key: cord-0749420-reli9fku
authors: Wang, Zhen; Wei, Ling; Qi, Jianjun; Qian, Ting
title: Attribute reduction of SE-ISI concept lattices for incomplete contexts
date: 2020-09-06
journal: Soft comput
DOI: 10.1007/s00500-020-05271-2
sha: 33beb5a004ac3b491c3462c03b3cc1b0281dbb97
doc_id: 749420
cord_uid: reli9fku

Three-way concept analysis in incomplete contexts lays the theory dealing with the data in incomplete contexts, especially three kinds of partially known formal concepts including SE-ISI formal concept, ISE-SI formal concept and ISE-ISI formal concept. Generally speaking, not every attribute is essential in an incomplete context since the purpose of research is different. Thus, we propose four kinds of attribute reduction of SE-ISI concept lattices based on different criteria. Then, we discuss the relationships among the four kinds of attribute reduction, including the relationships among the consistent sets and relationships among the reducts. Finally, based on discernibility matrices and discernibility functions, the approaches to obtaining these attribute reduction are presented.

is obtained by a pair of derivation operators. All the formal concepts can form a complete lattice called a concept lattice, which is the basic structure of FCA. As an effective mathematical tool for conceptual data analysis and knowledge processing, both the theoretical researches and practical applications of FCA have been promoted. For the promotion of theoretical researches, many scholars have studied attribute reduction (Zhang et al. 2005; Wang and Ma 2006; Wei et al. 2008; Wang and Zhang 2008; Wu et al. 2009; Liu et al. 2009; Qi 2009; Li and Wu 2011; Medina 2012; Shao et al. 2013; Liang et al. 2013; Li et al. 2013a; Kumar et al. 2015; Ganter and Obiedkov 2016; Shao and Li 2016; Dias and Vieira 2017; Chen et al. 2018; Li and Zhang 2019) , rules acquisition (Wille 1989; Missaoui et al. 1994; Li et al. 2013b; Shao et al. 2014) , concept lattice construction (Nourine and Raynaud 1999; Djouadi and Prade 2010; Qian et al. 2017) and so on. For the development of practical applications, FCA has been used in disease control, chemistry, information retrieval, smart city, etc. (Tilley 2004; Kumar and Srinivs 2010; Quintero and Restrepo 2017; Xie et al. 2018) .

In fact, being one of the key issues in knowledge discovery, attribute reduction has been extensively studied in different fields, since it can decrease the dimension and make data analysis easier. Attribute reduction is also an interesting topic in FCA, and many significant results about it have been obtained. For example, Ganter and Obiedkov (2016) proposed attribute reduction which refers to omitting attributes/objects that are equivalent to combinations of other attributes/objects. Zhang et al. (2005) discussed the attribute reduction that can keep the original concept lattice. Within such framework, Zhang et al. (2005) constructed judgment theorems of consistent sets, and developed approaches to attribute reduction based on discernibility matrix, which was further simplified by Qi (2009) from the viewpoint of parentchild concepts. Moreover, based on irreducible elements that play key roles in lattice construction, Wang and Ma (2006) and Li et al. (2013a) proposed the attribute reduction preserving the extents of meet-irreducible and join-irreducible elements in the original concept lattices, respectively. From the viewpoint of granular computing, Wu et al. (2009) proposed granular reduction. In general, the attribute reduction in References (Zhang et al. 2005; Wang and Ma 2006; Wu et al. 2009; Li et al. 2013a ) is taken as four basic kinds of attribute reduction in FCA. Additionally, Liu et al. (2009) studied the attribute reduction of object oriented concept lattices and property oriented concept lattices. Furthermore, Wang and Zhang (2008) discussed the relationship between the attribute reduction of object and property oriented concept lattices. Besides, Li and Wu (2011) and Shao et al. (2013) studied attribute reduction from the perspectives of covering rough set theory and linear dependence of vectors, respectively.

