模糊信息距离及其若干应用
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摘要
随着社会的快速发展，人们所面临的问题愈来愈复杂。由于客观世界自身的不确定性以及人类对现实世界有限的认知能力，在不确定环境下做出合理的识别和决策已经成为一个有着强烈需求的实际问题。作为处理不确定性问题的有力工具，模糊集理论已经被广泛应用。在模糊集理论应用于实际的过程中，如何度量目标之间的信息距离是一个关键环节。本文主要基于模糊集，型-2模糊集理论进行研究，利用模糊信息构造距离公式并将其应用于模式识别、图像分割以及决策分析等领域。本文的主要工作概括如下：
1.针对一般的模糊集进行研究，主要讨论模糊散度的构造与应用。基于α型相对信息，提出两类新的模糊散度并采用f散度的观点研究其性质。最后，将这两类模糊散度公式应用于图像分割领域，给出基于阈值的分割算法以显示其有效性。
2.研究一类特殊的模糊集，即，模糊数上的距离公式。主要针对三角模糊数进行讨论，利用三角模糊数的截集信息构造一族带参数的距离公式，并使用该距离给出一种基于TOPSIS的方法来处理三角模糊数上的多属性决策问题。此外，定义三角模糊数上新的完备距离公式，并将该距离应用于模糊聚类分析当中，解决三角模糊数型数据的聚类问题。
3.针对型-2模糊集的一种特例，即，Vague集进行讨论。首先利用Vague集自身的信息给出Vague集上含参数的信息距离公式，由该距离诱导的相似度与相关的几个相似度在数据集上的比较显示出其具有一定的优势；其次，依据Vague集与模糊集之间的转换关系提出Vague集上的积分型距离公式；最后将两种距离公式应用于模式分类与医疗诊断。
4.由于区间值模糊集（即，Vague集）的投票模型在反映信息方面存在不足，没有充分表达赞同票和不反对票在决策中的倾向性影响，为此借鉴三参数区间数进一步增强模糊信息的表示，提出了三参数区间值模糊集的概念。当重心点为区间中点时，三参数区间值模糊集退化为区间值模糊集（即，Vague集）。论文构造了三参数区间值模糊集上的信息距离，并将其应用于多属性决策。最后，提出基于记分函数的排序方法以得到备选方案的严格序关系。
5.提出模糊数型-2模糊集的概念。在实际决策过程中，往往需要领域专家对备选方案在各属性下的特性进行评价。由于各属性的类型，取值范围以及量纲等均存在差异，很难用确定的数值表示，在实际中往往采用语言变量进行描述。然而语言变量不适合量化计算，因此采用模糊数对语言变量进行描述是一种常见的处理方式。由此可见，模糊数型-2模糊集可为这种问题提供一个较为合适的数学模型。论文提出三角模糊数型-2模糊集的距离公式，该公式结合了Hamming距离和Hausdorff距离的特点，在模式识别中的应用显示出其有效性。
With the rapid development of society, the complexity of the handling issuesincrease. According to the objective world’s own uncertainties and the human being’sfinite knowledge on the external world, making the corresponding decision withuncertain background becomes a real problem for decision makers to handle. As a veryuseful tool for handling the uncertain problems, fuzzy sets have already been applieddeeply and widely. In real applications of the fuzzy sets theory, measuring the distanceor similarity between objects is a key step. In this dissertation, fuzzy sets theory andtype-2fuzzy sets theory will be mainly discussed. The novel distance measures arepresented by using fuzzy information, and applied in many areas such as patternrecognition, image segmentation and decision-making analysis, etc. The main work ofthis dissertation is following:
1. For the general fuzzy sets, the construction of fuzzy divergences and itsapplications are mainly discussed. Based on the relative information of typeα, twonew classes of fuzzy divergence are proposed. The properties of the two classes offuzzy divergence are discussed in the view of f divergence. Finally, the twoclasses of fuzzy divergence are applied to deal with image segmentation, thealgorithm based on threshoulding method are given to show the effectiveness of thefuzzy divergences.
