Continuity and completeness of lattice-valued possibilistic measures

Properties such as continuity from above and from below and various kinds of completeness are analysed when investigating set functions, in particular probabilistic and possibilistic measures and the relations of these properties to properties such as (σ)-additivity of probability measures or (complete) maxitivity of possibilistic measures are proved. In this work, the notions of continuity from above and from below are introduced for non-numerical possibilistic measures, taking their values in a complete lattice and at least for some relations valid for real-valued possibilistic measures of their analogies for lattice-valued possibilistic measures are stated and proved.     

A computable approach to measure and integration theory

We introduce a computable framework for Lebesgue’s measure and integration theory in the spirit of domain theory. For an effectively given second countable locally compact Hausdorff space and an effectively given finite Borel measure on the space, we define a recursive measurable set, which extends the corresponding notion due to S?anin for the Lebesgue measure on the real line. We also introduce the stronger notion of a computable measurable set, where a measurable set is approximated from inside and outside by sequences of closed and open subsets, respectively. The more refined property of computable measurable sets give rise to the idea of partial measurable subsets, which naturally form a domain for measurable subsets. We then introduce interval-valued measurable functions and develop the notion of recursive and computable measurable functions using interval-valued simple functions. This leads us to the interval versions of the main results in classical measure theory. The Lebesgue integral is shown to be a continuous operator on the domain of interval-valued measurable functions and the interval-valued Lebesgue integral provides a computable framework for integration.     

An Outer-Measure Approach for Resource-Bounded Measure

Lutz developed a general theory of resource-bounded measurability and measure on suitable complexity classes C⊆C (see Proceedings of the 13th IEEE Conference on Computational Complexity, pp. 236–248, 1998), where Cantor Space C is the class of all decision problems, and classes C include various exponential time and space complexity classes, the class of all decidable languages, and the Cantor space C itself. In this paper, a different general theory of resource-bounded measurability and measure on those complexity classes is developed. Our approach is parallel to the Carathéodory outer measure approach in classical Lebesgue measure theory. We shall show that many nice properties in the classical Lebesgue measure theory hold in the resource-bounded case also. The Carathéodory approach gives short and easy proofs of theorems in the resource-bounded case as well as in the classical case. The class of measurable sets in our paper is strictly larger than that of Lutz, and the two resource-bounded measures assign the same measure for a set if the set is measurable in the sense of Lutz.     

区间复模糊软集的距离测度及其应用

基于区间复模糊软集的概念,定义了多种区间复模糊软集的距离测度公式,包含Hausdorff距离、Hamming距离、Euclidean距离、广义Hausdorff距离、广义Euclidean距离、广义加权Hausdorff距离、广义加权Euclidean距离、加权Hausdorff距离、加权Hamming距离、加权Euclidean距离。提出了除交、并、补运算外的区间复模糊软集的加法、乘法、部分隶属度和部分非隶属度运算以及距离测度之间的运算性质。基于区间复模糊软集距离测度构造了一种TOPSIS决策方法,并将这种决策方法应用于经济分析中,验证了所提方法的可行性。     

The information content of rules and rule sets and its application

The information content of rules is categorized into inner mutual information content and outer impartation information content. Actually, the conventional objective interestingness measures based on information theory are all inner mutual information, which represent the confidence of rules and the mutual information between the antecedent and consequent. Moreover, almost all of these measures lose sight of the outer impartation information, which is conveyed to the user and help the user to make decisions. We put forward the viewpoint that the outer impartation information content of rules and rule sets can be represented by the relations from input universe to output universe. By binary relations, the interaction of rules in a rule set can be easily represented by operators： union and intersection. Based on the entropy of relations, the outer impartation information content of rules and rule sets are well measured. Then, the conditional information content of rules and rule sets, the independence of rules and rule sets and the inconsistent knowledge of rule sets are defined and measured. The properties of these new measures are discussed and some interesting results are proven, such as the information content of a rule set may be bigger than the sum of the information content of rules in the rule set, and the conditional information content of rules may be negative. At last, the applications of these new measures are discussed. The new method for the appraisement of rule mining algorithm, and two rule pruning algorithms, λ-choice and RPClC, are put forward. These new methods and algorithms have predominance in satisfying the need of more efficient decision information.     

