| The rough set theory is a very useful mathematical tool to address imprecise, uncertain and vague information, it has enjoyed widespread success in many research areas, such as data mining, knowledge discovery, intelligent information processing, machine learning, clustering analysis and pattern classification.Classical rough set model assumes the classification must be fully correct or certain. Hence, it cannot effectively deal with data sets which have noisy data. To alleviate this problem, some probabilistic rough set models, such as 0.5 probabilistic rough set model, variable precision rough set model, Bayesian rough set model, game rough set model and decision-theoretic rough set model, are proposed by introducing probabilistic threshold values.Attribute reduction is one of the most fundamental and important topics in rough set theory. Attribute reduction can simplify the data analysis, improve the accuracy of a classification algorithm and reduce the dimensionality of the feature space. Uncertainty measures of rough set are very important in attribute reduction. In classical rough set model, uncertainty measures have the monotonicity with respect to the granularity of partition, the monotonicity is very important for constructing attribution reduction algorithms. However, when uncertainty measures in classical rough set model are directly extended to probabilistic rough set model, the monotonicity of uncertainty measures with respect to the granularity of partition does not hold because of introducing probabilistic threshold values in probabilistic rough set model. The non-monotonicity of uncertainty measures will increase the complexity of an attribute reduction algorithm in probabilistic rough set model. Thus, it is necessary to study the monotonic uncertainty measures and attribute reduction based on monotonic uncertainty measures in probabilistic rough set model.On the other hand, decision-theoretic rough set model is a typical probabilistic rough set model. However, the monotonicity of decision regions (positive region or non-negative region) with respect to set inclusion of attributes does not hold in decision-theoretic rough set model. Hence, some existing definitions of attribute rcducts may change decision regions in decision-theoretic rough set model. In addition, attribute reduction algorithm must check all subsets of an attribute set, or it may find a super reduct, which complicates the algorithm design.This dissertation focuses on the theory and methods of attribute reduction in probabilistic rough set model. The main contributions of this dissertation are summarized as follows.(1) Three basic uncertainty measures and three expected granularity-based uncertainty measures are proposed in probabilistic rough set model, and the monotonicity of these measures are proven. The findings provide theoretic basis for attribute reduction in probabilistic rough set model.The rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information. Uncertainty measure is a main research topic in rough set theory. In classical rough set model, the monotonicity is a basic property of uncertainty measures. However, the monotonicity of uncertainty measures does not hold in probabilistic rough set model, which complicates the algorithm design. Firstly, we analyze the non-monotonicity of uncertainty measures in probabilistic rough set. Secondly, we propose three basic uncertainty measures and three expected granularity-based uncertainty measures, the monotonicity of these measures are proven, and the relationships between uncertainty measures in classical rough set model and uncertainty measures in probabilistic rough set model are also discussed. The results of experimental analysis are included to validate the effectiveness of the proposed uncertainty measures. The findings will be helpful for attribute reduction in probabilistic rough set model. (Chapter 2)(2) The applications of uncertainty measures in attribute reduction are studied, and the attribution reduction approaches based on monotonic uncertainty measures are provided in probabilistic rough set model.Attribute reduction is one of the most extensively studied problems in rough set theory. A major application of uncertainty measures is constructing attribute reduction algorithms. A new reduct is defined in probabilistic rough set model based on the proposed monotonic uncertainty measures. The core and attribute significance are also defined. Heuristic reduction algorithms are then developed. The results show that the new reduct can achieve better classification performance. (Chapter 3)(3) Decision region (positive region or non-negative region) distribution preservation reducts are introduced into decision-theoretic rough set model. Heuristic reduction algorithms for the calculation of decision region distribution preservation reducts are developed.In classical rough set model, the positive region and the non-negative region are monotonic with respect to the set inclusion of attributes. However, the monotonicity property of the decision regions (positive region or non-negative region) with respect to the set inclusion of attributes does not hold in decision-theoretic rough set model. Therefore, the decision regions may be changed after attribute reduction based on quantitative preservation or qualitative preservation of decision regions. This effect is observed partly because decision regions are defined by introducing the probabilistic threshold values. In addition, heuristic reduction algorithms based on decision regions may find super reducts because of the non-monotonicity of decision regions. To address the above issues, this paper proposes solutions to the attribute reduction problem based on decision region preservation in the decision-theoretic rough set model. First, the (α, β) positive region distribution preservation reduct and the (α,β) non-negative region distribution preservation reduct are introduced into the decision-theoretic rough set model. Second, two new monotonic measures are constructed by considering variants of the conditional information entropy, from which we can obtain the heuristic reduction algorithms. The results of the experimental analysis validate the monotonicity of new measures and verify the effectiveness of decision region distribution preservation reducts. (Chapter 4)(4) Heuristic computing methods of decision region distribution preservation reducts are proposed by considering variants of the condition information content, and minimum attribute reduction problem is studied. On this basis, heuristic genetic algorithm to decision region distribution preservation reducts is developed.In decision-theoretic rough set model, since decision regions (positive region or non-negative region) are defined by allowing some extent misclassification, the monotonicity of decision regions with respect to the set inclusion of attributes does not hold. Attribute reduction algorithms must search all possible subsets of an attribute set due to the non-monotonicity of decision regions. To simplify the algorithm design, the positive region and non-negative distribution condition information contents are presented, which are used to design heuristic algorithms to two types of decision region distribution preservation reducts. In a bid to find the minimum attribute reduct, heuristic genetic algorithm to decision region distribution preservation reducts is proposed. A new operator, which is called modify operator, is constructed by using two types of decision region distribution condition information contents so that genetic algorithm can find decision region distribution preservation reducts. Experimental results verify the effectiveness of decision region distribution preservation reducts and show the efficiency of the genetic algorithm to find the minimum attribute reduct. (Chapter 5)... |
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