C is a non-empty finite set of condition attributes and d is a decision attribute with C {d} =Ø; V = V C V d, where V C is the set of condition attribute values and V d is the set of decision attribute values; Table A set-valued information system. U a a 2 a 3 a 4 x {} {} {, 2} {, 2} x 2 {,, 2} {,, 2} {, 2} {,, 2} x 3 {, 2} {} {, 2} {, 2} x 4 {, } {, 2} {, 2} {} x 5 {, 2} {, 2} {, 2} {} x 6 {} {} {, } {, }

3 622 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) f is a mapping from U (C {d}) to V such that f : U C 2 V C is a set-valued mapping and f : U {d} V d is a single-valued mapping. Following are two common ways to present a semantic interpretation of the set-valued information systems [4,5,25 28]. Type A: For x U, b A, f (x, b) is interpreted conjunctively. For instance, suppose b is the attribute speaking a language", then f (x, b) ={Chinese, Korean, Japanese} can be interpreted as: x speaks Chinese, Korean, and Japanese. Furthermore, when considering the attribute feeding habits" of animals, if we denote the attribute value of herbivore as " and carnivore as ", then animals possessing attribute value {, } are considered as possessing both herbivorous and carnivorous nature. Type B: For x U, b A, f (x, b) is interpreted disjunctively. For instance, suppose b is the attribute speaking a language", then f (x, b) ={Chinese, Korean, Japanese} can be interpreted as: x speaks Chinese, Korean or Japanese, and x can speak only one of them. Incomplete information systems with some unknown attribute values or partial known attribute values [6,7] are such a type of set-valued information systems. Definition [4,29]. In the set-valued information system (U, A, V, f ), forb A, the tolerance relation T b is defined as: T b ={(x, y) f (x, b) f (y, b) = }. () and for B A, the tolerance relation is defined as follows: ={(x, y) b B, f (x, b) f (y, b) = }= b B T b. (2) When (x, y),wecallx and y are indiscernible or x tolerant with y w.r.t. B.Let (x) ={y y U, y x},wecall (x) thetoleranceclassforx w.r.t.. Example. A set-valued information system is presented in Table. Let B = A. Here, tolerance classes of objects in U can be computed by Definition. Therefore, (x ) ={x, x 2, x 4 }, (x 2 ) ={x, x 2, x 3, x 4, x 5, x 6 }, (x 3 ) = (x 6 ) = {x 2, x 3, x 5, x 6 }, (x 4 ) ={x, x 2, x 5, x 6 }, (x 5 ) ={x 2, x 3, x 4, x 5, x 6 }. Definition 2 [29]. Given a set-valued information system IS = (U, A, V, f ), X U, B A, the lower and upper approximations of X in terms of tolerance relation are defined as (X) ={x U (x) X} (X) ={x U (x) X = } (3) (4) Intuitively, these two approximations divide the universe U into three disjoint regions: the positive region POS TB (X),the boundary region BND TB (X) and the negative region NEG TB (X), respectively. POS TB (X) = (X) BND TB (X) = (X) (X) (5) NEG TB (X) = U (X) The positive region POS TB (X) is defined by the lower approximation, the boundary region BND TB (X) is defined by the difference between the upper and lower approximations, and the negative region NEG TB (X) is defined by the complement of the upper approximation. Example 2 (Continuation of Example ). Let X ={x, x 2, x 3, x 4, x 5 }.Since (x ) X, (x 2 ) X, (x 3 ) X, (x 4 ) X, (x 5 ) X and (x 6 ) X, then (X) ={x, x 4 };since i {, 2, 3, 4, 5, 6}, (x i ) X =,then (X) = {x, x 2, x 3, x 4, x 5, x 6 }.Hence,POS TB (X) = (X) ={x, x 4 }, BND TB (X) = (X) (X) ={x 2, x 3, x 5, x 6 } and NEG TB (X) = U (X) = Ø. However, the above definitions are not robust enough for tolerating the noisy samples in real applications. Followed by the idea of probabilistic rough sets [3], the rough sets can also be generalized by introducing a measure of inclusion degree. Given two sets X and Y in the universe U, X s inclusion degree in Y is defined as I(X, Y) = X Y, X where stands for the cardinal number of a set and X = Ø.

