Knowledge structure,knowledge granulation and knowledge distance in a knowledge base

One of the strengths of rough set theory is the fact that an unknown target concept can be approximately characterized by existing knowledge structures in a knowledge base. Knowledge structures in knowledge bases have two categories: complete and incomplete. In this paper, through uniformly expressing these two kinds of knowledge structures, we first address four operators on a knowledge base, which are adequate for generating new knowledge structures through using known knowledge structures. Then, an axiom definition of knowledge granulation in knowledge bases is presented, under which some existing knowledge granulations become its special forms. Finally, we introduce the concept of a knowledge distance for calculating the difference between two knowledge structures in the same knowledge base. Noting that the knowledge distance satisfies the three properties of a distance space on all knowledge structures induced by a given universe. These results will be very helpful for knowledge discovery from knowledge bases and significant for establishing a framework of granular computing in knowledge bases.     

Information granulation and uncertainty measures in interval-valued intuitionistic fuzzy information systems

Information granulation and entropy are main approaches for investigating the uncertainty of information systems, which have been widely employed in many practical domains. In this paper, information granulation and uncertainty measures for interval-valued intuitionistic fuzzy binary granular structures are addressed. First, we propose the representation of interval-valued intuitionistic fuzzy information granules and examine some operations of interval-valued intuitionistic fuzzy granular structures. Second, the interval-valued intuitionistic fuzzy information granularity is introduced to depict the distinguishment ability of an interval-valued intuitionistic fuzzy granular structure (IIFGS), which is a natural extension of fuzzy information granularity. Third, we discuss how to scale the uncertainty of an IIFGS using the extended information entropy and the uncertainty among interval-valued intuitionistic fuzzy granular structures using the expanded mutual information derived from the presented intuitionistic fuzzy information entropy. Fourth, we discovery the relationship between the developed interval-valued intuitionistic fuzzy information entropy and the intuitionistic fuzzy information granularity presented in this paper.     

广义直觉模糊信息粒度

直觉模糊信息粒度是度量直觉模糊粒结构不确定性的一种重要方法。本文针对直觉模糊信息粒度在度量直觉模糊粒结构上存在的不足进行分析。首先,介绍直觉模糊粒结构距离。其次,改进了基于偏序关系提出的直觉模糊信息粒度的公理化定义,从直觉模糊粒结构距离观点出发,以最细的直觉模糊粒结构为参照物,计算每个直觉模糊粒结构与最细的直觉模糊粒结构之间的距离,距离的值越大,这个直觉模糊粒结构的信息粒度越大。最后,通过例子验证结论的合理性。     

一种基于Rough理论的知识推理冲突消解策略

知识推理中的冲突消解,不仅受到知识粒度的影响,还与所采用的具体方法有关.首先讨论了应用Rough理论对规则的粒度进行客观刻画,在此基础上采用距离概念构造了刻画规则相近性的规则距离公式,并在公式中引入了Rough理论中属性重要性算子,然后在分析传统冲突消解策略的基础上,提出了知识推理过程中规则集进化的概念,最后通过实例说明了此策略在知识推理中的具体应用.     

Allocation of information granularity in optimization and decision-making models: Towards building the foundations of Granular Computing

The highly diversified conceptual and algorithmic landscape of Granular Computing calls for the formation of sound fundamentals of the discipline, which cut across the diversity of formal frameworks (fuzzy sets, sets, rough sets) in which information granules are formed and processed. The study addresses this quest by introducing an idea of granular models – generalizations of numeric models that are formed as a result of an optimal allocation (distribution) of information granularity. Information granularity is regarded as a crucial design asset, which helps establish a better rapport of the resulting granular model with the system under modeling. A suite of modeling situations is elaborated on; they offer convincing examples behind the emergence of granular models. Pertinent problems showing how information granularity is distributed throughout the parameters of numeric functions (and resulting in granular mappings) are formulated as optimization tasks. A set of associated information granularity distribution protocols is discussed. We also provide a number of illustrative examples.     

基于覆盖上近似算子的拟阵结构及其关系

Covering-based rough sets,as a technique of granular computing,can be a useful tool for dealing with inexact,uncertain or vague knowledge in information systems.Matroids generalize linear independence in vector spaces,graph theory and provide well established platforms for greedy algorithm design.In this paper,we construct three types of matroidal structures of covering-based rough sets.Moreover,through these three types of matroids,we study the relationships among these matroids induced by six types of covering-based upper approximation operators.First,we construct three families of sets by indiscernible neighborhoods,neighborhoods and close friends,respectively.Moreover,we prove that they satisfy independent set axioms of matroids.In this way,three types of matroidal structures of covering-based rough sets are constructed.Secondly,we study some characteristics of the three types of matroid,such as dependent sets,circuits,rank function and closure.Finally,by comparing independent sets,we study relationships among these matroids induced by six types of covering-based upper approximation operators.     

