Maximal Consistent Blocks Research Articles

Mining incomplete data using global and saturated probabilistic approximations based on characteristic sets and maximal consistent blocks

In this paper, we discuss a rough set approach to missing attribute values. Among many ways of interpreting missing values, in this paper we focus on two interpretations, lost values and “do not care” conditions. Using these interpretations, global and saturated probabilistic approximations are constructed with two types of granules: characteristic sets and maximal consistent blocks. We compare eight approaches, combining two interpretations of missing attribute values, two types of probabilistic approximations with two types of granules using an error rate that is computed as a result of ten-fold cross-validation. Using a 5% level of statistical significance, we present the experimental results for these eight approaches, showing statistically significant differences between all approaches to mining incomplete data. The results also show that no one method and approach is the best for every data set and that all eight approaches should be attempted. The final section of the paper presents the idea of concept-compatible data sets. We show that for these types of data sets, global and saturated probabilistic approximations for a concept are identical to the concept. We also show that for an incomplete data set with no duplicate rows using the lost interpretation of missing attribute values, the data set is concept-compatible.

Fuzzy and rough approach to the problem of missing data in fall detection system

Two new methods for mining incomplete data based on interval-valued fuzzy set theory and rough set theory, particularly maximal consistent blocks were proposed, and their application to the fall detection system, exactly to the posture detection system. The suggested methods are based on interval-valued aggregation operators supporting decision-making involving uncertainty. Additionally, the new measure of knowledge was used in two aspects: in the imputation of missing data and during the inference process. The usage of the knowledge measure allowed for a reduction of the uncertainty due to incompleteness and imprecision of data, while increasing the efficiency of the decision-making process, through the additional selection of objects for which the degree of the information measure is the highest. The developed hybrid approach to mining incomplete data and its application in a real decision-making problem yield stable performance despite the increase in missing data.

Handling the Complexity of Computing Maximal Consistent Blocks

The maximal consistent blocks technique, adopted from discrete mathematics, describes the maximal collection of objects, in which all objects are indiscernible in terms of available information. In this paper, we estimate the total possible number of maximal consistent blocks and prove that the number of such blocks may grow exponentially with respect to the number of attributes for incomplete data with “do not care” conditions. Results indicate that the time complexity of some known algorithms for computing maximal consistent blocks has been underestimated so far. Taking into account the complexity, for the practical usage of such blocks, we propose a performance improvement involving the parallelization of the maximal consistent blocks construction method.

TFD-IIS-CRMCB: Telecom Fraud Detection for Incomplete Information Systems Based on Correlated Relation and Maximal Consistent Block

Telecom fraud detection is of great significance in online social networks. Yet the massive, redundant, incomplete, and uncertain network information makes it a challenging task to handle. Hence, this paper mainly uses the correlation of attributes by entropy function to optimize the data quality and then solves the problem of telecommunication fraud detection with incomplete information. First, to filter out redundancy and noise, we propose an attribute reduction algorithm based on max-correlation and max-independence rate (MCIR) to improve data quality. Then, we design a rough-gain anomaly detection algorithm (MCIR-RGAD) using the idea of maximal consistent blocks to deal with missing incomplete data. Finally, the experimental results on authentic telecommunication fraud data and UCI data show that the MCIR-RGAD algorithm provides an effective solution for reducing the computation time, improving the data quality, and processing incomplete data.

Global and saturated probabilistic approximations based on generalized maximal consistent blocks

Abstract In this paper incomplete data sets, or data sets with missing attribute values, have three interpretations, lost values, attribute-concept values and ‘do not care’ conditions. Additionally, the process of data mining is based on two types of probabilistic approximations, global and saturated. We present results of experiments on mining incomplete data sets using six approaches, combining three interpretations of missing attribute values with two types of probabilistic approximations. We compare our six approaches, using the error rate computed as a result of ten-fold cross validation as a criterion of quality. We show that for some data sets the error rate is significantly smaller (5% level of significance) for lost values, for some data sets the smaller error rate is associated with attribute-concept values, and sometimes with ‘do not care’ conditions. Again, for some approaches the error rate is significantly smaller for saturated probabilistic approximations than for global probabilistic approximations, while for some approaches it is the other way around. Thus, for an incomplete data set, the best approach to data mining should be chosen by trying all six approaches.

A New Approach to Constructing Maximal Consistent Blocks for Mining Incomplete Data

A maximal consistent block technique is used for mining incomplete data sets. Initially, the idea of maximal consistent blocks was introduced for data sets with missing attribute values interpreted as “do not care” conditions. In such data, missing attribute values may be replaced by any existing attribute value. In this paper, we analyze existing algorithms for computing maximal consistent blocks and show that some algorithms may output blocks that are not maximal, especially for data sets with non-transitive characteristic relations. We also introduce a new approach for computing maximal consistent blocks from incomplete data, with two interpretations of missing attribute values: lost values and “do not care” conditions. We indicated that the obtained consistent blocks are maximal and our approach is faster than the commonly used method of generating maximal consistent blocks based on characteristic relation.