4 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) Definition 3. Given a set-valued information system IS = (U, A, V, f ), X U, B A, the(α, β)-probabilistic lower and upper approximation of X in terms of tolerance relation are defined as T (α,β) B (X) ={x U I( (x), X) α} T (α,β) B (X) ={x U I( (x), X) >β} (6) (7) Similarly, the (α, β)-probabilistic positive, boundary and negative regions can be defined by the (α, β)-probabilistic lower and upper approximations, respectively. POS (α,β) (X) ={x U I( (x), X) α} BND (α,β) (X) ={x U β <I( (x), X) <α} NEG (α,β) (X) ={x U I( (x), X) β} Clearly, the parameters α and β allow certain acceptable level of errors, and the probabilistic neighborhood rough set model makes the process of decision making more reasonable. Specially, it degrades to the classical rough model if α = and β =. In addition, the values of α and β can be automatically computed by using the Bayesian minimum conditional risk criterion in the decision-theoretic rough set model [3]. (8) 3. Matrix representation of the approximations In this section, we give the matrix representation of the lower and upper approximations in the set-valued information system. We propose a matrix approach for computing the approximations in the set-valued information system. Definition 4. Given two μ ν matrices Y = (y ij ) μ ν and Z = (z ij ) μ ν, preference relations " and " aredefinedas follows respectively. () Preference relations ": Y Z if and only if y ij z ij, i =, 2,...,μ, j =, 2,...,ν; (2) Preference relations ": Y Z if and only if y ij z ij, i =, 2,...,μ, j =, 2,...,ν; (3) Minimum min: min(y, Z) = (min(y ij, z ij )) μ ν. Definition 5 [23]. Let U ={x, x 2,...,x n }, and X be a subset of U. The characteristic function G(X) = (g, g 2,...,g n ) T (T denotes the transpose operation) is defined as, x i X g i = (9), x i / X where G(X) assigns to an element that belongs to X and to an element that does not belong to X. Example 3. Let U ={x, x 2, x 3, x 4, x 5, x 6 } and X ={x, x 2, x 3, x 4, x 5 }.ThenG(X) = (,,,,, ) T. Definition 6 [32]. Given a set-valued information system IS = (U, A, V, f ). Letb A and T b be a tolerance relation on U, M T b n n = (ϕ ij ) n n be an n n matrix representing T b, called the relation matrix w.r.t. b. Then, (x i, x j ) T b ϕ ij = (), (x i, x j ) T b Definition 7 [32]. Let B A and be a tolerance relation on U, M n n = (m ij ) n n be an n n matrix representing, called the relation matrix w.r.t. B.Then, (x i, x j ) m ij = (), (x i, x j ) Corollary [33]. Let M n n = (m ij ) n n and be a tolerance relation on U. Then m ii = and m ij = m ji, i, j n.

5 624 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) Lemma. Let b, b 2 AandT {b,b 2 },T b,t b2 be the tolerance relation on U. Then M T {b,b 2 } n n { } = min M T b n n, M T b 2 n n. Corollary 2. Let B A. Then M n n = min b B M T b n n. Example 4. We continue Example.LetB = A ={a, a 2, a 3, a 4 }.Then M T a M T a 3, M T a 4, M T a 2 { } Here, it is easy to verify that M min M T a 6 6, MT a 2 6 6, MT a 3 6 6, MT a , and M. Definition 8. Let B A and be a tolerance relation on U, n n be an induced diagonal matrix of M n n = (m ij ) n n.then ( n n = diag,,..., ) ( ) = diag (2) λ λ 2 λ n where λ i = n j= m ij, i n. Corollary 3. n n = (x ) (x 2 ) (x n ) n, j= m j n,..., j= m 2j n j= m nj and T B (x i ) n, i n. Example 5 Continuation of Example 4. Here we compute the induced diagonal matrix of M 6 6 by Definition 8. diag(/3, /6, /4, /4, /5, /4). Definition 9. The n-column vector called basic vector, denoted by H(X), isdefinedas: H(X) = n n (M n n G(X)) = n n n (3) where is dot product of matrices and n = M n n G(X). Corollary 4. Suppose H(X) = (h, h 2,, h n ) T.Let n n = diag( λ, λ 2,..., λ ),M n n n = (m ij ) n n,g(x) = (g, g 2,...,g n ) T, n = (ω,ω 2,...,ω n ) T,whereλ i = n j= m ij, i n. Then, i {, 2,...,n}, ω i = n j= m ijg j, h i = ω n i j= λ i = m ijg j and h i. n j= m ij