In the framework of FCA, the relationship between objects and attributes is discussed from the perspectives of "commonly possess" and "be commonly possessed." A formal concept has good semantic, since it can show some kind of "balance" between extent and intent. That is, all objects in the extent commonly possess all attributes in the intent, and all attributes in the intent are shared by all objects in the extent. Actually, the formal context also offers us the information whether an object (attribute) does not possess (is not shared by) an attribute (object), but this meaning is not reflected in formal concepts. Then, Qi et al. (2014 Qi et al. ( , 2016 generalized FCA to three-way concept analysis (3WCA) through considering the "negative" property shown in formal contexts and introducing the idea of three-way decisions (Yao 2012 (Yao , 2016 Fujita et al. 2016) . Two key notions in 3WCA are object/attribute-induced three-way concepts and object/attribute-induced three-way concept lattices. Based on two kinds of three-way concept lattices, Ren and Wei (2016) defined four kinds of basic attribute reduction, and studied their relationships.

Both FCA and 3WCA are based on formal contexts, in which the relations between every object and every attribute are definite. However, we often encounter some situations with missing information in the real world; therefore, Burmeister and Holzer (2000) proposed incomplete contexts to reflect such situations. As for incomplete contexts, Djouadi et al. (2009) and Li et al. (2013c) defined ill-known formal concepts and approximate concepts, respectively. And then Yao (2017) gave a general framework for incomplete contexts, and defined three kinds of partially known formal concepts, including SE-ISI formal concept (i.e., approximate concept), ISE-SI formal concept and ISE-ISI formal concept (i.e., ill-known formal concept). To distinguish from threeway concepts and partially known formal concepts, formal concepts in FCA can be called classical formal concepts or SE-SI formal concepts. Furthermore, Ren et al. (2018) presented an analysis of the relationships among the three kinds of partially known formal concepts and SE-SI formal concepts.

With regard to attribute reduction of incomplete contexts, Li and Wang (2016) defined three-way approximate concept lattice reduction. Table 1 is a summary of some current works about attribute reduction.

In the above three kinds of partially known formal concepts, we think that SE-ISI formal concept has better semantic interpretation than the other two. Because the extent of an SE-ISI formal concept is a crisp set that is in more accordance with human cognition, and the intent of it is an interval set that can effectively process the uncertain information. Thus, for the extensive usages of SE-ISI formal concepts, we should make the discovery and representation of implicit knowledge in SE-ISI concept lattices easier and simpler. Therefore, based on different criteria, we propose four kinds of attribute reduction of SE-ISI concept lattices.

SE-ISI concept lattices constructed by SE-ISI formal concepts and their hierarchy relations can be viewed as the knowledge generated from the contexts. Thus, if an attribute reduct can preserve the basic structure of the lattice, then we consider that all the knowledge of the context is preserved. We know that the meet(join)-irreducible elements in a lattice play important roles; therefore, we propose two types of reduction, which can preserve all the extents of the two kinds of irreducible elements. Further, all the elements in SE-ISI concept lattices can be obtained by the join of SE-ISI object concepts induced by an object; therefore, SE-ISI object concepts can be viewed as information granules. Hence, we propose the reduction which can keep all the extents of such special concepts.

The main contributions of this paper include: (1) four kinds of attribute reduction of SE-ISI concept lattices are defined; (2) the relationships among four kinds of attribute reduction of SE-ISI concept lattices are analyzed; (3) the approaches to obtaining four kinds of attribute reduction of SE-ISI concept lattices are presented.

The rest of the paper is organized as follows. In Sect. 2, the basic knowledge of formal concept analysis and threeway concept analysis in incomplete contexts is reviewed. In Sect. 3, four different kinds of attribute reduction of SE-ISI concept lattices are defined, and their relationships including the relationships among the corresponding consistent sets and reducts are also discussed. Then, the approaches to computing the attribute reduction are presented in Sect. 4. Next, Lattice reduction (Zhang et al. 2005) MIE-preserving reduction (Wang and Ma 2006) JIE-preserving reduction (Li et al. 2013a) Granular reduction (Wu et al. 2009) 3WCA OE(AE)-lattice reduction (Ren and Wei 2016) Three-way approximate concept lattice reduction (Li and Wang 2016) OE(AE)-MIE-preserving reduction (Ren and Wei 2016) OE(AE)-JIE-preserving reduction (Ren and Wei 2016) OE-granular reduction (Ren and Wei 2016) an empirical case is shown in Sect. 5. Finally, conclusions and future studies are given in Sect. 6.

