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Cluster ensemble is a very important tool that aggregates several base clusterings to generate a single output clustering with improved robustness and stability. However, the quality of the final clustering is often affected by uncertainties on the generation and integration of base clusterings. In this paper, we develop an information-theoretical framework which makes an effort to obtain a final clustering with high consensus on both the original data set and the base clustering set by minimizing the two uncertainties of cluster ensemble. In this framework, we provide a weighted consensus measure based on information entropy to evaluate the quality of a clustering, the similarity between clusters and the similarity between objects. Based on the measure, we propose three weighted cluster ensemble algorithms with different ensemble strategies in the framework, including the weighted feature consensus algorithm, the weighted relabeling consensus algorithm and the weighted pairwise-similarity consensus algorithm. In the experimental analysis, we compare the proposed algorithms with other existing clustering ensemble algorithms on several data sets. The comparison results illustrate the proposed algorithms are very effective and robust. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • A cautious ranking methodology is built, which involves data representation, feature selection and ranking mechanism. • The form of interval data is introduced for reflecting the fluctuation of indicators. • An ordered-relevance-preserving feature selection method is constructed in a systematic ranking methodology. • Case studies show that the proposed systematic approach is efficient and effective. • This study provides a new view and thinking for ranking decision. Abstract To reasonably and effectively solve the ranking decision problem, there currently exist various meaningful methods that are focused on different perspectives. However, the ranking decision problem is a systematic issue that involves data representation, the dominance relation, feature selection, and ranking mechanism. In this study, we aimed to build a novel ranking methodology by taking into account both the inherent multicriteria nature of practical decision situations and cautious decision makers’ preferences. In order to better reveal the entirety of the data set, the form of interval data is introduced to characterize the ranges of attribute values. For the purpose of improving the decision performance, we develop a measurement called interval ordered conditional entropy to extract the most representative condition attributes having significant ordered relevance to the decision attribute. Based on the cautious dominance relation introduced for interval data, a two-step ranking mechanism with cautious characteristics is introduced that utilizes an interval ordered information table organized according to the previously selected informative attributes. In addition, the validity of this ranking method is tested through a detailed case study on stock screening decisions involving three successive rounds of tests. The corresponding results indicate the effectiveness of the methodological approach proposed in this paper. [ABSTRACT FROM AUTHOR]

A fuzzy granular structure refers to a mathematical structure of the collection of fuzzy information granules granulated from a dataset, while a fuzzy information granularity is used to measure its uncertainty. However, the existing forms of fuzzy information granularity have two limitations. One is that when the fuzzy information granularity of one fuzzy granular structure equals that of the other, one can say that these two fuzzy granular structures possess the same uncertainty, but these two fuzzy granular structures may be not equivalent to each other. The other limitation is that existing axiomatic approaches to fuzzy information granularity are still not complete, under which when the partial order relation among fuzzy granular structures cannot be found, their coarseness/fineness relationships will not be revealed. To address these issues, a so-called fuzzy granular structure distance is proposed in this study, which can well discriminate the difference between any two fuzzy granular structures. Besides this advantage, the fuzzy granular structure distance has another important benefit: It can be used to establish a generalized axiomatic constraint for fuzzy information granularity. By using the axiomatic constraint, the coarseness/fineness of any two fuzzy granular structures can be distinguished. In addition, through taking the fuzzy granular structure distances of a fuzzy granular structure to the finest one and the coarsest one into account, we also can build a bridge between fuzzy information granularity and fuzzy information entropy. The applicable analysis on 12 real-world datasets shows that the fuzzy granular structure distance and the generalized fuzzy information granularity have much better performance than existing methods. [ABSTRACT FROM AUTHOR]

Many real data increase dynamically in size. This phenomenon occurs in several fields including economics, population studies, and medical research. As an effective and efficient mechanism to deal with such data, incremental technique has been proposed in the literature and attracted much attention, which stimulates the result in this paper. When a group of objects are added to a decision table, we first introduce incremental mechanisms for three representative information entropies and then develop a group incremental rough feature selection algorithm based on information entropy. When multiple objects are added to a decision table, the algorithm aims to find the new feature subset in a much shorter time. Experiments have been carried out on eight UCI data sets and the experimental results show that the algorithm is effective and efficient. [ABSTRACT FROM AUTHOR]

Abstract: Many real data sets in databases may vary dynamically. With the rapid development of data processing tools, databases increase quickly not only in rows (objects) but also in columns (attributes) nowadays. This phenomena occurs in several fields including image processing, gene sequencing and risk prediction in management. Rough set theory has been conceived as a valid mathematical tool to analyze various types of data. A key problem in rough set theory is executing attribute reduction for a data set. This paper focuses on attribute reduction for data sets with dynamically-increasing attributes. Information entropy is a common measure of uncertainty and has been widely used to construct attribute reduction algorithms. Based on three representative entropies, this paper develops a dimension incremental strategy for redcut computation. When an attribute set is added to a decision table, the developed algorithm can find a new reduct in a much shorter time. Experiments on six data sets downloaded from UCI show that, compared with the traditional non-incremental reduction algorithm, the developed algorithm is effective and efficient. [Copyright &y& Elsevier]