6 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) Definition. Let μ ν. Four cut matrices of H(X), denotedbyh [μ,ν] (X), H (μ,ν] (X), H [μ,ν) (X) and H (μ,ν) (X), are defined as follows. () H [μ,ν] (X) = (h i ) n h =, μ h i ν i, else (4) (2) H (μ,ν] (X) = (h i ) n h =, μ<h i ν i, else (5) (3) H [μ,ν) (X) = (h i ) n h =, μ h i <ν i, else (6) (4) H (μ,ν) (X) = (h i ) n h =, μ<h i <ν i, else (7) Remark. These four cut matrices are Boolean matrices. Lemma 2. Given any subset X U in a set-valued information system IS = (U, A, V, f ), whereu ={x, x 2,...,x n }.B A and is a tolerance relation on U. H(X) = (h, h 2,...,h n ) T is the basic vector. Then the lower and upper approximations of X in the set-valued information system can be computed from the cut matrix of H(X) as follows. () The n-column boolean vector G( (X)) of the lower approximation (X): G( (X)) = H [,] (X) (8) (2) The n-column boolean vector G( (X)) of the upper approximation (X): G( (X)) = H (,] (X) (9) Proof. Suppose G(X) = (g, g 2,...,g n ) T, G( (X)) = (g, g 2,...,g n )T, H [,] (X) = (h, h 2,...,h n )T. () ": i {, 2,...,n},ifg i =, namely, x i (X),then (x i ) X. Therefore, x j (x i ),wehave(x j, x i ) n j= and x j X, which means m ij =, g j =. That is, m ij = m ij g j. According to Corollary 4 and Definition, h i = m ijg j = n j= m ij and h = i. Hence, i {, 2,...,n}, g i h i, that is, G((X)) H [,] (X). ": i {, 2,, n}, ifh n j= i =, then h i =, namely, m ijg j n = j= m h i =. Obviously, j {, 2,...,n}, ifm ij =, ij then g j =, in other words, x j (x i ),wehavex j X. Then, (x i ) X, namely, x i (X) and g i =. Thus i {, 2,...,n}, g i h i. Therefore, G((X)) H [,] (X). Thus, G( (X)) = H [,] (X). (2) The proof is similar to that of (). Corollary 5. The positive region POS TB (X), the boundary region BND TB (X), and the negative region NEG TB (X) can also be generated from the cut matrix of H(X), respectively as follows. () The n-column boolean vector G(POS TB (X)) of the positive region: G(POS TB (X)) = H [,] (X) (2) (2) The n-column boolean vector G(BND TB (X)) of the boundary region: G(BND TB (X)) = H (,) (X) (2)

7 626 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) (3) The n-column boolean vector G(NEG TB (X)) of the negative region: G(NEG TB (X)) = H [,] (X) (22) Proof. The proof is similar to that of Lemma 2. Example 6 Continuation of Examples 4 and 5. Since X and M 6 6 have been constructed in Examples 3 and 4, respectively. Then H(X) = 6 6 (M 6 6 X) = diag(/3, /6, /4, /4, /5, /4) (3, 5, 3, 4, 4, 3)T = (, 5/6, 3/4,, 4/5, 3/4) T. We obtain n-column boolean vectors of approximations from H(X) as follows. H(X) G( (X)) = H [,] (X) 5/6 3/4 4/5 3/4 G( (X)) = H (,] (X) G(POS TB (X)) = H [,] (X) G(BND TB (X)) = H (,) (X) G(NEG TB (X)) = H [,] (X) In other words, (X) ={x, x 4 }, (X) ={x, x 2, x 3, x 4, x 5, x 6 }, POS TB (X) ={x, x 4 }, BND TB (X) ={x 2, x 3, x 5, x 6 } and NEG TB (X) = Ø, respectively. Corollary 6. The (α, β)-probabilistic lower and upper approximations and (α, β)-probabilistic positive, boundary and negative regions are generated from the cut matrix of H(X) as follows. () The