一种新的Vague集距离测度及其应用

在给出Vague集及其距离测度的概念后,将单个Vague值作为一个区间,运用Vague集向Fuzzy集转化的思想将区间离化为点,通过求点的欧式距离得出两个Vague值之间的距离测度。分析了该公式的性能,并讨论了它在模糊识别中的应用。     

结合类内和类间距离的可能聚类分割算法

目的 为了进一步提高噪声图像分割的抗噪性和准确性,提出一种结合类内距离和类间距离的改进可能聚类算法并将其应用于图像分割。方法 该算法避免了传统可能性聚类分割算法中仅仅考虑以样本点到聚类中心的距离作为算法的测度,将类内距离与类间距离相结合作为算法的新测度,即考虑了类内紧密程度又考虑了类间离散程度,以便对不同的聚类结构有较强的稳定性和更好的抗噪能力,并且将直方图融入可能模糊聚类分割算法中提出快速可能模糊聚类分割算法,使其对各种较复杂图像的分割具有即时性。结果 通过人工合成图像和实际遥感图像分割测试结果表明,本文改进可能聚类算法是有效的,其分割轮廓清晰,分类准确且噪声较小,其误分率相比其他算法至少降低了2个百分点,同时能获得更满意的分割效果。结论 针对模糊C-均值聚类分割算法和可能性聚类分割算法对于背景和目标颜色相近的图像分类不准确的缺陷,将类内距离与类间距离相结合作为算法的测度有效的解决了图像分割归类问题,并且结合直方图提出快速可能模糊聚类分割算法使其对于大篇幅复杂图像也具有适用性。     

Some cosine similarity measures and distance measures between q-rung orthopair fuzzy sets

In this paper, we consider some cosine similarity measures and distance measures between q-rung orthopair fuzzy sets (q-ROFSs). First, we define a cosine similarity measure and a Euclidean distance measure of q-ROFSs, their properties are also studied. Considering that the cosine measure does not satisfy the axiom of similarity measure, then we propose a method to construct other similarity measures between q-ROFSs based on the proposed cosine similarity and Euclidean distance measures, and it satisfies with the axiom of the similarity measure. Furthermore, we obtain a cosine distance measure between q-ROFSs by using the relationship between the similarity and distance measures, then we extend technique for order of preference by similarity to the ideal solution method to the proposed cosine distance measure, which can deal with the related decision-making problems not only from the point of view of geometry but also from the point of view of algebra. Finally, we give a practical example to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods.     

Possibilistic support vector machines

We propose new support vector machines (SVMs) that incorporate the geometric distribution of an input data set by associating each data point with a possibilistic membership, which measures the relative strength of the self class membership. By using a possibilistic distance measure based on the possibilistic membership, we reformulate conventional SVMs in three ways. The proposed methods are shown to have better classification performance than conventional SVMs in various tests.     

模糊集之间的距离测度

介绍了Vague集距离测度公理化的定义,引入了一些新的距离测度并给出了这些距离测度性质的证明。讨论了相似测度与距离测度之间的关系。最后,指出Vague集距离测度在模式识别中的应用。     

New hesitation-based distance and similarity measures on intuitionistic fuzzy sets and their applications

In this paper, we present new definitions on distance and similarity measures between intuitionistic fuzzy sets (IFSs) by combining with hesitation degree. First, we discuss the limitations in traditional distance and similarity measures, which are caused by the neglect of hesitation degree's influence. Even though a vector-valued similarity measure was proposed, which has two components indicating similarity and hesitation aspects, it still cannot perform well in practical applications because hesitation works only when the values of similarity measures are equal. In order to overcome the limitations, we propose new definitions on hesitation, distance and similarity measures, and research some theorems which satisfy the requirements of the proposed definitions. Meanwhile, we investigate the relationships among hesitation, distance, similarity and entropy of IFSs to verify the consistency of our work and previous research. Finally, we analyse and discuss the advantages and disadvantages of the proposed similarity measure in detail, and then we apply the proposed measures (dH and SH) to deal with pattern recognition problems, and demonstrate that they outperform state-of-the-art distance and similarity measures.     

Value集的模糊嫡、相似度量和距离测度的关系

Vague集理论在各个领域中的广泛应用引起越来越多学者的注意,而模糊墒、相似度量和距离测度是其中的3个关键技术。目前已提出多种关于Vague集的模糊嫡、相似度量和距离测度的计算方法,但这些研究都没有讨论这3个基本概念之间的联系。基于Vague集的模糊嫡、相似度量和距离测度的公理化定义,给出了三者之间的相互诱导关系,建立了模糊墒、相似度量和距离测度之间的联系。     

基于可能性测度的计算树逻辑CTL*与可能性互模拟

提出了基于可能性测度的计算树逻辑CTL*(PoCTL*)的概念。给出了在可能的Kripke结构中可能性互模拟的定义并对其性质进行了详细的探讨。对商可能性Kripke结构及其相关构造进行了特别的研究。     

Inclusion measures of probabilistic linguistic term sets and their application in classifying cities in the Economic Zone of Chengdu Plain

The probabilistic linguistic term set is a powerful tool to express and characterize people’s cognitive complex information and thus has obtained a great development in the last several years. To better use the probabilistic linguistic term sets in decision making, information measures such as the distance measure, similarity measure, entropy measure and correlation measure should be defined. However, as an important kind of information measure, the inclusion measure has not been defined by scholars. This study aims to propose the inclusion measure for probabilistic linguistic term sets. Formulas to calculate the inclusion degrees are put forward Then, we introduce the normalized axiomatic definitions of the distance, similarity and entropy measures of probabilistic linguistic term sets to construct a unified framework of information measures for probabilistic linguistic term sets. Based on these definitions, we present the relationships and transformation functions among the distance, similarity, entropy and inclusion measures. We believe that more formulas to calculate the distance, similarity, inclusion degree and entropy can be induced based on these transformation functions. Finally, we put forward an orthogonal clustering algorithm based on the inclusion measure and use it in classifying cities in the Economic Zone of Chengdu Plain, China.     

