arXiv subjects

William Zhu

Publications and source records attributed to William Zhu.

At least 37 records · Page 2Linked to original sources

arXiv2013

Matrix approach to rough sets through vector matroids over a field

Rough sets were proposed to deal with the vagueness and incompleteness of knowledge in information systems. There are may optimization issues in this field such as attribute reduction. Matroids generalized from matrices are widely used in optimization. Therefore, it is necessary to connect matroids with rough sets. In this paper, we take field into consideration and introduce matrix to study rough sets through vector matroids. First, a matrix representation of an equivalence relation is proposed, and then a matroidal structure of rough sets over a field is presented by the matrix. Second, the properties of the matroidal structure including circuits, bases and so on are studied through two special matrix solution spaces, especially null space. Third, over a binary field, we construct an equivalence relation from matrix null space, and establish an algebra isomorphism from the collection of equivalence relations to the collection of sets, which any member is a family of the minimal non-empty sets that are supports of members of null space of a binary dependence matrix. In a word, matrix provides a new viewpoint to study rough sets.

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arXiv2013

Characteristic matrix of covering and its application to boolean matrix decomposition and axiomatization

Covering is an important type of data structure while covering-based rough sets provide an efficient and systematic theory to deal with covering data. In this paper, we use boolean matrices to represent and axiomatize three types of covering approximation operators. First, we define two types of characteristic matrices of a covering which are essentially square boolean ones, and their properties are studied. Through the characteristic matrices, three important types of covering approximation operators are concisely equivalently represented. Second, matrix representations of covering approximation operators are used in boolean matrix decomposition. We provide a sufficient and necessary condition for a square boolean matrix to decompose into the boolean product of another one and its transpose. And we develop an algorithm for this boolean matrix decomposition. Finally, based on the above results, these three types of covering approximation operators are axiomatized using boolean matrices. In a word, this work borrows extensively from boolean matrices and present a new view to study covering-based rough sets.

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arXiv2012

Matroidal structure of generalized rough sets based on symmetric and transitive relations

Rough sets are efficient for data pre-process in data mining. Lower and upper approximations are two core concepts of rough sets. This paper studies generalized rough sets based on symmetric and transitive relations from the operator-oriented view by matroidal approaches. We firstly construct a matroidal structure of generalized rough sets based on symmetric and transitive relations, and provide an approach to study the matroid induced by a symmetric and transitive relation. Secondly, this paper establishes a close relationship between matroids and generalized rough sets. Approximation quality and roughness of generalized rough sets can be computed by the circuit of matroid theory. At last, a symmetric and transitive relation can be constructed by a matroid with some special properties.

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arXiv2012

A Comparative Study of Discretization Approaches for Granular Association Rule Mining

Granular association rule mining is a new relational data mining approach to reveal patterns hidden in multiple tables. The current research of granular association rule mining considers only nominal data. In this paper, we study the impact of discretization approaches on mining semantically richer and stronger rules from numeric data. Specifically, the Equal Width approach and the Equal Frequency approach are adopted and compared. The setting of interval numbers is a key issue in discretization approaches, so we compare different settings through experiments on a well-known real life data set. Experimental results show that: 1) discretization is an effective preprocessing technique in mining stronger rules; 2) the Equal Frequency approach helps generating more rules than the Equal Width approach; 3) with certain settings of interval numbers, we can obtain much more rules than others.

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arXiv2012

Covering matroid

In this paper, we propose a new type of matroids, namely covering matroids, and investigate the connections with the second type of covering-based rough sets and some existing special matroids. Firstly, as an extension of partitions, coverings are more natural combinatorial objects and can sometimes be more efficient to deal with problems in the real world. Through extending partitions to coverings, we propose a new type of matroids called covering matroids and prove them to be an extension of partition matroids. Secondly, since some researchers have successfully applied partition matroids to classical rough sets, we study the relationships between covering matroids and covering-based rough sets which are an extension of classical rough sets. Thirdly, in matroid theory, there are many special matroids, such as transversal matroids, partition matroids, 2-circuit matroid and partition-circuit matroids. The relationships among several special matroids and covering matroids are studied.

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arXiv2012

Matroidal structure of rough sets based on serial and transitive relations

The theory of rough sets is concerned with the lower and upper approximations of objects through a binary relation on a universe. It has been applied to machine learning, knowledge discovery and data mining. The theory of matroids is a generalization of linear independence in vector spaces. It has been used in combinatorial optimization and algorithm design. In order to take advantages of both rough sets and matroids, in this paper we propose a matroidal structure of rough sets based on a serial and transitive relation on a universe. We define the family of all minimal neighborhoods of a relation on a universe, and prove it satisfy the circuit axioms of matroids when the relation is serial and transitive. In order to further study this matroidal structure, we investigate the inverse of this construction: inducing a relation by a matroid. The relationships between the upper approximation operators of rough sets based on relations and the closure operators of matroids in the above two constructions are studied. Moreover, we investigate the connections between the above two constructions.