n-column boolean vector G(T (α,β) B (X)) of the lower approximation T (α,β) B (X): ( G T (α,β) B (X) ) = H [α,] (X) (23) (α,β) (α,β) (2) The n-column boolean vector G( (X)) of the upper approximation TB (X): ( ) (α,β) G (X) = H (β,] (X) (3) The n-column boolean vector G(POS (α,β) ( G POS (α,β) (X) ) = H [α,] (X) (4) The n-column boolean vector G(BND (α,β) ( G BND (α,β) (X) ) = H (α,β) (X) (5) The n-column boolean vector G(NEG (α,β) ( G NEG (α,β) (X) ) = H [,β] (X) (X)) of the positive region: (X)) of the boundary region: (X)) of the negative region: (24) (25) (26) (27) Proof. The proof is similar to that of Lemma Updating approximations in set-valued information systems incrementally based on the matrix In this section, we consider the problem of updating approximations of a subset X of U in terms of variation of the attribute set in set-valued information systems. The variation of the attribute set, also called the attribute generalization [9], contains two situations, adding attributes and removing attributes, respectively. Since the lower approximation, upper approximation, positive, boundary and negative regions can be induced from the basic vector H(X) directly.the key step for computing basic vector H(X) is to construct the relation matrix. Thus, it reduces running time if we can renew the relation

8 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) matrix with an incremental updating strategy rather than reconstructing it from scratch. In the following, we discuss how to update the relation matrix incrementally while the attribute set varies. Lemma 3. Let P and Q be two attribute sets. Then M T P n n M T P Q n n,mt Q n n M T P Q n n. 4.. Updating approximations incrementally when adding an attribute set Corollary 7. Let P, Q AandQ P =. Suppose M T P Q n n = (m ) ij n n is a relation matrix representing T P Q,andM T P n n = (m ij ) n n a relation matrix representing T P. The relation matrix by adding Q to P can be updated as: m ij =, m ij = (x i, x j ) T Q, m ij = (x i, x j ) T Q Remark 2. When adding Q to P, m ij is constant while m ij =. Example 7 (Continuation of Example 4). Let P ={a, a 4 }, Q ={a 2, a 3 }. Here, we update the matrix by adding Q to P. According to Definition 7, wehavem T P. According to Corollary 7, ifm ij =, then m ij =. Moreover, m ii = and m ij = m ji by Corollary. Hence, we just update the m ij which value is equal to in M T P and i n, i < j n. Hence,M T P Q, where the four s are changed. Corollary 8. Let P, Q A, Q P = and T P ( diag,,..., λ λ 2 λ n exclusive disjunction (exclusive or). n n ( ) = diag λ, λ 2,..., λ n,whereλ i (28) 6 6 = n j= m ij. Suppose T P Q n n = ).Thenλ i λ i, i n. Moreover, λ i = λ i n j= m ij m ij,where is the logical operation Proof. According to Remark 2, m ij is constant while m ij = when adding Q to P.Hence,m ij is constant or m ij changes from to.(a)whenm ij is constant, λ i is constant; (b) When m ij changes from to, λ i decreases because λ i = n j= m ij.thus, = λ i n j= m ij m ij. λ i Example 8 (Continuation of Example 7). T P ( ) diag λ, λ 2,, λ 6 = diag(/3, /6, /5, /6, /5, /5). After adding Q to P, by Corollary 8, wehaveλ = λ = 3, λ 2 = λ 2 = 6, λ 3 = λ 3 = 4, λ 4 = λ 4 2 = 4, λ 5 = λ 5 = 5 and λ 6 = λ 6 = 4. Hence, T P Q 6 6 = diag(/3, /6, /4, /4, /5, /4). Corollary 9. Let P, Q A, Q P = and T P n = MT P n n G(X) = (ω,ω 2,...,ω n ) T,whereω i = n j= m ijg j and G(X) = {g, g 2,...,g n }. Suppose T P Q n n = (ω,ω 2,...,ω n ) T.Thenω i ω i, i n, and ω i = ω i n j= ((m ij m ) ij g j).

