基于模糊格的超盒粒计算分类器

粒的表示、粒之间的关系和运算是粒计算的主要研究内容。利用向量表示超盒粒, 分析向量之间的偏序关 系和超盒粒之间的偏序关系的不一致性, 并引入保序函数消除该不一致性。利用格和其对偶格之间的非线性正评价函数和保序函数构造超盒粒之间模糊包含关系。为得到不同粒度的粒, 设计超盒粒之间的合并算子和分解算子, 证明由超盒粒集、超盒粒之间的模糊包含关系、合并算子、分解算子构成的代数系统是模糊格, 构造基于模糊格的超盒粒计算分类器。用机器学习数据集中的分类问题, 验证该分类器具有和模糊格推理分类器相同的推广能力并减少超盒粒的数量。     

FL-GrCCA: A granular computing classification algorithm based on fuzzy lattices

Defining a relation between granules and computing ever-changing granules are two important issues in granular computing. In view of this, this work proposes a partial order relation and lattice computing, respectively, for dealing with the aforementioned issues. A fuzzy lattice granular computing classification algorithm, or FL-GrCCA for short, is proposed here in the framework of fuzzy lattices. Algorithm FL-GrCCA computes a fuzzy inclusion relation between granules by using an inclusion measure function based on both a nonlinear positive valuation function, namely arctan, and an isomorphic mapping between lattices. Changeable classification granules are computed with a dilation operator using, conditionally, both the fuzzy inclusion relation between two granules and the size of a dilated granule. We compare the performance of FL-GrCCA with the performance of popular classification algorithms, including support vector machines (SVMs) and the fuzzy lattice reasoning (FLR) classifier, for a number of two-class problems and multi-class problems. Our computational experiments showed that FL-GrCCA can both speed up training and achieve comparable generalization performance.     

用模糊包含度构造超盒粒分类器

如何将训练集分割成大小不一的超盒粒是粒计算领域的关键问题之一。引入非线性正评价函数并用于构造超盒粒之间的模糊包含度函数,通过粒度阈值,对两个超盒粒有条件合并,构造含有大小不同超盒粒的分类器。实验结果表明超盒粒分类器与模糊格推理分类器相比提高了测试精度,与支持向量机相比加快了训练速度且提高了测试精度。     

模糊集、粗糙集和商空间理论的比较研究

针对模糊集、粗糙集及商空间理论,从对智能的理解、粒的表示、粒度的定义和粒的关系这4个方面进行比较。分析比较结果可知,它们的共同之处有：用集合定义粒,用粒描述知识;不同之处有：词计算和粗糙集理论分别从微观角度研究词的推理和属性的约简,而商空间理论是从宏观角度研究粒度的变化规律。     

二进制粒计算模型

粒计算是一种处理不确定性数据的理论方法,涵盖粗糙集、模糊集、商空间、词计算等。目前,数据的粒化与粒的计算主要涉及集合的运算与度量,集合运算的低效制约着粒计算相关算法的应用领域。为此,提出了一种二进制粒计算模型,给出了粒的三层结构,包括粒子、粒群与粒库,并定义了二进制粒子及二进制粒子的运算,将传统的集合运算转化为二进制数的计算,进一步给出了二进制粒子的距离度量,将等价类的集合表示方式转化为粒子的距离度量表示方式,给出了粒子距离的相关性质。该模型定义了二进制粒群距离的概念,给出了二进制粒群距离的计算方法,提出了基于二进制粒群距离的属性约简方法,证明了该方法与经典粗糙集约简方法的等价性,并以二进制粒群距离作为启发式信息,给出了两种约简算法。     

粒的特征及形式化表示研究

为了能为粒计算的粒提供一种统一的表示形式,研究了粒的特征的基本含义、描述及其关系,分析了粒的对象、特征、关系及状态等四个基本要素,提出了基于数据层面的粒的四元组的形式化表示形式,即由对象集、特征集、关系集和约束集四元组组成。接着给出了几个特殊粒的表示形式,然后说明了该表示形式的统一性及其优点,并结合具体的例子进行了研究。该形式化表示形式具有重要的方法论意义,很好地解决了基于数据层面的粒的表示问题,有利于问题求解和粒计算理论的研究。     

粒计算下的粗糙集模型比较*

提出了几种组合粒下的粗糙集模型,并将其与单一粒下的粗糙集进行了比较,同时与粒逻辑运算下的粗糙集模型进行了比对,创造性地得到了组合粒、单一粒以及粒逻辑运算下的粗糙集模型之间的关系。结果表明,组合粒与粒逻辑运算组成了一个链结构,这为研究基于信息粒的知识获取以及动态粒的推理奠定了基础。     

Foreword to the Special Section on Granular Computing

The papers in this special section focus on granular computing, defined by Lotfi A. Zadeh as "information granulation involves partitioning a class of objects (points) into granules, with a granule being a clump of objects (points) which are drawn together by indistinguishability, similarity, or functionality."     