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arXiv2012

Relation matroid and its relationship with generalized rough set based on relation

Recently, the relationship between matroids and generalized rough sets based on relations has been studied from the viewpoint of linear independence of matrices. In this paper, we reveal more relationships by the predecessor and successor neighborhoods from relations. First, through these two neighborhoods, we propose a pair of matroids, namely predecessor relation matroid and successor relation matroid, respectively. Basic characteristics of this pair of matroids, such as dependent sets, circuits, the rank function and the closure operator, are described by the predecessor and successor neighborhoods from relations. Second, we induce a relation from a matroid through the circuits of the matroid. We prove that the induced relation is always an equivalence relation. With these two inductions, a relation induces a relation matroid, and the relation matroid induces an equivalence relation, then the connection between the original relation and the induced equivalence relation is studied. Moreover, the relationships between the upper approximation operator in generalized rough sets and the closure operator in matroids are investigated.

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arXiv2012

Cost-sensitive C4.5 with post-pruning and competition

Decision tree is an effective classification approach in data mining and machine learning. In applications, test costs and misclassification costs should be considered while inducing decision trees. Recently, some cost-sensitive learning algorithms based on ID3 such as CS-ID3, IDX, λ-ID3 have been proposed to deal with the issue. These algorithms deal with only symbolic data. In this paper, we develop a decision tree algorithm inspired by C4.5 for numeric data. There are two major issues for our algorithm. First, we develop the test cost weighted information gain ratio as the heuristic information. According to this heuristic information, our algorithm is to pick the attribute that provides more gain ratio and costs less for each selection. Second, we design a post-pruning strategy through considering the tradeoff between test costs and misclassification costs of the generated decision tree. In this way, the total cost is reduced. Experimental results indicate that (1) our algorithm is stable and effective; (2) the post-pruning technique reduces the total cost significantly; (3) the competition strategy is effective to obtain a cost-sensitive decision tree with low cost.

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arXiv2012

Characteristic of partition-circuit matroid through approximation number

Rough set theory is a useful tool to deal with uncertain, granular and incomplete knowledge in information systems. And it is based on equivalence relations or partitions. Matroid theory is a structure that generalizes linear independence in vector spaces, and has a variety of applications in many fields. In this paper, we propose a new type of matroids, namely, partition-circuit matroids, which are induced by partitions. Firstly, a partition satisfies circuit axioms in matroid theory, then it can induce a matroid which is called a partition-circuit matroid. A partition and an equivalence relation on the same universe are one-to-one corresponding, then some characteristics of partition-circuit matroids are studied through rough sets. Secondly, similar to the upper approximation number which is proposed by Wang and Zhu, we define the lower approximation number. Some characteristics of partition-circuit matroids and the dual matroids of them are investigated through the lower approximation number and the upper approximation number.

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arXiv2012

Relationship between the second type of covering-based rough set and matroid via closure operator

Recently, in order to broad the application and theoretical areas of rough sets and matroids, some authors have combined them from many different viewpoints, such as circuits, rank function, spanning sets and so on. In this paper, we connect the second type of covering-based rough sets and matroids from the view of closure operators. On one hand, we establish a closure system through the fixed point family of the second type of covering lower approximation operator, and then construct a closure operator. For a covering of a universe, the closure operator is a closure one of a matroid if and only if the reduct of the covering is a partition of the universe. On the other hand, we investigate the sufficient and necessary condition that the second type of covering upper approximation operation is a closure one of a matroid.

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arXiv2012

Topological characterizations to three types of covering approximation operators

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Topology, one of the most important subjects in mathematics, provides mathematical tools and interesting topics in studying information systems and rough sets. In this paper, we present the topological characterizations to three types of covering approximation operators. First, we study the properties of topology induced by the sixth type of covering lower approximation operator. Second, some topological characterizations to the covering lower approximation operator to be an interior operator are established. We find that the topologies induced by this operator and by the sixth type of covering lower approximation operator are the same. Third, we study the conditions which make the first type of covering upper approximation operator be a closure operator, and find that the topology induced by the operator is the same as the topology induced by the fifth type of covering upper approximation operator. Forth, the conditions of the second type of covering upper approximation operator to be a closure operator and the properties of topology induced by it are established. Finally, these three topologies space are compared. In a word, topology provides a useful method to study the covering-based rough sets.

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arXiv2012

Geometric lattice structure of covering-based rough sets through matroids

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Geometric lattice has widely used in diverse fields, especially search algorithm design which plays important role in covering reductions. In this paper, we construct four geometric lattice structures of covering-based rough sets through matroids, and compare their relationships. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by the covering, and its characteristics including atoms, modular elements and modular pairs are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, sufficient and necessary conditions for three types of covering upper approximation operators to be closure operators of matroids are presented. We exhibit three types of matroids through closure axioms, and then obtain three geometric lattice structures of covering-based rough sets. Third, these four geometric lattice structures are compared. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely geometric lattice, to study covering-based rough sets.

