Knowledge Reduction in Generalized Consistent Decision Formal Contexts

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.

Knowledge reduction is one of the basic issues in knowledge presentation and data mining. In this study, an order-preserving mapping between the set of all the extensions of the conditional concept lattice and that of the decision concept lattice is defined to classify formal decision contexts into consistent and inconsistent categories. Then, methods of knowledge reduction for both the consistent and the inconsistent formal decision contexts are formulated by constructing proper discernibility matrices and their associated Boolean functions. For the consistent formal decision contexts, the proposed reduction method can avoid redundancy subject to maintaining consistency, while for the inconsistent formal decision contexts, the reduction method can make the set of all the compact non-redundant decision rules complete in the initial formal decision context.

Knowledge reduction is one of challenging problems in rough set theory, many static knowledge reduction algorithms have been proposed, however, since the objects in the actual information system are often changed, these static algorithms are computationally time-consuming. In order to overcome this shortcoming, the concept of simplified information system is introduced, and it is proved that knowledge reduction based on the simplified discernibility matrix is equivalent to that based on skowron’s discernibility matrix, On this condition, an efficient incremental updating algorithm for knowledge reduction is designed, which can be used to improve the efficiency of knowledge reduction, when a new object is added to information system. Finally, Theoretical analysis and example results show that illustrate the efficiency and feasibility of the algorithm.

Formal concept analysis of binary relations has proved to be useful in resolve of many problems of theoretical or interest. This paper deals with approach to knowledge reduction in formal context based on object oriented concept lattice and attribute oriented concept lattice. The concept of consistent set in formal context is first introduced and its equivalent definitions are given. The judgment theorem and discernibility matrix associated with formal context reducts are established, from which we can provide the approach to knowledge reduction in formal context.

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

Knowledge reduction is one of critical problems in data mining and information processing. It can simplify the structure of the lattice during the construction of fuzzy-crisp concept lattice. In terms of fuzzy-crisp concept, we develop an order-class matrix to represent extents and intents of concepts, respectively. In order to improve the computing efficiency, it is necessary to reduce the size of lattices as much as possible. Therefore the judgement theorem of meet-irreducible elements is proposed. To deal with attribute reductions, we develop a discernibility Boolean matrix in formal fuzzy contexts by preserving extents of meet-irreducible elements via order-class matrix. A heuristic attribute-reduction algorithm is proposed. Then we extend the proposed model to consistent formal fuzzy decision contexts. Our methods present a new framework for knowledge reduction in formal fuzzy contexts.

Skowron's discernibility matrix is one of representative approaches in computing relative core and relative reducts, while redundant information is also involved. To decrease the complexity of computation, the idea of granular computing is applied to lower the rank of discernibility matrix. In addition, the absorptivity based on bit-vector computation is proposed to simplify computation of relative core and relative reducts

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.

Rough set theory and formal concept analysis are two complementary mathematical tools for data analysis. In this paper, based on object oriented concept lattice, we first propose one kind of consistency of conditional concepts and decision concepts. Subsequently, notions of consistent set and reduct set of attributes are introduced, and some judgment theorems for consistent set are presented. Finally, discernibility matrix is employed to determine consistent sets and calculate reduct sets in consistent formal decision contexts.

Rough sets serves as a tool for data analysis and knowledge discovery from data databases. Attribute reduction is a basic issue in knowledge representation and data mining. This paper deals with distribution reduction in an inconsistent interval ordered information systems. The distribution reduction and maximum distribution reduction are proposed in inconsistent interval ordered information systems. Moreover, properties and relationship between them are discussed. Furthermore, judgement theorem and discernibility matrix are obtained, from which approaches to distribution reductions can be provided in inconsistent interval ordered information systems.

Knowledge reduction in formal contexts using non-negative matrix factorization

In this paper, two new concepts of lower approximation reduction and upper approximation reduction are introduced. Lower approximation reduction is the smallest attribute subset that preserves the lower approximations of all decision classes, and upper approximation reduction is the smallest attribute subset that preserves the upper approximations of all decision classes. For an inconsistent DT, an upper approximation consistent set must be a lower approximation consistent set, but the converse is not true. For a consistent DT, they are equivalent. After giving their equivalence definitions, we examine the judgement theorem and discernibility matrices associated with the two reducts, from which we can obtain approaches to knowledge reduction in inconsistent decision tables.

Knowledge reduction in real decision formal contexts

Formal concept analysis and rough set theory provide two different methods for data analysis and knowledge processing. Knowledge reduct in this paper combines the two models. For an initial data sets described by formal context, look for absolute necessary attribute sets by applying rough set theory. The sets can image the concepts and hiberarchy structure completely. Then calculate the value cores of attributes values for all objects and delete redundant attributes. At last, delete repeated instances and get the minimum formal context. Construct the concept lattice of the minimum formal context can diminish the size of concept lattice of the initial table at a certain extent.

Some disadvantages should be discussed deeply for the current reduction algorithms. To eliminate these limitations of classical algorithms based on positive region and conditional information entropy, a new conditional entropy, which could reflect the change of decision ability objectively, was defined with separating consistent objects form inconsistent objects. To select optimal attribute reduction, the judgment theorem of reduction with an inequality was investigated. Condition attributes were considered to estimate the significance for decision classes, and a complete heuristic algorithm was designed and implemented. Finally, through analyzing the given example, the proposed heuristic information is better and more efficient than the others. Comparing the proposed algorithm with these current algorithms through discrete data sets from UCI Machine Learning Repository, the experimental results prove its validity, which enlarges the applied area of rough set. from inconsistent objects, and the corresponding judgment theorem with an inequality is proposed for searching the minimal or optimal reduction of knowledge. With this new conditional entropy, we design a new heuristic algorithm of attribute reduction, which can reduce the computational complexity. Its time complexity is O(|C| 2 |U|), where |C| is the number of condition attributes, and |U| is the number of objects. In the large decision tables, since there usually is |U| >> |C|, the new attribute reduction algorithm is better than the presented algorithms above. Finally, the implemented algorithms are experimented with an example and four data sets from UCI Machine Learning Repository are given to illustrate the effectiveness of the new proposed algorithm through intercomparsion. The experiment results demonstrate its usefulness and are analyzed for further research.