The modeling of time series based on fuzzy information granules

A lot of research has resulted in many time series models with high precision forecasting realized at the numerical level. However, in the real world, higher numerical precision may not be necessary for the perception, reasoning and decision-making of human. Model of time series with an ability of humans to perceive and process abstract entities (rather than numeric entities) is more adaptable for some problems of decision-making. With this regard, information granules and granular computing play a primordial role. Fox example, if change range (intervals) of stock prices for a certain period in the future is regarded as information granule, constructing model that can forecast change ranges (intervals) of stock prices for a period in the future is better able to help stock investors make reasonable decisions in comparison with those based upon specific forecasting numerical value of stock price. In this paper, we propose a new modeling approach to realize interval prediction, in which the idea of information granules and granular computing is integrated with the classical Chen’s method. The proposed method is to segment an original numeric time series into a collection of time windows first, and then build fuzzy granules expressed as a certain fuzzy set over each time windows by exploiting the principle of justifiable granularity. Finally, fuzzy granular model can be constructed by mining fuzzy logical relationships of adjacent granules. The constructed model can carry out interval prediction by degranulation operation. Two benchmark time series are used to validate the feasibility and effectiveness of the proposed approach. The obtained results demonstrate the effectiveness of the approach. Besides, for modeling and prediction of large-scale time series, the proposed approach exhibit a clear advantage of reducing computation overhead of modeling and simplifying forecasting.     

粒及粒计算在逻辑推理中的应用

讨论了信息粒的结构及其实例。基于Rough集方法定义了决策规则粒,构造了决策规则粒库,它被用作逻辑推理。定义了粒语言,描述了这种语言的语法、语义、粒语句的运算法则和粒之相关的几个性质。定义了粒之间的相互包含(inclusion)和相似(closeness)。基于这些概念,构造了一种逻辑推理的新模型。这种推理模式的特点在于它既是逻辑的又是集合论的。所谓逻辑的就是说推理是遵循一种逻辑运算;所谓集合论的是指这种推理可利用对应于这种逻辑公式的意义集的运算进行推理,还用实例说明了这种推理模式是可行和有效的。     

Rough Mereological Calculi of Granules: A Rough Set Approach To Computation

Rough Mereology is a paradigm allowing for a synthesis of main ideas of two potent paradigms for reasoning under uncertainty: Fuzzy Set Theory and Rough Set Theory. Approximate reasoning is based in this paradigm on the predicate of being a part to a degree . We present applications of Rough Mereology to the important theoretical idea put forth by Lotfi Zadeh (1996, 1997), i.e., Granularity of Knowledge: We define granules of knowledge by means of the operator of mereological class and we extend the idea of a granule over complex objects like decision rules as well as decision algorithms. We apply these notions and methods in the distributed environment discussing complex problems of knowledge and granule fusion. We express the mechanism of complex granule formation by means of a formal grammar called Synthesis Grammar defined over granules of knowledge, granules of classifying rules, or over granules of classifying algorithms. We finally propose hybrid rough-neural schemes bridging rough and neural computations.     

二维近似空间上基于粒计算的数据识别

利用逻辑推理和粒计算相互融合的研究成果,通过扩展近似空间,在形成的二维近似空间上定义了逻辑公式和公式确定的粒。为数据识别的需要,特别构造了一类重要的二维近似空间——矩形近似空间。经对矩形近似空间上公式确定的粒进行粒计算,得到矩形近似空间上公式粗糙描述的概念,由此可有效地实现对公式中隐含信息的识别,以达到数据保密传送的目的。这种基于粒计算的数据识别不仅表明前期理论研究的重要性,也表明理论结果的应用价值。     

基于粒计算的Apriori算法及其在图书管理系统中的应用

粒计算作为一种新的信息和知识处理的方法近来已经被许多研究者所重视,以及在许多领域中的得到应用。本质上,粒计算能够表示存储在系统中的数据的语义信息,因此粒计算能作为用于探索数据性质的一种方法,如挖掘数据库中的关联规则。本文在分析经典Apriori算法的基础上,从信息粒的角度出发,提出基于粒计算生成k-频繁项目集算法。分析了对给定问题,当用粒计算模型求解时需要解决的几个基本问题。最后通过实例说明如何通过信息粒的二进制表示,并基于粒计算k-频繁项目集生成算法来获取隐藏在图书借还记录中的有关关联规则。可以看出该算法具有实际应用价值。     

Building the fundamentals of granular computing: A principle of justifiable granularity

The study introduces and discusses a principle of justifiable granularity, which supports a coherent way of designing information granules in presence of experimental evidence (either of numerical or granular character). The term “justifiable” pertains to the construction of the information granule, which is formed in such a way that it is (a) highly legitimate (justified) in light of the experimental evidence, and (b) specific enough meaning it comes with a well-articulated semantics (meaning). The design process associates with a well-defined optimization problem with the two requirements of experimental justification and specificity. A series of experiments is provided as well as a number of constructs carried for various formalisms of information granules (intervals, fuzzy sets, rough sets, and shadowed sets) are discussed as well.     