2. The distance measures on a class of special fuzzy sets, which are fuzzynumbers, are researched. Among the all kinds of fuzzy number, triangular fuzzynumber is mainly discussed. A novel class of distance with parameters on the set oftriangular fuzzy numbers is defined by using the information of cut-sets. Using theprovided distance, a method based on TOPSIS is presented to deal with themulti-criteria decision-making problem on triangular fuzzy numbers. Furthermore,another novel complete distance measure between triangular fuzzy numbers isdefined. And two clustering algorithms based on the distance are provided to dealwith clustering of the data of triangular fuzzy numbers.
3. The distance measures between vague sets, which are special form of thetype-2fuzzy sets, are discussed. First of all, the distance measures with parametersare presented by taking into account the IFSs’ own information. Comparing withseveral existent similarity measures on some data, the clustering results show theeffectiveness of the similarity measures induced by the proposed distance measures. Moreover, using the transform relationship between an intuitionistic fuzzy set and afuzzy set, new distance measures on intuitionistic fuzzy sets named integral-typedistance measures is defined. Finally, the applications to pattern classification andmedical diagnosis are shown.
4. As we all know the drawback of the vote model of the interval-valued fuzzysets, i.e. vague sets, is that the tendentious affect of the favor and the abstentionsdoes not represent sufficiently. Therefore, the model of the interval numbers ofthree parameters can be used to represent fuzzy information. The notion of threeparameters interval-valued fuzzy set is presented. While the point with being mostlikely value of the object is just the midpoint of interval, a three parametersinterval-valued fuzzy set is regarded as an interval-valued fuzzy set, i.e. vague set.A distance on the three parameters interval-valued fuzzy values is provided andapplied to deal with the decision-making model on the three parametersinterval-valued fuzzy sets. Finally, a decision-making approach based on scorefunction is presented, and a strictly order on alternatives can be obtained by theproposed method.
5. The notion of fuzzy-number type-2fuzzy sets is presented. In real makingdecision, it is often needed that the field experts estimate the criteria values of thealternatives. Due to the differences of the criteria’s type, range and dimension, it ishard to use the accurate values to express the evaluation of the alternatives. So thelangrage variables are often used to express the criteria values. However it is hardfor langrage variables to be computed, there is a natural way to handle withlangrage variables by using fuzzy numbers. Therefore, the notion of fuzzy-numbertype-2fuzzy sets is able to provide with a proper model for this case. In this chapter,a distance on the fuzzy-number type-2fuzzy sets is presented. The distancecombines the characteristic of Hamming metric with that of Hausdorff metric. Andthe effectiveness of the distance is shown by its application to pattern recognition.
1.针对一般的模糊集进行研究，主要讨论模糊散度的构造与应用。基于α型相对信息，提出两类新的模糊散度并采用f散度的观点研究其性质。最后，将这两类模糊散度公式应用于图像分割领域，给出基于阈值的分割算法以显示其有效性。
2.研究一类特殊的模糊集，即，模糊数上的距离公式。主要针对三角模糊数进行讨论，利用三角模糊数的截集信息构造一族带参数的距离公式，并使用该距离给出一种基于TOPSIS的方法来处理三角模糊数上的多属性决策问题。此外，定义三角模糊数上新的完备距离公式，并将该距离应用于模糊聚类分析当中，解决三角模糊数型数据的聚类问题。
3.针对型-2模糊集的一种特例，即，Vague集进行讨论。首先利用Vague集自身的信息给出Vague集上含参数的信息距离公式，由该距离诱导的相似度与相关的几个相似度在数据集上的比较显示出其具有一定的优势；其次，依据Vague集与模糊集之间的转换关系提出Vague集上的积分型距离公式；最后将两种距离公式应用于模式分类与医疗诊断。
4.由于区间值模糊集（即，Vague集）的投票模型在反映信息方面存在不足，没有充分表达赞同票和不反对票在决策中的倾向性影响，为此借鉴三参数区间数进一步增强模糊信息的表示，提出了三参数区间值模糊集的概念。当重心点为区间中点时，三参数区间值模糊集退化为区间值模糊集（即，Vague集）。论文构造了三参数区间值模糊集上的信息距离，并将其应用于多属性决策。最后，提出基于记分函数的排序方法以得到备选方案的严格序关系。
5.提出模糊数型-2模糊集的概念。在实际决策过程中，往往需要领域专家对备选方案在各属性下的特性进行评价。由于各属性的类型，取值范围以及量纲等均存在差异，很难用确定的数值表示，在实际中往往采用语言变量进行描述。然而语言变量不适合量化计算，因此采用模糊数对语言变量进行描述是一种常见的处理方式。由此可见，模糊数型-2模糊集可为这种问题提供一个较为合适的数学模型。论文提出三角模糊数型-2模糊集的距离公式，该公式结合了Hamming距离和Hausdorff距离的特点，在模式识别中的应用显示出其有效性。