9 628 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) Proof. The proof is similar to that of Corollary 8. Example 9 (Continuation of Example 7). Since G(X) = (,,,,, ) T,then T P 6 = MT P 6 6 G(X) = (ω,ω 2,,ω 6 ) T = (3, 5, 4, 5, 4, 4) T. When adding Q to P, according to Corollary 9, ω = ω = 3, ω 2 = ω 2 = 5, ω 3 = ω 3 = 3, ω 4 = ω 4 2 = 4, ω 5 = ω 5 = 4 and ω 6 = ω 6 = 3. Thus, T P Q 6 = (3, 5, 3, 4, 4, 3)T. In the subsection, we discuss about how to update M T P n n, T P n n and T P n when adding an attribute set Q to P. Then, H(X) can be obtained by Corollary 4 and approximations can be computed from the cut matrix of H(X) Updating approximations incrementally when deleting an attribute set Corollary. Let Q P A. Suppose M T P Q n n = (m ) ij n n is a relation matrix representing T P Q,andM T P n n = (m ij ) n n is a relation matrix representing T P. The relation matrix by deleting Q from P can be updated as: m, m ij = (x i, x j ) T P Q ij = (29), m ij = (x i, x j ) T P Q Remark 3. When deleting Q from P, m ij is constant while m ij =. Example. We continue Example 4.LetP ={a, a 2, a 3, a 4 }, Q ={a 2, a 3 }. Here, we update the matrix by deleting Q from P. ByExample4, M T P. Then, m ii = and m ij = m ji by Corollary. Furthermore, according to Corollary, m ij = when m ij =. Therefore, we just update the m ij whose value is equal to in M T P i n, i < j n.wehavem T P Q, where the four s are changed. λ n 6 6 and ( ) Corollary. Let Q P Aand T P n n = diag λ, λ 2,..., λ n,whereλ i = ( n j= m ij. Suppose T P Q n n = diag,,..., ) λ λ 2.Thenλ i λ i, i n. Moreover, λ i = λ i + n j= m ij m ij. Proof. The proof is similar to that of Corollary 8. Example (Continuation of Example ). T P ( ) diag λ, λ 2,..., λ 6 = diag(/3, /6, /4, /4, /5, /4). After deleting Q from P, by Corollary,wehaveλ = λ = 3, λ 2 = λ 2 = 6, λ 3 = λ 3 + = 5, λ 4 = λ = 6, λ 5 = λ 5 = 5 and λ 6 = λ 6 + = 5. Hence, T P Q 6 6 = diag(/3, /6, /5, /6, /5, /5).

10 Corollary 2. Let Q P Aand T P n = MT P {g, g 2,...,g n }. Suppose T P Q n n = ( ω,ω 2,...,ω n Proof. The proof is similar to that of Corollary 8. J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) = n j= m ijg j and G(X) = n n G(X) = (ω,ω 2,...,ω n ) T,whereω i ) T.Thenω i ω i, i n, and ω i = ω i + n j= ((m ij m ) ij g j). Example 2 (Continuation of Example ). Since G(X) = (,,,,, ) T,then T P 6 = MT P 6 6 G(X) = (ω,ω 2,, ω 6 ) T = (3, 5, 3, 4, 4, 3) T.WhendeletingQ from P, according to Corollary 2, ω = ω = 3, ω 2 = ω 2 = 5, ω 3 = ω 3 + = 3, ω 4 = ω = 4, ω 5 = ω 5 = 4 and ω 6 = ω 6 + = 3. Thus, T P Q 6 = (3, 5, 4, 5, 4, 4)T. In this subsection, we proposed the methods of updating M T P n n, T P n n and T P n whendeletinganattributesetq from P. Then, H(X) can be obtained by Corollary 4 and approximations can be computed from the cut matrix of H(X). 5. Development of static and dynamic algorithms for computing approximations based on the matrix In this section, we design static and dynamic algorithms for computing approximations based on the matrix in the set-valued decision information system corresponding to Sections 3 and 4, respectively. 5.. Algorithm (The static algorithm for computing approximations based on the matrix) Algorithm : The static algorithm for computing approximations of decision classes in the set-valued decision information system based on the matrix () Input: () A set-valued decision information system with a condition attribute set P and a decision attribute d; (2) The thresholds α and β. Output: The lower and upper approximations of the decision classes in the set-valued decision information system. begin 2 3 for i = to n do m ii = ; // Construct the relation matrix with respect to T P, M T P n n = (m ij) n n // According Corollary, m ii = 4 5 for j = i + to n do if (x i, x j ) T P then m ij = m ji = ; // According to Definitions and 7 6 else m ij = m ji = ; 7 end 8 end 9 for i = to n do // Compute the induced diagonal matrix of M T P n n : = diag(/λ, /λ 2,, /λ n ) λ i = n j= m ij ; end 2 From the set-valued decision information system, 3 for each class X k in the decision attribute do // Compute the approximations of the decision classes 4 Construct the n-column Boolean vector of X k : X k = (g k, g k2,...,g kn ) T ; 5 for i = to n do 6 ω ki = n j= m ij g kj ; // Compute intermediate values = (ω ki ) 7 end 8 Let H(X k ) = (h k, h k2,...,h kn ) T ; 9 for i = to n do 2 h ki = ω ki λ ; i 2 end 22 Compute the n-column Boolean vector of lower approximation: G(T P (α,β) (X k )) = H [α,] (X k ); 23 Compute the n-column Boolean vector of upper approximation: G(T P (α,β) (Xk )) = H (β,] (X k ); // Compute H(X k ) 24 Output the approximations T (α,β) (α,β) P (X k ) and T P (Xk ) by G(T (α,β) (α,β) P (X k )) and G(T P (Xk )), respectively. 