粗糙集理论若干问题的研究与应用
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摘要
粗糙集理论是处理不精确、不确定和不完整的一种新的数学工具。本文介绍了粗糙集的基本概念和研究现状,在此基础上研究粗糙集的两个因子、不确定测量方法、属性约简和属性细化以及粗糙集方法在软集合参数约简中的应用问题。具体内容介绍如下：
(1)针对原确定增量算子与不确定减量算子表达式没有清晰表达这两个算子真实含义的问题,重新定义该算子,并从理论上证明两类算子定义方式是等价的。在此基础上提出多集合确定增量算子与不确定减量算子定义,并研究两类多集合算子的性质。
(2)提出一般关系下的信息熵和条件熵的概念,并从理论上证明一般关系下的信息熵是等价关系与相容关系下的信息熵的扩展,且一般关系下的条件熵是等价关系下的条件熵各项分量的加权和。其次提出粗覆盖下的条件熵与互信息熵的概念,并研究了它们的性质。
(3)对已有和声搜索算法的两个重要调节参数进行改进,利用每次迭代目标函数值的最大差值来调节这两个参数,提出自适应和声搜索算法,为验证算法的有效性,利用五个标准测试函数且与其它三种优化算法作比较,仿真结果表明自适应和声搜索算法具有跳出局部极小值的能力和较强的鲁棒性。最后把该算法应用到粗糙集属性约简中,利用属性重要度作为属性约简的启发式信息,得到比较好的结果。
(4)把粗糙集不可分辨关系引入到软集合中,提出软集合正则参数约简的定义,并利用软集合参数重要度得到正则参数约简的必要条件,且给出正则参数约简算法。在此基础上研究模糊软集合正则参数约简问题。最后研究模糊软集合方法在决策问题上的应用,举例说明目前已有方法的不足,并对已有方法进行修正。
(5)提出了有效等价细化和有效集合细化的概念,研究条件属性细化程度与上近似、下近似、近似分类精度、近似分类质量、规则数目、相对约简和属性必要性的关系。
Rough set theory is a new mathematical tool to deal with imperfectness, uncertainty and vagueness. Firstly, the basic concepts and present develoments of rough set theory are introduced. In the following, five regions including two operators, uncertain measure method, attribute reduction, attribute subdivision and parameter reduction of soft set based on rough set theory are mainly discussed. The contributions of this thesis are as follows:
(1)Two new operators are redefined to express the essential characters of certain increment operator and uncertain decrement operator. It is proved theoretically that the two new operators are equivalent to the original two operators. Then the multiple certain increment operator and multiple uncertain decrement operator are introduced based on two new operators, and these properties are discussed.
(2)The information entropy and conditional entropy based on general relation are introduced. And it is proved that the information entropy based on general relation is the extension of information entropy based on equivalent relation and compatible relation. Furthermore, the conditional entropy based on general relation is equal to the weighted sums for components of conditional entropy based on equivalent relation. At last, the conditional entropy and mutual information entropy based on generalized covering are introduced and these properties are discussed.
(3)Two important parameters are improved by the difference between maximum and minimum of objective values, and a new adaptive harmony search algorithm is proposed. The new algorithm is compared with three algorithms and tested by five test functions. The simulation results show that AHS has the strong robustness and can escape the local minimum. Then attribute reduction method based on AHS is constructed by introduce the attribute significance. And the test results show the validity and feasibility of this method.
(4)The normal parameter reduction is introduced by the indiscemibility relation of rough set theory. And a necessary condition of normal parameter reduction is given by the parameter importance degree of soft set. Then the parameter reduction algorithm is proposed. Finally, a fuzzy soft set approach to decision making problem is discussed. A counter example is introduced and the correction is made for deficiency of the existing approach.
(5)The effectively equivalent class subdivision and effective set subdivision are introduced. Then the relationships between the degrees of attribute subdivision and the upper approximation, lower approximation, accuracy of approximate classification, quality of approximate classification, decision-making rule, relative reduction and necessity of attributes are discussed.
(1)针对原确定增量算子与不确定减量算子表达式没有清晰表达这两个算子真实含义的问题,重新定义该算子,并从理论上证明两类算子定义方式是等价的。在此基础上提出多集合确定增量算子与不确定减量算子定义,并研究两类多集合算子的性质。
(2)提出一般关系下的信息熵和条件熵的概念,并从理论上证明一般关系下的信息熵是等价关系与相容关系下的信息熵的扩展,且一般关系下的条件熵是等价关系下的条件熵各项分量的加权和。其次提出粗覆盖下的条件熵与互信息熵的概念,并研究了它们的性质。
(3)对已有和声搜索算法的两个重要调节参数进行改进,利用每次迭代目标函数值的最大差值来调节这两个参数,提出自适应和声搜索算法,为验证算法的有效性,利用五个标准测试函数且与其它三种优化算法作比较,仿真结果表明自适应和声搜索算法具有跳出局部极小值的能力和较强的鲁棒性。最后把该算法应用到粗糙集属性约简中,利用属性重要度作为属性约简的启发式信息,得到比较好的结果。
(4)把粗糙集不可分辨关系引入到软集合中,提出软集合正则参数约简的定义,并利用软集合参数重要度得到正则参数约简的必要条件,且给出正则参数约简算法。在此基础上研究模糊软集合正则参数约简问题。最后研究模糊软集合方法在决策问题上的应用,举例说明目前已有方法的不足,并对已有方法进行修正。
(5)提出了有效等价细化和有效集合细化的概念,研究条件属性细化程度与上近似、下近似、近似分类精度、近似分类质量、规则数目、相对约简和属性必要性的关系。
Rough set theory is a new mathematical tool to deal with imperfectness, uncertainty and vagueness. Firstly, the basic concepts and present develoments of rough set theory are introduced. In the following, five regions including two operators, uncertain measure method, attribute reduction, attribute subdivision and parameter reduction of soft set based on rough set theory are mainly discussed. The contributions of this thesis are as follows:
(1)Two new operators are redefined to express the essential characters of certain increment operator and uncertain decrement operator. It is proved theoretically that the two new operators are equivalent to the original two operators. Then the multiple certain increment operator and multiple uncertain decrement operator are introduced based on two new operators, and these properties are discussed.
(2)The information entropy and conditional entropy based on general relation are introduced. And it is proved that the information entropy based on general relation is the extension of information entropy based on equivalent relation and compatible relation. Furthermore, the conditional entropy based on general relation is equal to the weighted sums for components of conditional entropy based on equivalent relation. At last, the conditional entropy and mutual information entropy based on generalized covering are introduced and these properties are discussed.
(3)Two important parameters are improved by the difference between maximum and minimum of objective values, and a new adaptive harmony search algorithm is proposed. The new algorithm is compared with three algorithms and tested by five test functions. The simulation results show that AHS has the strong robustness and can escape the local minimum. Then attribute reduction method based on AHS is constructed by introduce the attribute significance. And the test results show the validity and feasibility of this method.
(4)The normal parameter reduction is introduced by the indiscemibility relation of rough set theory. And a necessary condition of normal parameter reduction is given by the parameter importance degree of soft set. Then the parameter reduction algorithm is proposed. Finally, a fuzzy soft set approach to decision making problem is discussed. A counter example is introduced and the correction is made for deficiency of the existing approach.
(5)The effectively equivalent class subdivision and effective set subdivision are introduced. Then the relationships between the degrees of attribute subdivision and the upper approximation, lower approximation, accuracy of approximate classification, quality of approximate classification, decision-making rule, relative reduction and necessity of attributes are discussed.
引文
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