Granular fuzzy models: a study in knowledge management in fuzzy modeling

In system modeling, knowledge management comes vividly into the picture when dealing with a collection of individual models. These models being considered as sources of knowledge, are engaged in some collective pursuits of a collaborative development to establish modeling outcomes of global character. The result comes in the form of a so-called granular fuzzy model, which directly reflects upon and quantifies the diversity of the available sources of knowledge (local models) involved in knowledge management. In this study, several detailed algorithmic schemes are presented along with related computational aspects associated with Granular Computing. It is also shown how the construction of information granules completed through the use of the principle of justifiable granularity becomes advantageous in the realization of granular fuzzy models and a quantification of the quality (specificity) of the results of modeling. We focus on the design of granular fuzzy models considering that the locally available models are those fuzzy rule-based. It is shown that the model quantified in terms of two conflicting criteria, that is (a) a coverage criterion expressing to which extent the resulting information granules “cover” include data and (b) specificity criterion articulating how detailed (specific) the obtained information granules are. The overall quality of the granular model is also assessed by determining an area under curve (AUC) where the curve is formed in the coverage-specificity coordinates. Numeric results are discussed with intent of displaying the most essential features of the proposed methodology and algorithmic developments.     

Axiomatic characterizations of dual concept lattices

Formal concept analysis is an algebraic model based on a Galois connection. It is used for symbolic knowledge exploration from an elementary form of a formal context. This paper mainly presents a general framework for concept lattice in which axiomatic approaches are used. The relationship between concept lattice and dual concept lattice is first studied. Based on set-theoretic operators, generalized concept systems are established. And properties of them are examined. By using axiomatic approaches, a pair of dual concept lattices is characterized by different sets of axioms. The connections between 0-1 binary relations and generalized concept systems are examined. And generalized dual concept systems can be constructed by a pair of dual set-theoretic operators. Axiomatic characterizations of the generalized concept systems guarantee the existence of a binary relation producing a formal context.     

Algorithm engineering for optimal alignment of protein structure distance matrices

Protein structural alignment is an important problem in computational biology. In this paper, we present first successes on provably optimal pairwise alignment of protein inter-residue distance matrices, using the popular dali scoring function. We introduce the structural alignment problem formally, which enables us to express a variety of scoring functions used in previous work as special cases in a unified framework. Further, we propose the first mathematical model for computing optimal structural alignments based on dense inter-residue distance matrices. We therefore reformulate the problem as a special graph problem and give a tight integer linear programming model. We then present algorithm engineering techniques to handle the huge integer linear programs of real-life distance matrix alignment problems. Applying these techniques, we can compute provably optimal dali alignments for the very first time.     

Numerical analysis of differential operators on raw point clouds

3D acquisition devices acquire object surfaces with growing accuracy by obtaining 3D point samples of the surface. This sampling depends on the geometry of the device and of the scanned object and is therefore very irregular. Many numerical schemes have been proposed for applying PDEs to regularly meshed 3D data. Nevertheless, for high precision applications it remains necessary to compute differential operators on raw point clouds prior to any meshing. Indeed differential operators such as the mean curvature or the principal curvatures provide crucial information for the orientation and meshing process itself. This paper reviews a half dozen local schemes which have been proposed to compute discrete curvature-like shape indicators on raw point clouds. All of them will be analyzed mathematically in a unified framework by computing their asymptotic form when the size of the neighborhood tends to zero. They are given in terms of the principal curvatures or of higher order intrinsic differential operators which, in return, characterize the discrete operators. All considered local schemes are of two kinds: either they perform a polynomial local regression, or they compute directly local moments. But the polynomial regression of order 1 is demonstrated to play a special role, because its iterations yield a scale space. This analysis is completed with numerical experiments comparing the accuracies of these schemes. We demonstrate that this accuracy is enhanced for all operators by applying previously the scale space.     