In order to make this paper self-contained, we first review some basic notions about formal concept analysis.

Definition 1 (Ganter and Wille 1999) A formal context K = (U , A, I ) consists of two sets U and A and a relation I between U and A. The elements of U are called the objects and the elements of A are called the attributes of the context. In order to express that an object x is in a relation I with an attribute a, we write x I a or (x, a) ∈ I .

Given a formal context K = (U , A, I ), a pair of derivation operators can be defined on X ⊆ U , B ⊆ A by

Based on the pair of derivation operators, a formal concept is defined as follows:

Definition 2 (Ganter and Wille 1999) Let K = (U , A, I ) be a formal context. For any X ⊆ U , B ⊆ A, if X * = B, B * = X , then we call the pair (X , B) a formal concept, and call X the extent and B the intent of (X , B), respectively.

We use L(K) or L(U , A, I ) to denote the set of all the formal concepts of K = (U , A, I ). If any two formal concepts

and the infimum and supremum of them are defined by

then the set L(K) is a complete lattice, which is called the concept lattice of K. Definition 4 (Davey and Priestley 1990) Let L be a lattice.

An element x ∈ L is called join-irreducible, if

And the meet-irreducible element can be defined dually. Table 2 shows a formal context K = (U , A, I ), in which U = {1, 2, 3, 4} and A = {a, b, c, d, e}, where (x, a i ) = + means x I a i and (x, a i ) = − means x I c a i for any x ∈ U and a i ∈ A. Figure 1 is the corresponding concept lattice of K.

For convenience, we use the element sequence of a set to represent the set itself in a formal concept except the universal 

Firstly, we introduce the notion of interval set in set theory. An interval set on the finite set U is defined as: 

In particularly,

The intersection ( ), union ( ) and difference (−) in I (2 A ) are defined, respectively, as follows:

In the rest of this section, the notions about three-way concept analysis in incomplete contexts are presented, and illustrated by an example. Based on an incomplete context IK = (U , A, V , I ) , the derivation operators R and R are defined on O ∈ 2 U and B ∈ 2 A by

Then, extended derivation operators are defined in the next definition based on the derivation operators R and R.

Definition 6 (Yao 2017) Let IK = (U , A, V , I ) be an incomplete context, two extended derivation operators, : 2 U → I (2 A ), and :

Given an incomplete context IK = (U , A, V , I ), then for

the following statements hold Ren et al. (2018) .

Based on the pair of extended derivation operators, an SE-ISI formal concept is defined as follows:

Definition 7 (Yao 2017) In an incomplete context IK = (U , A, V , I ), a pair of a set of objects and an interval set of attributes, (O, [B, B] ), is called a partially known formal concept with a set extent and an interval-set intent, or simply a partially known SE-ISI formal concept, if the following conditions hold:

We call a partially known SE-ISI formal concept an SE-ISI formal concept in this paper. And in particular, for any

and the infimum and supremum of them are defined by

then the set L SE-ISI (IK) is a complete lattice, called the lattice of partially known formal concepts with a set extent and an interval-set intent, or simply partially known SE-ISI concept lattice. And we call it SE-ISI concept lattice for short.

Furthermore, we define the parent-child relation in an SE-ISI concept lattice as follows: if 

. Table 3 shows an incomplete context IK = (U , A, V , I ) in Anderson et al. (2004) , in which the object set U = {1, 2, 3, 4} is a set of four patients who suffer from severe acute respiratory syndrome (SARS), and the attribute set A={a, b, c, d, e, f , g} is a set of seven symptoms (Fever, Cough, Headache, Difficulty Breathing, Diarrhea, Arrhythmia and Insomnia). Figure 3 is the corresponding SE-ISI concept lattice of IK.