Componentwise decomposition of some lattice-valued fuzzy integrals

Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set, the top element of the lattice to the entire universe and satisfy the property of monotonicity. If the lattice is complete then a lattice-valued fuzzy integral of Sugeno type, with similar properties such as the Sugeno integral in its original form, can be introduced in a natural way. The main result of the paper is a componentwise decomposition theorem of an L-valued fuzzy integral to its L-valued fuzzy integrals components, where L is a complete lattice with negation and is organized as a complete lattice too. This result is useful to obtain the properties of L-valued fuzzy integrals from the properties of L-valued fuzzy integrals and to calculate in a simple way the values of some integrals from the values of the components. The important case L = [0, 1], when L becomes the lattice of the intuitionistic fuzzy values is distinctly discussed. An idea of application to synthetic evaluation of objects is also suggested.     

计算Vague集贴近度的一种新方法

指出了Chen给出的Vague集的贴近度存在的缺陷,并分析了李凡[1]提出的贴近度。在此基础上,引入了Vague集(值)的距离概念,进而引入并研究了Vague集(值)的贴近度,并证明了这一新的贴近度的性质。     

Divergence and fuzziness measures

In an axiomatic way a divergence between fuzzy sets is introduced which extends the symmetric difference between crisp sets. Any fuzzy measure of the divergence between two fuzzy sets weighs their “distance”. The distance between a fuzzy set and the family of crisp sets is fuzziness measure.     

Vague集（值）间的距离测度

1 引言自从Zadeh提出fuzzy集理论以来,fuzzy集理论就在不断地发展和完善,并在许多领域里得到了成功的应用。但在实际应用中我们也发现它存在一些问题,例如,fuzzy集的隶属度是一个单值,该单值既包含了支持u∈U(U为论域)的证据,也包含了反对u∈U的证据,它不可能表示其中的一个,更不可能同时表示支持和反对的证据。为此,Gau等在文中提出了vague集的概念,用一个真隶属函数t_A和一个假隶属函数f_A来描述vague集,真假隶属函数形成了隶属函     

A review of: “CYBERNETICS: THEORY AND APPLICATIONS”, by Robert Trappl,Hemisphere, Washington,D.C., 1983. $80, xii + 455 pages.

The greatest portion of papers dealing with the Dempster-Shafer theory consider the case when the basic universe is a finite set, so that all the numerical characteristics introduced and investigated in the D-S theory, including the believeability and plausibility functions as the most important ones, can be easily defined by well-known combinatoric formulas outgoing from a simple probability distribution (basic belief assignment, in the terms of D-S theory) on the power-set 𝒫(S) of all subsets of S. The obvious fact that these numerical characteristics can be equivalently defined also by appropriate set-valued random variables becomes to be of greater importance in the case when S is infinite. We investigate, in this paper, the case when the power-set 𝒫(S) over an infinite set S is equipped by a nonempty σ-field  ? 𝒫(𝒫(S)) andwhen the belief and plausibility functions are defined by a set-valued random variable (i.e., -measurable mapping) which takes a given probability space into the measurable space (𝒫(S),) In general, the values of the two functions in question need not be defined for each subset T of S. Therefore, we define four extensions of these functions to whole the (S) based on the well-known concepts of inner and outer measure, and investigate their properties; interesting enough, just one of them respect the philosophy of the D-S approach to uncertainty quantification and processing and keeps the properties possessed by believeability and plausibility functions defined over finite spaces.     

Learning indistinguishability from data

 In this paper we revisit the idea of interpreting fuzzy sets as representations of vague values. In this context a fuzzy set is induced by a crisp value and the membership degree of an element is understood as the similarity degree between this element and the crisp value that determines the fuzzy set. Similarity is assumed to be a notion of distance. This means that fuzzy sets are induced by crisp values and an appropriate distance function. This distance function can be described in terms of scaling the ordinary distance between real numbers. With this interpretation in mind, the task of designing a fuzzy system corresponds to determining suitable crisp values and appropriate scaling functions for the distance. When we want to generate a fuzzy model from data, the parameters have to be fitted to the data. This leads to an optimisation problem that is very similar to the optimisation task to be solved in objective function based clustering. We borrow ideas from the alternating optimisation schemes applied in fuzzy clustering in order to develop a new technique to determine our set of parameters from data, supporting the interpretability of the fuzzy system.