Abstract: Many real data sets in databases may vary dynamically. With such data sets, one has to run a knowledge acquisition algorithm repeatedly in order to acquire new knowledge. This is a very time-consuming process. To overcome this deficiency, several approaches have been developed to deal with dynamic databases. They mainly address knowledge updating from three aspects: the expansion of data, the increasing number of attributes and the variation of data values. This paper focuses on attribute reduction for data sets with dynamically varying data values. Information entropy is a common measure of uncertainty and has been widely used to construct attribute reduction algorithms. Based on three representative entropies, this paper develops an attribute reduction algorithm for data sets with dynamically varying data values. When a part of data in a given data set is replaced by some new data, compared with the classic reduction algorithms based on the three entropies, the developed algorithm can find a new reduct in a much shorter time. Experiments on six data sets downloaded from UCI show that the algorithm is effective and efficient. [Copyright &y& Elsevier]

Abstract: In cluster analysis, one of the most challenging and difficult problems is the determination of the number of clusters in a data set, which is a basic input parameter for most clustering algorithms. To solve this problem, many algorithms have been proposed for either numerical or categorical data sets. However, these algorithms are not very effective for a mixed data set containing both numerical attributes and categorical attributes. To overcome this deficiency, a generalized mechanism is presented in this paper by integrating Rényi entropy and complement entropy together. The mechanism is able to uniformly characterize within-cluster entropy and between-cluster entropy and to identify the worst cluster in a mixed data set. In order to evaluate the clustering results for mixed data, an effective cluster validity index is also defined in this paper. Furthermore, by introducing a new dissimilarity measure into the k-prototypes algorithm, we develop an algorithm to determine the number of clusters in a mixed data set. The performance of the algorithm has been studied on several synthetic and real world data sets. The comparisons with other clustering algorithms show that the proposed algorithm is more effective in detecting the optimal number of clusters and generates better clustering results. [Copyright &y& Elsevier]

Abstract: Ranking decision for interval data is a very important issue in decision making analysis. In recent years, several ranking approaches based on dominance relations have been developed. In these approaches, a dominance degree and an entire dominance degree are employed. However, one cannot obtain the complete rank of objects. To address this problem, this work will propose a two-grade approach to ranking interval data. In this approach, we keep the ranking result induced by the entire dominance degree in the first grade, and then refine the objects that cannot be ranked through introducing a so-called entire directional distance index. An example and a real case are employed to verify the effectivity of the two-grade ranking approach proposed in this paper. [Copyright &y& Elsevier]

Zadeh’s seminal work in theory of fuzzy-information granulation in human reasoning is inspired by the ways in which humans granulate information and reason with it. This has led to an interesting research topic: granular computing (GrC). Although many excellent research contributions have been made, there remains an important issue to be addressed: What is the essence of measuring a fuzzy-information granularity of a fuzzy-granular structure? What is needed to answer this question is an axiomatic constraint with a partial-order relation that is defined in terms of the size of each fuzzy-information granule from a fuzzy-binary granular structure. This viewpoint is demonstrated for fuzzy-binary granular structure, which is called the binary GrC model by Lin. We study this viewpoint from from five aspects in this study, which are fuzzy BINARY-granular-structure operators, partial-order relations, measures for fuzzy-information granularity, an axiomatic approach to fuzzy-information granularity, and fuzzy-information entropies. [ABSTRACT FROM AUTHOR]

The given attribute reduction approach decides the decision performance of a reduced decision table, which can give a guidance for selecting one rule-extraction method in practical applications. The objective of this study is to compare the decision performance of positive-region reduction, Shannon entropy reduction and Liang entropy reduction. In this paper, the relationships between positive-region reduction, Shannon entropy reduction and Liang entropy reduction are first investigated. Then, by means of three evaluation indices (certainty measure, consistency measure and support measure), we systemically analyse these change mechanisms for decision performance of a decision table induced by each of these three types of reduction approaches. Finally, by numerical experiments, these change mechanisms of a decision table's decision performance are verified for the above-mentioned three attribute reductions. [ABSTRACT FROM AUTHOR]

Ensemble clustering has attracted much attention for its robustness, stability, and accuracy in academic and industry communities. In order to yield base clusterings with high quality and diversity simultaneously in ensemble clustering, many efforts have been done by exploiting different clustering models and data information. However, these methods neglect correlation between different base clusterings during the process of base clusterings generation, which is important to obtain a quality and diverse clustering decision. To overcome this deficiency, a sequential ensemble clusterings generation algorithm for mixed data is developed in this paper based on information entropy. The first high quality base clustering is yield by maximizing the entropy-based criterion. Afterward, a sequential paradigm is utilized to incrementally find more base clusterings, in which the diversity between a new base clustering and the former base partitions is measured by the normalized mutual information. Extensive experiments conducted on various data sets have demonstrated the superiority of our proposal as compared to several existing base clusterings generation algorithms. [ABSTRACT FROM AUTHOR]