Complexity of rule sets in mining incomplete data using characteristic sets and generalized maximal consistent blocks

Abstract In this paper, missing attribute values in incomplete data sets have three possible interpretations: lost values, attribute-concept values and ‘do not care’ conditions. For rule induction, we use characteristic sets and generalized maximal consistent blocks. Therefore, we apply six different approaches for data mining. As follows from our previous experiments, where we used an error rate evaluated by ten-fold cross validation as the main criterion of quality, no approach is universally the best. Thus, we decided to compare our six approaches using complexity of rule sets induced from incomplete data sets. We show that the smallest rule sets are induced from incomplete data sets with attribute-concept values, while the most complicated rule sets are induced from data sets with lost values. The choice between interpretations of missing attribute values is more important than the choice between characteristic sets and generalized maximal consistent blocks.

Complexity of Rule Sets Mined from Incomplete Data Using Probabilistic Approximations Based on Generalized Maximal Consistent Blocks

Abstract In this paper, incomplete data sets have two kinds of missing attribute vales: lost values and “do not care” conditions. Lost values are interpreted as erased or as not inserted into the data set, while “do not care” conditions may be replaced by any specified attribute value. In addition, we use two kinds of probabilistic approximations, global and saturated. Both probabilistic approximations are constructed from generalized maximal consistent blocks. Since we are using two kinds of missing attribute values and two kinds of probabilistic approximations, we use four different ways of data mining. In our previous study, it was shown that pairwise differences in an error rate, evaluated by ten-fold cross validation between these four ways of data mining are statistically insignificant (5% level of significance). Hence, we explore the next important problem: when the rule sets will be the simplest. We show that the total number of rules is the smallest when missing attribute values are interpreted as “do not care” conditions. The difference between using both kinds of probabilistic approximations is insignificant.

Characteristic sets and generalized maximal consistent blocks in mining incomplete data

Mining incomplete data using approximations based on characteristic sets is a well-established technique. It is applicable to incomplete data sets with a few interpretations of missing attribute values, e.g., lost values and “do not care” conditions. On the other hand, maximal consistent blocks were introduced for incomplete data sets with only “do not care” conditions, using only lower and upper approximations. In this paper we introduce an extension of the maximal consistent blocks to incomplete data sets with any interpretation of missing attribute values and with probabilistic approximations. We prove new results on probabilistic approximations based on generalized maximal consistent blocks. Additionally, we present results of experiments on mining incomplete data using both characteristic sets and maximal consistent blocks and using two interpretations of missing attribute values: lost values and “do not care” conditions. We show that there is some evidence that the best approach is using middle probabilistic approximations based on characteristic sets or on maximal consistent blocks.

Tolerance-based multigranulation rough sets in incomplete systems

Presently, the notion of multigranulation has been brought to our attention. In this paper, the multigranulation technique is introduced into incomplete information systems. Both tolerance relations and maximal consistent blocks are used to construct multigranulation rough sets. Not only are the basic properties about these models studied, but also the relationships between different multigranulation rough sets are explored. It is shown that by using maximal consistent blocks, the greater lower approximation and the same upper approximation as from tolerance relations can be obtained. Such a result is consistent with that of a single-granulation framework.

Neighborhood Systems Based Rough Sets in Grey Fuzzy Information System

Pawlak’s rough set theory offers a formal theoretical framework to deal with categorical data. Later on, Dubois and Prade introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets. However, this model is not applicable to interval data, which widely exist in real world applications. In this work, we provide a new rough set approach to the grey fuzzy information system by the neighborhood system theory. By using grey lattice operation and Hausdorff distance, three forms of rough set models are constructed based on coverings, maximal consistent blocks and neighborhood system. Moreover, three numerical examples are employed to substantiate the conceptual arguments.

NEIGHBORHOOD SYSTEM BASED ROUGH SET: MODELS AND ATTRIBUTE REDUCTIONS

The neighborhood system based rough set is a generalization of Pawlak's rough set model since the former uses the neighborhood system instead of the partition for constructing target approximation. In this paper, the neighborhood system based rough set approach is employed to deal with the incomplete information system. By the coverings induced by the maximal consistent blocks and the support sets of the descriptors, respectively, two neighborhood systems based rough sets are explored. By comparing with the original maximal consistent block and descriptor based rough sets, the neighborhood system based rough sets hold the same lower approximations and the smaller upper approximations. Furthermore, the concept of attribute reduction is introduced into the neighborhood systems and the corresponding rough sets. The judgement theorems and discernibility functions to compute reducts are also presented. Some numerical examples are employed to substantiate the conceptual arguments.

Consistency measure, inclusion degree and fuzzy measure in decision tables

Classical consistency degree has some limitations for measuring the consistency of a decision table, in which the lower approximation of a target decision is only taken into consideration. In this paper, we focus on how to measure the consistencies of a target concept and a decision table and the fuzziness of a rough set and a rough decision in rough set theory. For three types of decision tables (complete, incomplete and maximal consistent blocks), the membership functions of an object are defined through using the equivalence class, tolerance class and maximal consistent blocks including itself, respectively. Based on these membership functions, we introduce consistency measures to assess the consistencies of a target set and a decision table, and define fuzziness measures to compute the fuzziness of a rough set and a rough decision in these three types of decision tables. In addition, the relationships among the consistency, inclusion degree and fuzzy measure are established as well. These results will be helpful for understanding the essence of the uncertainty in decision tables and can be applied for rule extraction and rough classification in practical decision issues.