An Efficient Incremental Updating Algorithm for Knowledge Reduction in Information System

Firstly, the concept of simplified information system is introduced, the notion of simplified discerniblity matrix is put forward and the method for computing core based on the simplified discerniblity matrix, which can well deal with inconsistent information system. And it is proved that core based on the simplified discernibility matrix is equivalent to that based on the previous one, on the basis of that, an efficient algorithm incremental updating for core is presented, which only need to analyze the updating parts of discernibility matrix and doesn’t need to re-calculate discerniblily matrix, when a new object is added to information system. Finally, Theoretical analysis and example results show that the algorithm is efficient and feasible.

Rough set theory has been considered as a useful tool to model the vagueness, imprecision, and uncertainty, and has been applied successfully in many fields. Knowledge reduction is one of the most important problems in rough set theory. However, in real-world most of information systems are based on dominance relations in stead of the classical rough set because of various factors. To acquire brief decision rules from systems based on dominance relations, knowledge reductions are needed. The main aim of this paper is to study the problem. The basic concepts and properties of knowledge reduction based on evidence reasoning theory are discussed. Furthermore, the characterization and knowledge reduction approaches based on evidence reasoning theory are obtained with examples in several kinds of ordered information system, which is every useful in future research works of the ordered information systems.

Knowledge reduction is an important problem in rough set theory. In this paper, an improved measurement is given to measure the roughness of rough set. Based on improved rough entropy, reduction theory and algorithm design are studied. Additionally, appling the weight's idea in fuzzy theory, the conditional attribute weight in the decision table is investigated. Combing the conditional attribute weight with rough entropy, simple knowledge reduction algorithm and examples are given. Theoretical analysis and examples indicate that the complexity of this reduction algorithm is less than that based on the current positive region and the conditional information entropy.

The disadvantages of the recent reduction algorithms are analyzed deeply. A new measure to knowledge and rough set is introduced to discuss the rough entropy of knowledge and the roughness of rough set. Based on this entropy, the new significance of attribute is defined and a heuristic algorithm of knowledge reduction is proposed and compared with two methods of attribute reduction which are based on the positive region and the conditional information entropy respectively. The result shows that the proposed heuristic information is better and more efficient than the others, and is greatly effective and feasible in searching the minimal or optimal reduction. Theoretical analysis and experimental results indicate that the time complexity of this reduction algorithm is less than that based on the current positive region and the conditional information entropy.

Since the definition of attribute reduction based on classic discernibility matrix is different from the definition of attribute reduction based on positive region, simplified discernibility matrix and the corresponding definition of attribute reduction are proposed. At the same time, it is proved that the proposed definition of attribute reduction is identical to the definition of attribution reduction based on positive region. For computing simplified discernibility matrix, IND(C) should usually be calculated at first, so a new algorithm for computing IND(C) is designed, whose temporal complexity is cut down to O(|C||U|). Furthermore, an efficient attribute reduction algorithm is proposed, whose temporal complexity and spatial complexity are cut down to \(\max(O(|C|^2|U'_{pos}||U'|, O(|U||C|))\) and max (O(|C||U′ pos ||U′|, O(|U|)) respectively. At last, an example is used to illustrate the efficiency of the new algorithms.KeywordsRough setdiscernibility matrixsimplified discernibility matrixattribute reductioncomplexity

Reduction of knowledge is one of the important issues in the research of rough set theory. It has been proven that the search for an optimal (minimal) reduction of decision table is NP-hard problem. Thus people have been trying to search for more efficient heuristic algorithms to get an approximate reduction of decision table all the time. Every heuristic must depends on a measure of the significance of attributes in decision table, because it is the key heuristic information to determine which attribute is reducible in decision table. It was analyzed by examples in this paper that the current four commonly used methods all have some drawbacks. Using them to a heuristic leads to wrong results. Then a new definition for measuring the significance of attributes in decision table is given based on discernibility matrix. The new definition can discern the difference of significance of no-core attributes what cannot be discerned or misled by current commonly used methods. The new definition is very useful in the practice of data mining and reduction of knowledge

Some scholars provided the discernibility matrix based on the positive region. The time complexity of the algorithm for computing core with this discernibility matrix is O(|CparU|2 ). For cutting down the time complexity of the algorithm, the definition of simplified discernibility matrix based on the positive region and the corresponding definition of the core are provided. At the same time, it is proved that this core is equivalent to the core based on the positive region. Since the partition of U/C is the key process for computing simplified discernibility matrix, a quick algorithm for computing U/C is designed with the idea of radix sorting. Its time complexity is O(|CparU|). On this condition, a new algorithm for computing core based on the positive region is designed with the simplified discernibility. Its time complexity is cut down to max{O(|CparU'posparU/C|), O(|CparU|)}.