25 end 26 end Algorithm (see Algorithm ) is a static (non-incremental) algorithm for computing approximations of the decision classes based on the matrix while the set-valued decision information system is constant. Steps 2-8 are to construct the relation matrix M T P n n by Definition 7 and Corollary. According to Definition 8, steps 9- are to compute the induced diagonal matrix. Step 4 is to construct the n-column Boolean vector of a concept X. Steps 5-7 are to compute the intermediate

11 63 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) values. Steps 8-2 are to compute the basic vector H(X) by Definition 9. Steps are to compute approximations by Corollary Algorithm DAUAM-A (The dynamic algorithm for updating approximations based on the matrix when adding an attribute set) Algorithm 2: The dynamic algorithm for updating approximations based on the matrix when adding an attribute set (DAUAM-A) Input: () A set-valued decision information system with a condition attribute set P and a decision attribute d; (2) The thresholds α and β; (3) Intermediate results: M = (m ij ) n n (relation matrix), = diag(/λ i ) (induced matrix), = (ω ki ), X k = (g ki ) T (n-column Boolean vector of X k ), which have been computed by Algorithm ; (4) Adding an attribute set Q and Q P = Ø. Output: Updated lower and upper approximations of the decision classes in the set-valued decision information system. begin 2 P P Q; // An attribute set Q is added to the attribute set P 3 for i = to n do 4 m ii = ; // According Corollary, m ii = is constant 5 for j = i + to n do 6 if m ij == then 7 m ij =m ji =;// According to Corollary 7, when adding Q to P, m ij is constant while m ij = 8 else 9 if (x i, x j ) T Q then m ij = m ji = ; // Update relation matrix: M = (m ij ) n n λ i = λ i andλ j = λ j ; // Update induced matrix: = diag(/λ i ) 2 3 for each k do if g ki == then ω ki = ω ki ; // Update intermediate result: = (ω ki ) 4 if g kj == then ω kj = ω kj ; 5 end 6 else 7 m ij = m ji = ; // According to Corollary 7 8 end 9 end 2 end 2 end 22 From the set-valued decision information system, 23 for each class X k in the decision attribute do // Compute the approximations of the decision classes 24 Let H(X k ) = (h k, h k2,...,h kn ) T ; 25 for i = to n do // Compute H(X k ) 26 h ki = ω ki λ ; i 27 end 28 Compute the n-column Boolean vector of lower approximation: G(T P (α,β) (X k )) = H [α,] (X k ); 29 Compute the n-column Boolean vector of upper approximation: G(T P (α,β) (Xk )) = H (β,] (X k ); 3 Output the approximations T (α,β) (α,β) P (X k ) and T P (Xk ) by G(T (α,β) (α,β) P (X k )) and G(T P (Xk )), respectively. 3 end 32 end Algorithm DAUAM-A (see Algorithm 2) is a dynamic (incremental) algorithm for updating approximations while adding an attribute set. Step 2 is to update the condition attribute set P. Steps 3-2 are to update intermediate results: M = (m ij ) n n (relation matrix), = diag(/λ i ) (induced matrix), = (ω ki ) when adding some attributes, which are the key steps in the Algorithm DAUAM-A. Specially, Step is to update the relation matrix, Step is to update the induced matrix and Steps 2-5 are to update the intermediate result according to Corollaries 7, 8 and 9, respectively. Steps 22-3 are to compute basic vectors of the decision classes and approximations by Corollary Algorithm DAUAM-D (The dynamic algorithm for updating approximations based on the matrix when deleting an attribute set) Algorithm DAUAM-D (see Algorithm 3) is a dynamic (incremental) algorithm for updating approximations while deleting an attribute set. Step 2 is to update the condition attribute set P. Steps 3-2 are to update intermediate results: M = (m ij ) n n (relation matrix), = diag(/λ i ) (induced matrix), = (ω ki ) when deleting some attributes, which are the key steps in the Algorithm DAUAM-A. Specially, Step is to update the relation matrix, Step is to update the induced matrix and