In fact, Yao (2017) has defined three kinds of partially known formal concepts. That is, SE-ISI formal concept, ISE-SI formal concept and ISE-ISI formal concept, and the differences of partially known formal concepts from SE-SI formal concept are that extents or/and intents of partially known formal concepts are interval sets. The details are briefly described in the following part and summarized in Table 4 .

Given an incomplete context IK = (U , A, V , I ), another two pairs of extended derivation operators and , and can be defined by 

is called a partially known formal concept with an interval-set extent and a set intent, or simply a partially known ISE-SI formal concept, if the following conditions hold:

is called a partially known formal concept with an interval-set extent and an intervalset intent, or simply a partially known ISE-ISI formal concept, if the following conditions hold:

Example 3 (Continued with Example 2) It can be verified that ([14, 124] , ab) is an ISE-SI formal concept and ([24, 24] , [c, abc f ]) is an ISE-ISI formal concept of the incomplete context IK in Table 3 .

As we have introduced, SE-ISI formal concept has the better semantics and applications than the other two kinds of partially known formal concepts. Therefore, we just focus on SE-ISI formal concepts to study the attribute reduction of incomplete contexts in this paper.

In this section, four kinds of attribute reduction of incomplete contexts are first proposed from different perspectives, that is, the structure of SE-ISI concept lattices, the construction of SE-ISI concept lattices and granular computing. Then, the relationships among the four kinds of attribute reduction are analyzed.

First, we denote the set of all the extents of concepts, the meetirreducible concepts, the join-irreducible concepts of the SE-ISI concept lattice

Definition 8 Let IK = (U , A, V , I ) be an incomplete context, and D ⊆ A.

Furthermore, if D is an SE-ISIL(resp. SE-ISIM, SE-ISIJ, SE-ISIG) consistent set, and no proper subset of D is, then D is called an SE-ISIL(resp. SE-ISIM, SE-ISIJ, SE-ISIG) reduct.

The reasons why four kinds of attribute reduction are proposed are as follows: First, we can see that all the information of an incomplete context can be reflected by the corresponding SE-ISI concept lattice straightforwardly, then SE-ISIL reduction is defined based on the structure of the lattices.

Then, based on lattice theory, the set of all the meet(join)irreducible elements of a finite lattice is infimum(supremum)dense in the lattice, that is, every element in the lattice can be represented by the meet(join) of meet(join)-irreducible elements, then these elements are the basic ones in lattice construction. Therefore, SE-ISIM reduction and SE-ISIJ reduction are defined from the viewpoint of lattice construction.

Finally, for the peculiarity of SE-ISI object concepts, all the SE-ISI formal concepts can be represented by the join of them, then SE-ISI object concepts can be regarded as information granules. Then, we propose the notion of SE-ISIG reduction from the viewpoint of granular computing.

In some application occasions, the equivalent description of SE-ISIL reduction may be more useful; we present it in the next lemma. First, the isomorphic relation between two SE-ISI concept lattices are defined as follows: Example 4 (Continued with Example 2) For the incomplete context IK in Table 3 , the attribute subset A 1 = {a, d, f , g} is an SE-ISIL reduct, and its corresponding concept lattice Fig. 4 which is obviously isomorphic to L SE-ISI (IK) in Fig. 3 . A 2 = {b, c, d, g} is an SE-ISIM reduct preserving all the extents of meet-irreducible elements (i.e., C 2 , C 3 , C 4 , C 6 and C 7 ) in L SE-ISI (IK), and its concept lattice L SE-ISI (U , A 2 , V , I A 2 ) is shown in Fig. 5 . A 3 = {d, f , g} is both an SE-ISIJ and SE-ISIG reduct, since it can preserve all the extents of join-irreducible elements (i.e., C 4 , C 8 , C 9 and C 10 ) and SE-ISI object concepts (i.e., C 4 , C 8 , C 9 and C 10 ) in L SE-ISI (IK), and the concept lattice of it is shown in Fig. 6. 

Because the aforementioned four kinds of attribute reduction of SE-ISI concept lattices are proposed from different perspectives, the relationships among them deserve being discussed.