Granular computing based on fuzzy similarity relations

Rough sets and fuzzy rough sets serve as important approaches to granular computing, but the granular structure of fuzzy rough sets is not as clear as that of classical rough sets since lower and upper approximations in fuzzy rough sets are defined in terms of membership functions, while lower and upper approximations in classical rough sets are defined in terms of union of some basic granules. This limits further investigation of the existing fuzzy rough sets. To bring to light the innate granular structure of fuzzy rough sets, we develop a theory of granular computing based on fuzzy relations in this paper. We propose the concept of granular fuzzy sets based on fuzzy similarity relations, investigate the properties of the proposed granular fuzzy sets using constructive and axiomatic approaches, and study the relationship between granular fuzzy sets and fuzzy relations. We then use the granular fuzzy sets to describe the granular structures of lower and upper approximations of a fuzzy set within the framework of granular computing. Finally, we characterize the structure of attribute reduction in terms of granular fuzzy sets, and two examples are also employed to illustrate our idea in this paper.     

基于模糊粗糙集的粒度计算

粒度计算是一种新的智能计算的理论和方法,目前受到很多学者的关注。但是,具体可行的粒表示模型和不同粒的推理方法研究相对较少。本文将模糊粗糙集纳入粒度计算这种新的理论框架,对于处理复杂信息系统,求解复杂问题无疑具有重要的意义。首先利用笛卡尔积,构建了模糊关系下的信息粒;然后给出不同粒度下模糊粗糙算子的表示方法,进而形成一个分层递阶结构;最后考虑了对于模糊信息系统粒度粗细的选择问题,并给出一个实例,从而为粒度计算提供一个具体而实用的框架。     

A granular computing framework for self-organizing maps

When using granular computing for problem solving, one can focus on a specific level of understanding without looking at unwanted details of subsequent (more precise) levels. We present a granular computing framework for growing hierarchical self-organizing maps. This approach is ideal since the maps are arranged in a hierarchical manner and each is a complete abstraction of a pattern within data. The framework allows us to precisely define the connections between map levels. Formulating a neuron as a granule, the actions of granule construction and decomposition correspond to the growth and absorption of neurons in the previous model. In addition, we investigate the effects of updating granules with new information on both coarser and finer granules that have a derived relationship. Called bidirectional update propagation, the method ensures pattern consistency among data abstractions. An algorithm for the construction, decomposition, and updating of the granule-based self-organizing map is introduced. With examples, we demonstrate the effectiveness of this framework for abstracting patterns on many levels.     

Information systems in modeling interactive computations on granules

In this paper, we discuss the importance of information systems in modeling interactive computations performed on (complex) granules and we propose a formal approach to interactive computations based on generalized information systems and rough sets which can be combined with other soft computing paradigms such as fuzzy sets or evolutionary computing, but also with machine learning and data mining techniques. Information systems are treated as dynamic granules used for representing the results of the interaction of attributes with the environment. Two kinds of attributes are distinguished, namely, the perception attributes, including sensory attributes, and the action attributes. Sensory attributes are the basic perception attributes, other perception attributes are constructed on the basis of the sensory ones. Actions are activated when their guards, being often complex and vague concepts, are satisfied to a satisfactory degree. The guards can be approximated on the basis of measurements performed by sensory attributes rather than defined exactly. Satisfiability degrees for guards are results of reasoning called the adaptive judgment. The approximations are induced using hierarchical modeling. We show that information systems can be used for modeling more advanced forms of interactions in hierarchical modeling. The role of hierarchical interactions is emphasized in the modeling of interactive computations. Some illustrative examples of interactions used in the ACT-R 6.0 system are reported. ACT-R 6.0 is based on a cognitive architecture and can be treated as an example of a highly interactive complex granule which can be involved in hierarchical interactions. For modeling of interactive computations, we propose much more general information systems than the studied dynamic information systems (see, e.g., Ciucci (2010) [8] and Pa?asiński and Pancerz (2010) [32]). For example, the dynamic information systems are making it possible to consider incremental changes in information systems. However, they do not contain the perception and action attributes necessary for modeling interactive computations, in particular for modeling intrastep interactions.