12 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) Algorithm 3: The dynamic algorithm for updating approximations based on the matrix when deleting an attribute set (DAUAM-D) Input: () A set-valued decision information system with a condition attribute set P and a decision attribute d; (2) The thresholds α and β; (3) Intermediate result: M = (m ij ) n n (relation matrix), = diag(/λ i ) (induced matrix), = (ω ki ), X k = (g ki ) T (n-column Boolean vector of X k ), which have been computed by Algorithm ; (4) Deleting an attribute set Q and Q P. Output: Updated lower and upper approximations of the decision classes in the set-valued decision information system. begin 2 P P Q; // An attribute set Q is deleted from the attribute set P 3 for i = to n do 4 m ii = ; // According Corollary, m ii = is constant 5 for j = i + to n do 6 if m ij = then 7 m ij =m ji =;// According to Corollary, when deleting Q, m ij is constant while m ij = 8 else 9 if (x i, x j ) T P Q then m ij = m ji = ; // Update relation matrix: M = (m ij ) n n λ i = λ i + andλ j = λ j + ; // Update induced matrix: = diag(/λ i ) 2 3 for each k do if g ki == then ω ki = ω ki + ; // Update intermediate result: = (ω ki ) 4 if g kj == then ω kj = ω kj + ; 5 end 6 else 7 m ij = m ji = ; // According to Corollary 8 end 9 end 2 end 2 end 22 From the set-valued decision information system, 23 for each class X k in the decision attribute do // Compute the approximations of the decision classes 24 Let H(X k ) = (h k, h k2,...,h kn ) T ; 25 for i = to n do // Compute H(X k ) 26 h ki = ω ki λ ; i 27 end 28 Compute the n-column Boolean vector of lower approximation: G(T P (α,β) (X k )) = H [α,] (X k ); 29 Compute the n-column Boolean vector of upper approximation: G(T P (α,β) (Xk )) = H (β,] (X k ); 3 Output the approximations T (α,β) (α,β) P (X k ) and T P (Xk ) by G(T (α,β) (α,β) P (X k )) and G(T P (Xk )), respectively. 3 end 32 end Steps 2-5 are to update the intermediate result according to Corollaries, and 2, respectively. Steps 22-3 are to compute basic vectors of the decision classes and approximations by Corollary Experimental evaluations To test performance of Algorithms, DAUAM-A and DAUAM-D, we download five data sets from the machine learning data repository, University of California at Irvine [34], which are all symbolic data with missing values. Here, all missing values are represented by the set of all possible values of each attribute, then the incomplete information system is regarded as a special case of the set-valued information system [5]. Besides, the set-valued data generator is developed and four different set-valued data sets are generated. All these data sets are outlined in Table 2. In what follows, we compare the computational times of non-incremental and incremental algorithms. To distinguish the computational times, we divide each of these nine data sets into five parts of equal size. The first part is regarded as the st data set, the combination of the st data set and the second part is viewed as the 2 nd data set, the combination of the 2 nd data set and the third part is regarded as the 3 rd data set,, the combination of all five parts is viewed as the 5 th data set. These data sets can be used to calculate the computational time used by non-incremental and incremental algorithms and show it vis-a-vis the sizes of the universe. In addition, by Algorithms, DAUAM-A and DAUAM-D, since the (α, β)- probabilistic approximations can be computed from H(X) with two thresholds α, β directly, then different thresholds cost the same operations. Thus, we test performance of these three algorithms with the constant thresholds α = and β =. The computations are conducted on a PC with Windows 7 and Inter(R) Core(TM)2 CPU P84 and 2 GB memory. Algorithms are coded in C++ and the software being used is Microsoft Visual 28.