This paper deals with knowledge reduction in generalized consistent decision formal contexts. The concept of a generalized consistent decision formal context is first introduced. Its equivalent definitions are also examined. The judgement theorem and discernibility matrix which are helpful for computing reducts are then established, from which we can obtain an approach to knowledge reduction in generalized consistent decision formal contexts.

In many real-world applications, the costs of different errors are often unequal. Therefore, the inclusion of costs into learning, also named cost-sensitive learning, has been regarded as one of the most relevant topics of future machine learning research. Rough set theory is a powerful mathematic tool dealing with inconsistent information for attribute dependence analysis, knowledge reduction and decision rule extraction. However, it is insensitive to the costs of misclassification due to the absence of a mechanism of considering the subjective knowledge. This paper discusses problems connected with introducing the subjective knowledge into rough set learning and proposes a weighted rough set approach for cost-sensitive learning. In this method, weights are employed to represent the subjective knowledge of costs and a weighted information system is defined firstly. With the introduction of weights, weighted attribute dependence analysis is carried out and an index of weighted approximate quality is given. Furthermore, weighted attribute reduction algorithm and weighted rule extraction algorithm are designed to find the reducts and rules with the consideration of weights. Based on the proposed weighted rough set, a series of comparing experimentations with several familiar general techniques on cost-sensitive learning are constructed. The results show that the approach of weighted rough set produces averagely the minimum misclassification costs and the lowest high cost errors.KeywordsWeighted rough setknowledge reductionrule extractioncost-sensitive learning

The Rough Set Theory (RST) is a mathematical formalism for representing uncertainty, which can be considered an extension of the classical set theory. It has been used in many different research areas, including those related to inductive machine learning and reduction of knowledge in knowledge based systems. This chapter introduces the main concepts of the RST and presents a family of algorithms for implementing them.

Reduction is one of the key problem in rough set theory due to its applications in data mining, rule induction, classification, etc.. In this paper the reductions for a set-valued decision information system(DIS) are studied. The judgment theorem and the discernibility matrix of the generalized decision reduct in a set-valued DIS are given, and the relationships among the generalized decision reduct and alternative types of knowledge reduction in set-valued DIS are investigated. It is proved that the reduct in the consistent set-valued DIS is equivalent to the generalized decision reduct, and the possible reduct in the inconsistent set-valued DIS is equivalent to the generalized decision reduct. The judgment theorem and the discernibility matrix associated with the possible reduct are also established.

Rough set theory provides some principles that are used for data classification and knowledge reduction. Reduct is one of the main concepts that can be used for feature set reduction and for data classification. Finding the reduct set is computationally expensive for data sets with large number of attributes. Several heuristic approached have been proposed to extract reduct sets where some of the approached used the Discernibility Matrix (DM) concept to perform the reduct computation. In this paper the Johnson reduction algorithm and the Object Reduct using Attribute Weighting technique algorithm (ORAW) for reduct computation are evaluated. The two approaches aim at reducing the number of features in the dataset. To evaluate the two approaches several UCI standard datasets were used in the experiments. The results of the experiments showed that the ORAW approach gives better results in term of classification accuracy where the average classification accuracy over eight data sets achieved by the ORAW approach was 85.6%; while Johnson approach achieved 78.8% of accuracy. For further evaluation, the two approaches were compared with some other well known classification techniques.

Objects in an information system analyzed by the rough sets theory methods are characterized by attributes, which can take on a finite set of values only. In diagnostic experiments, condition attributes are usually treated as continuous variables, taking values from certain intervals. So, to use this theory in such problems, certain discretization (coding) of continuous variables is needed. The optimal classification properties of an information system are taken by the authors as base criteria for selecting discretization. The concepts of a random information system and of an expected values of classification quality are introduced. As a result of discretization of continuous attributes, one can get a finite number of regions called states. It is observed that the optimal number of states is not greater than the number of objects. >

The equivalence relation in the theory is defined by the complete equal relation, which will result in complexity of computation of the knowledge reduction in the real world in rough sets theory (RST). In order to overcome this shortage of general rough set theory, the elementary concept of tolerance rough sets theory is proposed in this paper, and the theory is employed to build objects' tolerance relations that can correctly classify objects in system. And the examples illustrating our approaches are presented in the paper for further confirmation.

The aim of this work is to show how data mining technology based on rough set can be used for efficient implementation. In this paper, we discuss rough set theory in detail. In order to discover new rules from databases based on rough set, an object-oriented software architecture being propitious to data mining is given. And the algorithms for data reduction, rule discovery and rule reduction are introduced. From information table defined by primitive database, discernibility matrix, reduction table and rule table are created. In the last, an example is given. This shows that our model and algorithms are feasible.