In this section, we investigate the relationship between SE-ISIL reduction and SE-ISIM reduction, including the D, V , I D ) . Then, D ∈ CS L can be obtained, that is, CS M ⊆ CS L .

From the above discussion, CS L = CS M is proved, which is the first part of the theorem. And the another part Red L = Red M can be naturally proved from the relations between consistent sets and reducts.

Theorem 1 shows that SE-ISIL reducts (consistent sets) equals to SE-ISIM reducts (consistent sets), which can be verified by the next example.

Example 5 (Continued with Example 2) For the incomplete context IK shown in Table 3 , we can verify that CS L = CS M ={{a, c, d, g}, {a, d, f , g}, {b, c, d, g}, {b, d, f , g}}, Red L = Red M ={{a, c, d, g}, {a, d, f , g}, {b, c, d, g}, {b, d, f , g}} 

In this section, the relationship between SE-ISIL reduction and SE-ISIG reduction is discussed. Proof First, we can obtain that CS L ⊆ CS G based on Lemma 3, and we also know that Red L ⊆ CS L . Thus, Red L ⊆ CS G can be proved straightforwardly.

From Theorem 2, we know that if D is an SE-ISIL consistent set (reduct), it must be an SE-ISIG consistent set, but it may not be an SE-ISIG reduct. And vice versa, if D is an SE-ISIG consistent set (reduct), it may not be an SE-ISIL consistent set (reduct). The above statement can be illustrated by Example 6.

Example 6 (Continued with Example 4) For the incomplete context IK shown in Table 3 , A 1 = {a, d, f , g} is both an SE-ISIL reduct and SE-ISIG consistent set of IK, but it is not an SE-ISIG reduct. Because there exists a proper subset A 3 = {d, f , g} ⊂ A 1 being an SE-ISIG reduct. And vice versa, the set A 3 is an SE-ISIG reduct rather than an SE-ISIL consistent set (reduct).

In this section, we present the relationship between SE-ISIL reduction and SE-ISIJ reduction.

Proof Based on the definitions of SE-ISIL and SE-ISIJ reduction, we can obtain that CS L ⊆ CS J , and we also have that Red L ⊆ CS L . Then, Red L ⊆ CS J holds.

From Theorem 3, we know that if D is an SE-ISIL consistent set (reduct), it must be an SE-ISIJ consistent set, but may not be an SE-ISIJ reduct. And vice versa, if D is an SE-ISIJ consistent set (reduct), it may not be an SE-ISIL consistent set (reduct). As shown in Example 6, A 3 is also an SE-ISIJ reduct, so the example can verify the statement too.

It is worth noting that there is actually no direct relationship between SE-ISIJ reduction and SE-ISIG reduction, which is explained in details in the next three examples. First, we state that an SE-ISIJ reduct may not be an SE-ISIG reduct. 

Then, we show that an SE-ISIG reduct may not be an SE-ISIJ reduct. Table 6 is an incomplete context IK 2 = (U 2 , A 6 , V , I A 6 ) whose object set U 2 = {1, 2, 3, 4} and attribute set A 6 = {a, b, c, d, e}. A 7 = {a, b, d, e} is an attribute subset, 

and SE-ISI concept lattices of IK 2 and (U 2 , A 7 , V , I A 7 ) are shown in Figs. 9 and 10, respectively. From the figures, we can know that A 7 = {a, b, d, e} is an SE-ISIG reduct rather than an SE-ISIJ consistent set or reduct, because the element C 2 in L SE-ISI (IK 2 ) is join-irreducible and C 2 = C 3 ∪ C 4 is join-reducible in

At last, we state that there exists an attribute subset being both an SE-ISIJ reduct and SE-ISIG reduct. From Examples 7 to 9, we can conclude that there is no precise relationship between SE-ISIJ reduction and SE-ISIG reduction.

Based on the above analysis, we can sum up the relationships among these four kinds of attribute reduction of SE-ISI concept lattices in Fig. 11 . Moreover, further explanations about the relationships among SE-ISI consistent sets and SE-ISI reducts are presented in Figs. 12 and 13, respectively. 