13 632 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) Table 2 A description of data sets. Data sets Abbreviation Samples Attributes Classes Source Congressional voting records CVR UCI 2 Mushroom Mushroom UCI 3 Soybean-large Soybean UCI 4 Audiology (Standardized) Audiology UCI 5 Dermatology Dermatology UCI 6 Set-valued data one SVD Data generator 7 Set-valued data two SVD2 2 4 Data generator 8 Set-valued data three SVD3 5 8 Data generator 9 Set-valued data four SVD Data generator Table 3 A description of adding attribute set. Data sets Attributes Original attribute set Adding attribute set CVR 6 {a 7, a 8,...,a 6 } {a, a 2,...,a 6 } 2 Mushroom 22 {a, a 2,...,a 5 } {a 6, a 7,...,a 22 } 3 Soybean 35 {a, a 2,...,a 3 } {a 3, a 32,...,a 35 } 4 Audiology 69 {a, a 2,...,a 2, a 3, a 32,...,a 69 } {a 2, a 22,, a 3 } 5 Dermatology 34 {a, a 2,...,a 27 } {a 28, a 29,...,a 34 } 6 SVD 5 {a, a 2, a 3 } {a 4, a 5 } 7 SVD2 {a, a 2,...,a } {a, a 2,...,a } 8 SVD3 {a, a 2, a 3, a 7,...,a } {a 4, a 5, a 6 } 9 SVD4 8 {a, a 3, a 4, a 6, a 7 } {a 2, a 5, a 8 } (a) CVR (b) Mushroom (c) Soybean (d) Audiology (e) Dermatology (f) SVD (g) SVD (h) SVD (i) SVD4 Fig.. A comparison of Algorithm and DAUAM-A versus the size of data when adding an attribute set.

14 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) Table 4 A description of deleting attribute set. Data sets Attributes Original attribute set Deleting attribute set CVR 6 {a, a 2,...,a 6 } {a, a 2,...,a 6 } 2 Mushroom 22 {a, a 2,...,a 22 } {a 6, a 7,...,a 22 } 3 Soybean 35 {a, a 2,...,a 35 } {a 3, a 32,...,a 35 } 4 Audiology 69 {a, a 2,...,a 69 } {a 2, a 22,...,a 3 } 5 Dermatology 34 {a, a 2,...,a 34 } {a 28, a 29,...,a 34 } 6 SVD 5 {a, a 2, a 3, a 4, a 5 } {a 4, a 5 } 7 SVD2 {a, a 2,...,a } {a, a 2,...,a } 8 SVD3 {a, a 2,...,a } {a 4, a 5, a 6 } 9 SVD4 8 {a, a 2,...,a 8 } {a 2, a 5, a 8 } (a) CVR (b) Mushroom (c) Soybean (d) Audiology (e) Dermatology (f) SVD (g) SVD (h) SVD (i) SVD4 Fig. 2. A comparison of Algorithm and DAUAM-D versus the size of data when deleting an attribute set. 6.. and DAUAM-A We compare Algorithm with DAUAM-A on the nine data sets shown in Table 2 when adding an attribute set to the original attribute set. The original and adding attribute sets and the experimental results of these nine data sets are shown in Table 3 and Figure, respectively. In Figure, we display detailed change trend line of each of two algorithms with the increasing size of the data set. In each of these sub-figures (a) (i), the x-coordinate pertains to the size of the data set (the five data sets starting from the smallest one), while the y-coordinate concerns the computational time. From Figure, the computational time of each of these two algorithms increases with the increasing size of data. As shown in each sub-figureof Figure, Algorithm DAUAM-A consistently is faster than Algorithm on updating approximations

15 634 J. Zhang et al. / International Journal of Approximate Reasoning 53 (22) when the same attribute set is added to the original attribute set in the set-valued decision information system. Moreover, the differences become larger and larger when the size of the data set increases and DAUAM-D We compare Algorithm with DAUAM-D on the nine data sets shown in Table 2 whendeletinganattributeset.the original and deleting attribute sets and the experimental results of these nine data sets are shown in Table 4 and Figure 2, respectively. The computational time of each of these two algorithms increases with the increasing size of data from Figure 2. In each sub-figure of Figure 2, Algorithm DAUAM-D is faster than Algorithm on updating approximations when the same attribute set is deleted from the original attribute set in the set-valued decision information system. Furthermore, the differences become larger and larger when the size of the data set increases. 7. Conclusions In this paper, we defined a basic vector H(X) and four cut matrices of H(X) which were used to derive the approximations, positive, boundaryand negativeregions intuitively. They canalsobeusedtoderivethe (α, β)-probabilistic approximations, positive, boundary and negative regions directly. The key for generating H(X) is to construct the relation matrix. We presented a basic method for constructing the relation matrix. Since the system will vary with time, the attribute set may vary simultaneously. Then, we obtained an incremental approach for updating the relation matrix by investigating the properties about the variation of the relation matrix while attribute set varies. We also proposed the corresponding algorithms for updating approximations. Experimental studies pertaining to different UCI and user-defined data sets showed that the proposed algorithms effectively reduce computational time of updating approximations. 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