Simulation for lattice-valued doubly labeled transition systems

During the last decades, a large amount of multi-valued transition systems, whose transitions or states are labeled with specific weights, have been proposed to analyze quantitative behaviors of reactive systems. To set up a unified framework to model and analyze systems with quantitative information, in this paper, we present an extension of doubly labeled transition systems in the framework of residuated lattices, which we will refer to as lattice-valued doubly labeled transition systems (LDLTSs). Our model can be specialized to fuzzy automata over complete residuated lattices, fuzzy transition systems, and multi-valued Kripke structures. In contrast to the traditional yes/no approach to similarity, we then introduce lattice-valued similarity between LDLTSs to measure the degree of closeness of two systems, which is a value from a residuated lattice. Further, we explore the properties of robustness and compositionality of the lattice-valued similarity. Finally, we extend the Hennessy–Milner logic to the residuate lattice-valued setting and show that the obtained logic is adequate and expressive with lattice-valued similarity.     

Granular fuzzy models: Analysis,design, and evaluation

The study is concerned with a design of granular fuzzy models. We exploit a concept of information granularity by developing a model coming as a network of intuitively structured collection of interval information granules described in the output space and a family of induced information granules (in the form of fuzzy sets) formed in the input space. In contrast to most fuzzy models encountered in the literature, the results produced by granular models are information granules rather than plain numeric entities. The design of the model concentrates on a construction of information granules that form a backbone of the overall construct. Interval information granules positioned in the output space are built by considering intervals of equal length, equal probability, and developing an optimized version of the intervals. The induced fuzzy information granules localized in the input space are realized by running a conditional Fuzzy C-Means (FCM). The performance of the model is assessed by considering criteria of coverage and information specificity (information granularity). Further optimization of the model is proposed along the line of an optimal re-distribution of input information granules induced by the individual interval information granules located in the output space. Experimental results involve some synthetic low-dimensional data and publicly available benchmark data sets.     

基于泛系粗糙集等价算子上的粒计算模型

标准粗糙集使用等价类作为粒来描述概念.本文弱化对等价关系的要求, 将更广泛的粒计算模型建立到泛系粗糙集上去.本文通过对全域的分割和覆盖来诱导出泛系粗糙集上的粒计算模型.     

单调概念格性质研究

概念格是知识表示和数据分析的重要工具,单调概念格是概念格的推广。本文就Deogun等提出的单调概念格进行了两方面的研究:一,指出Deogun等提出的单调概念格性质的错误并加以修正;二,证明单调概念格就是闭格,从而找到用拓扑闭包算子和拓扑交结构来表示单调概念格的两种格表示方法,并建立起单调概念格与有上界的拓扑交结构的范畴等价。本文所建立的单调概念格的拓扑表示方法将方便我们进一步研究单调概念格的构造算法、约简算法和实际应用,具有理论和实际的双重意义。     

LTV度量反馈控制问题的最优控制器

In this paper,we consider the measurement feedback control problem for discrete linear time-varying systems within the framework of nest algebra consisting of causal and bounded linear operators.Based on the inner-outer factorization of operators,we reduce the control problem to a distance from a certain operator to a special subspace of a nest algebra and show the existence of the optimal LTV controller in two different ways：one via the characteristic of the subspace in question directly,the other via the duality theory.The latter also gives a new formula for computing the optimal cost.     

A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts

Group decision making is a type of decision problem in which multiple experts acting collectively, analyze problems, evaluate alternatives, and select a solution from a collection of alternatives. As the natural language is the standard representation of those concepts that humans use for communication, it seems natural that they use words (linguistic terms) instead of numerical values to provide their opinions. However, while linguistic information is readily available, it is not operational and thus it has to be made usable though expressing it in terms of information granules. To do so, Granular Computing, which has emerged as a unified and coherent framework of designing, processing, and interpretation of information granules, can be used. The aim of this paper is to present an information granulation of the linguistic information used in group decision making problems defined in heterogeneous contexts, i.e., where the experts have associated importance degrees reflecting their ability to handle the problem. The granulation of the linguistic terms is formulated as an optimization problem, solved by using the particle swarm optimization, in which a performance index is maximized by a suitable mapping of the linguistic terms on information granules formalized as sets. This performance index is expressed as a weighted aggregation of the individual consistency achieved by each expert.     

THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS

1 Introduction Singular integral equations (SIEs) with Cauchy kernels Of the formoften arise in mathematical models of physical phenomena. Since closed-form solutions to SIEsare generally not available, much att.ntion has been focused on numerical methods of solution.In the past twenty years, various collocation methods for SIEs have been the topic of a greatmany of papers, most of which can be found in two surveys[213]. The early works in the fieldis to study tile numerical solutions for…