In Sect. 3, we introduced four different attribute reduction of SE-ISI concept lattices, and gave some examples to illustrate the definitions and the relationships about them. In this section, the approaches to computing the attribute reduction by discernibility matrices and discernibility functions are proposed.

The method of discernibility matrix put forward by Zhang et al. (2005) aims to complete contexts, we generalize it to incomplete contexts as follows:

is a join(meet)-irreducible element, then the matrix is called the SE-ISIJ(SE-ISIM) discernibility matrix of IK, denoted by

) is called the SE-ISIG discernibility matrix of IK, written as Λ G .

Example 10 (Continued with Example 2) The SE-ISIL discernibility matrix of IK in Table 3 is shown in Table 7 , in which the concepts of the 1st column(row) are child(parent) concepts. If an SE-ISIL discernibility attribute set is ∅, then we denote it as a space.

Based on SE-ISIL discernibility matrix, how to judge an attribute subset is an SE-ISIL consistent set or not will be illustrated in the next theorem. First, we give several lemmas which are useful in the proof of the theorem. Given an incomplete context, Lemma 4 actually presents an easier way to obtain the extents and intents of SE-ISI concepts of its subcontexts. For instance, for the subcon-

On the other hand, for

From the above analysis, we can obtain

holds. In order to prove this lemma, we only need to prove

can be obtained.

Lemma 6 actually describes the properties of SE-ISIL consistent sets. The statements in Example 4 can be used to verify this lemma. The set A 1 ={a, d, f , g} is an SE-ISIL consistent set of the incomplete context IK in Table 3 , then it can be verified that Lemma 6 holds in every pair of child-parent SE-ISI concepts in Fig. 3 .

The judging theorem of SE-ISIL consistent set is proposed based on the above lemmas as follows: Table 7 The SE-ISIL discernibility matrix of IK (1) D is an SE-ISIL consistent set.

(

Proof Firstly we prove that (1) ⇔ (2).

can be obtained.

(2) ⇒ (1). Suppose statement (2) 

Example 11 (Continued with Examples 4 and 9) Considering the incomplete context IK in Table 3 , the sets A 1 ={a, d, f , g} and A 3 ={d, f , g}, we can validate that the statements (2) and (3) of Theorem 4 hold on A 1 , then A 1 is an SE-ISIL consistent set according to the theorem. In contrast, (12, [∅, abg] 

, and there also exist an element ab ∈ Λ L such that ab ∩ A 3 =∅. Therefore, A 3 is not an SE-ISIL consistent set. On the other hand, the statements that A 1 ={a, d, f , g} is an SE-ISIL consistent set and A 3 ={d, f , g} is not an SE-ISIL consistent set of the incom-plete context are illustrated in Example 4 based on Figs. 3, 4 and 6.

On the basis of Theorem 4 that is the theoretic base of the reduction approaches, we define SE-ISI discernibility functions as follows to calculate all the SE-ISI reducts.

Definition 11 Let IK = (U , A, V , I ) be an incomplete context. The SE-ISIL(SE-ISIM, SE-ISIJ, SE-ISIG) discernibility function is defined as follows:

Based on the absorption law and distribute law of logical theory, every SE-ISI discernibility function f can be represented by a minimal disjunctive normal form, whose items are all SE-ISI reducts of IK.

Example 12 (Continued with Example 9) Based on the SE-ISIL discernibility matrix shown in Table 7 , we obtain the SE-ISIL reducts of IK in Table 3 as follows:

It means, there are four SE-ISIL (SE-ISIM) reducts, which are {a, c, d, g}, {a, d, f , g}, {b, c, d, g} and {b, d, f , g}. Specially, if we choose C i to be a join-irreducible SE-ISI formal concept or an SE-ISI object concept, then we can obtain SE-ISIJ discernibility matrix and SE-ISIG discernibility matrix, respectively. Thus, we can get the SE-ISIJ reducts and SE-ISIG reducts as follows:

Therefore, the SE-ISIJ reducts are {a, c, d, g}, {b, c, d, g} and {d, f , g}, the SE-ISIG reducts are {a, c, d, g}, {b, c, d, g} and {d, f , g}. 