With the rapid development of society, the complexity of the handling issuesincrease. According to the objective world’s own uncertainties and the human being’sfinite knowledge on the external world, making the corresponding decision withuncertain background becomes a real problem for decision makers to handle. As a veryuseful tool for handling the uncertain problems, fuzzy sets have already been applieddeeply and widely. In real applications of the fuzzy sets theory, measuring the distanceor similarity between objects is a key step. In this dissertation, fuzzy sets theory andtype-2fuzzy sets theory will be mainly discussed. The novel distance measures arepresented by using fuzzy information, and applied in many areas such as patternrecognition, image segmentation and decision-making analysis, etc. The main work ofthis dissertation is following:
1. For the general fuzzy sets, the construction of fuzzy divergences and itsapplications are mainly discussed. Based on the relative information of typeα, twonew classes of fuzzy divergence are proposed. The properties of the two classes offuzzy divergence are discussed in the view of f divergence. Finally, the twoclasses of fuzzy divergence are applied to deal with image segmentation, thealgorithm based on threshoulding method are given to show the effectiveness of thefuzzy divergences.
2. The distance measures on a class of special fuzzy sets, which are fuzzynumbers, are researched. Among the all kinds of fuzzy number, triangular fuzzynumber is mainly discussed. A novel class of distance with parameters on the set oftriangular fuzzy numbers is defined by using the information of cut-sets. Using theprovided distance, a method based on TOPSIS is presented to deal with themulti-criteria decision-making problem on triangular fuzzy numbers. Furthermore,another novel complete distance measure between triangular fuzzy numbers isdefined. And two clustering algorithms based on the distance are provided to dealwith clustering of the data of triangular fuzzy numbers.
3. The distance measures between vague sets, which are special form of thetype-2fuzzy sets, are discussed. First of all, the distance measures with parametersare presented by taking into account the IFSs’ own information. Comparing withseveral existent similarity measures on some data, the clustering results show theeffectiveness of the similarity measures induced by the proposed distance measures. Moreover, using the transform relationship between an intuitionistic fuzzy set and afuzzy set, new distance measures on intuitionistic fuzzy sets named integral-typedistance measures is defined. Finally, the applications to pattern classification andmedical diagnosis are shown.
4. As we all know the drawback of the vote model of the interval-valued fuzzysets, i.e. vague sets, is that the tendentious affect of the favor and the abstentionsdoes not represent sufficiently. Therefore, the model of the interval numbers ofthree parameters can be used to represent fuzzy information. The notion of threeparameters interval-valued fuzzy set is presented. While the point with being mostlikely value of the object is just the midpoint of interval, a three parametersinterval-valued fuzzy set is regarded as an interval-valued fuzzy set, i.e. vague set.A distance on the three parameters interval-valued fuzzy values is provided andapplied to deal with the decision-making model on the three parametersinterval-valued fuzzy sets. Finally, a decision-making approach based on scorefunction is presented, and a strictly order on alternatives can be obtained by theproposed method.
5. The notion of fuzzy-number type-2fuzzy sets is presented. In real makingdecision, it is often needed that the field experts estimate the criteria values of thealternatives. Due to the differences of the criteria’s type, range and dimension, it ishard to use the accurate values to express the evaluation of the alternatives. So thelangrage variables are often used to express the criteria values. However it is hardfor langrage variables to be computed, there is a natural way to handle withlangrage variables by using fuzzy numbers. Therefore, the notion of fuzzy-numbertype-2fuzzy sets is able to provide with a proper model for this case. In this chapter,a distance on the fuzzy-number type-2fuzzy sets is presented. The distancecombines the characteristic of Hamming metric with that of Hausdorff metric. Andthe effectiveness of the distance is shown by its application to pattern recognition.
引文
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