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arXiv2012

Some characteristics of matroids through rough sets

At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. Further, matroid theory borrows extensively from the terminology of linear algebra and graph theory. We can combine rough set theory with matroid theory through using rough sets to study some characteristics of matroids. In this paper, we apply rough sets to matroids through defining a family of sets which are constructed from the upper approximation operator with respect to an equivalence relation. First, we prove the family of sets satisfies the support set axioms of matroids, and then we obtain a matroid. We say the matroids induced by the equivalence relation and a type of matroid, namely support matroid, is induced. Second, through rough sets, some characteristics of matroids such as independent sets, support sets, bases, hyperplanes and closed sets are investigated.

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arXiv2012

Condition for neighborhoods in covering based rough sets to form a partition

Neighborhood is an important concept in covering based rough sets. That under what condition neighborhoods form a partition is a meaningful issue induced by this concept. Many scholars have paid attention to this issue and presented some necessary and sufficient conditions. However, there exists one common trait among these conditions, that is they are established on the basis of all neighborhoods have been obtained. In this paper, we provide a necessary and sufficient condition directly based on the covering itself. First, we investigate the influence of that there are reducible elements in the covering on neighborhoods. Second, we propose the definition of uniform block and obtain a sufficient condition from it. Third, we propose the definitions of repeat degree and excluded number. By means of the two concepts, we obtain a necessary and sufficient condition for neighborhoods to form a partition. In a word, we have gained a deeper and more direct understanding of the essence over that neighborhoods form a partition.

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arXiv2012

Rough sets and matroidal contraction

Rough sets are efficient for data pre-processing in data mining. As a generalization of the linear independence in vector spaces, matroids provide well-established platforms for greedy algorithms. In this paper, we apply rough sets to matroids and study the contraction of the dual of the corresponding matroid. First, for an equivalence relation on a universe, a matroidal structure of the rough set is established through the lower approximation operator. Second, the dual of the matroid and its properties such as independent sets, bases and rank function are investigated. Finally, the relationships between the contraction of the dual matroid to the complement of a single point set and the contraction of the dual matroid to the complement of the equivalence class of this point are studied.

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arXiv2012

Condition for neighborhoods induced by a covering to be equal to the covering itself

It is a meaningful issue that under what condition neighborhoods induced by a covering are equal to the covering itself. A necessary and sufficient condition for this issue has been provided by some scholars. In this paper, through a counter-example, we firstly point out the necessary and sufficient condition is false. Second, we present a necessary and sufficient condition for this issue. Third, we concentrate on the inverse issue of computing neighborhoods by a covering, namely giving an arbitrary covering, whether or not there exists another covering such that the neighborhoods induced by it is just the former covering. We present a necessary and sufficient condition for this issue as well. In a word, through the study on the two fundamental issues induced by neighborhoods, we have gained a deeper understanding of the relationship between neighborhoods and the covering which induce the neighborhoods.

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arXiv2012

Lattice structures of fixed points of the lower approximations of two types of covering-based rough sets

Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough sets. In this paper, we propose two family of sets and study the conditions that these two sets become some lattice structures. These two sets are consisted by the fixed point of the lower approximations of the first type and the sixth type of covering-based rough sets, respectively. These two sets are called the fixed point set of neighborhoods and the fixed point set of covering, respectively. First, for any covering, the fixed point set of neighborhoods is a complete and distributive lattice, at the same time, it is also a double p-algebra. Especially, when the neighborhood forms a partition of the universe, the fixed point set of neighborhoods is both a boolean lattice and a double Stone algebra. Second, for any covering, the fixed point set of covering is a complete lattice.When the covering is unary, the fixed point set of covering becomes a distributive lattice and a double p-algebra. a distributive lattice and a double p-algebra when the covering is unary. Especially, when the reduction of the covering forms a partition of the universe, the fixed point set of covering is both a boolean lattice and a double Stone algebra.

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arXiv2012

Feature selection with test cost constraint

Feature selection is an important preprocessing step in machine learning and data mining. In real-world applications, costs, including money, time and other resources, are required to acquire the features. In some cases, there is a test cost constraint due to limited resources. We shall deliberately select an informative and cheap feature subset for classification. This paper proposes the feature selection with test cost constraint problem for this issue. The new problem has a simple form while described as a constraint satisfaction problem (CSP). Backtracking is a general algorithm for CSP, and it is efficient in solving the new problem on medium-sized data. As the backtracking algorithm is not scalable to large datasets, a heuristic algorithm is also developed. Experimental results show that the heuristic algorithm can find the optimal solution in most cases. We also redefine some existing feature selection problems in rough sets, especially in decision-theoretic rough sets, from the viewpoint of CSP. These new definitions provide insight to some new research directions.

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