In this section, a real-life database is analyzed to illustrate the semantics and demonstrate the applications of the proposed attribute reduction methods. The presented database is about the patients after surgery and their physical indexes, the details of which are as follows. Table 9 An incomplete context IK 3 transformed from Table 8 U Table 8 and > 7) and I(≤ 7). The value of patient 5 on the attribute COMFORT is missing. Then, nominal scale Ganter and Wille (1999) is used to transform Table 8 into an incomplete context IK 3 =(U 3 , A 8 , V , I A 8 ) as shown in Table 9 , in which U 3 ={1, 2, 3, 4, 5, 6} is a set of six patients after surgery and A 8 ={a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , c 3 , d 1 , d 2 , d 3 } is a set of ten nominal attributes, that is a 1 is (L-SURF, high), a 2 is (L-SURF, low), b 1 is (L-O 2 , excellent), b 2 is (L-O 2 , good), c 1 is (L-BP, high), c 2 is (L-BP, mid), c 3 is (L-BP, low), d 1 is (COMFORT, III), d 2 is (COMFORT, II) and d 3 is (COMFORT, I). The SE-ISI concepts of the incomplete context IK 3 are listed in Table 10 and the corresponding SE-ISI concept lattice is shown in Fig. 14 .

In order to obtain all SE-ISI concepts and the corresponding SE-ISI concept lattice of an incomplete context, one needs to consider all the attributes in the incomplete context. As we can see from Table 10 , there are only ten attributes in this case, but the representations of all SE-ISI concepts are a little bit complicated. In fact, the data that we meet in real life are usually more complicated , then the attribute reduction of incomplete contexts is necessary to be considered.

Since the ordered hierarchical structure of all the SE-ISI concepts is reflected by the SE-ISI concept lattice, Table 10 The SE-ISI concepts corresponding to IK 3 in Table 9 Label Concept Label Concept The SE-ISI concepts of (U 3 , M 1 , V , I M1 )

and SE-ISI concept lattice is the core structure in threeway concept analysis in incomplete contexts, then SE-ISIL attribute reduction needs to be considered. Based on the proposed SE-ISIL discernibility matrix and SE-ISIL discernibility function in Sect. 4, we can obtain that the SE-ISIL (SE-ISIM) reduct of the incomplete context IK 3 is Table 11 , and their corresponding SE-ISI concept lattice is presented in Fig. 15 . From Figs. 14 and 15, we can see the SE-ISI concept lattice of (U 3 , M 1 , V , I M 1 ) is isomorphic to the SE-ISI concept lattice of IK 3 . Then, the classifications of patients and the hierarchy structure of SE-ISI concepts keep unchanged, but only smaller amount of attributes needs to be considered. For instance, only based on the two nominal attributes b 1 and d 2 , the patients 2, 3 and 5 can still be classified into one class. Then, the knowledge in SE-ISI concept is represented in a more concise way after SE-ISIL reduction without knowledge loss.

Then, based on the proposed SE-ISIJ discernibility matrix and SE-ISIJ discernibility function, the SE-ISIJ reducts are {a 1 , a 2 , b 1 , b 2 , d 2 , d 3 }, {a 1 , a 2 , b 2 , c 1 , d 2 , d 3 }, {a 1 , a 2 , b 1 , Table 12 The SE-ISI concepts of (U 3 , M 2 , V , I M2 )

The SE-ISI concepts of (U 3 , M 2 , V , I M 2 ) are listed in Table 12 , and their corresponding SE-ISI concept lattice of (U , M 2 , V , I M 2 ) is presented in Fig. 16 . Here, M 2 is the SE-ISIJ reduct {a 1 , a 2 , b 2 , c 1 , d 2 , d 3 }. From Fig. 16 , we can see that the SE-ISIJ reduct cannot preserve the structure of the SE-ISI concept lattice which means that some knowledge may be lost after SE-ISIJ reduction. For example, the patients 1, 2, 3 and 5 cannot belong to one class in Fig. 16 because the attribute b 1 has been removed after SE-ISIJ reduction. But the extents of all the join-irreducible elements (i.e., C 16 -C 21 ) of SE-ISI concept lattice in Fig. 14 don't change in Fig. 16 . Since the joinirreducible elements are the basic elements in the lattice construction, SE-ISIJ reduction is important in SE-ISI concept lattice construction.

Finally, based on the proposed SE-ISIG discernibility matrix and SE-ISIG discernibility function, the SE-ISIG reducts are {a 1 , a 2 , b 1 , b 2 , d 2 , d 3 }, {a 1 , a 2 , b 2 , c 1 , d 2 , d 3 },  {a 1 , a 2 , b 1 , b 2 , c 2 , d 1 , d 2 }, {a 1 , a 2 , b 1 , c 2 , d 1 , d 2 , d 3 }, {a 1 , a 2 ,  b 1 , c 2 , c 3 , d 2 , d 3 }, {a 1 , a 2 , c 1 , c 2 , c 3 , d 2 , d 3 }. The SE-ISI concepts of (U 3 , M 3 , V , I M 3 ) are listed in Table 13 , and their (1, [a 2 b 1 d 3 , a 2 b 1 d 3 ] )

C 7 (256, [a 1 , a 1 d 2 ]) C 18 (5, [a 1 b 1 , a 1 b 1 d 2 d 3 ] )

C 8 (13, [a 2 b 1 , a 2 corresponding SE-ISI concept lattice of (U , M 3 , V , I M 3 ) is presented in Fig. 17 . Here, M 3 is the SE-ISIG reduct {a 1 , a 2 , b 1 , b 2 , d 2 , d 3 }.

Similar to SE-ISIJ reduction, we can see that the SE-ISIG reduct cannot preserve the structure of the SE-ISI concept lattice from Fig. 17 , which means some knowledge may be lost after SE-ISIG reduction. But the extents of SE-ISI object concepts (i.e., C 16 -C 21 ) of SE-ISI concept lattice in Fig. 14 are unchanged in Fig. 17 . Since the SE-ISI object concepts are important in granular computing, SE-ISIG reduction has more applications in granular computing.

From the above discussion, we can see that the core semantic of the attribute reduction of an incomplete context is to find some minimal attribute subsets that can preserve some characteristics of the SE-ISI concept lattice of the incomplete context. And in this paper, four kinds of attribute reduction of incomplete contexts, that is SE-ISIL, SE-ISIM, SE-ISIJ and SE-ISIG are proposed, which can preserve all the extents of SE-ISI concepts, meet-irreducible SE-ISI concepts, joinirreducible SE-ISI concepts and SE-ISI object concepts, respectively. These different attribute reduction consider different perspectives or different information of the incomplete contexts and they can be applied in different occasions. If we want to preserve all the knowledge of the incomplete context, SE-ISIL reduction needs to be considered; if we want to preserve the basic elements of lattice construction, we need to use SE-ISIM or SE-ISIJ reduction; if we want to preserve the information granules, SE-ISIG reduction needs to be applied.

In this paper, attribute reduction of SE-ISI concept lattices has been systematically studied. First, we have defined four kinds of attribute reduction of SE-ISI concept lattices. Then, the relationships among these newly proposed attribute reduction have been investigated. Finally, the approaches to computing these attribute reducts have been presented.

The basic ideas of these attribute reduction are inspired by reduction theory based on a formal context in formal concept analysis and three-way concept analysis. However, all the attribute reduction in this paper are based on SE-ISI concept lattices of incomplete contexts, which are different from the existing reduction theory based on a completed formal context.

In order to apply the theories discussed in this paper to the real world, we will propose the corresponding algorithms of these different attribute reduction in the future, which may help us to deal with the big data problems in our daily lives conveniently, and can make our study more suitable in practices.

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Conflict of interest All authors declare that they have no conflict of interest.Ethical approval This manuscript does not contain any studies with human participants or animals performed by any of the authors. This manuscript is the authors' original work and has not been published nor has it been submitted simultaneously elsewhere.