Possibility theory and statistical reasoning

Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished from uncertainty due to incomplete information. This paper proposes an overview of numerical possibility theory. Its aim is to show that some notions in statistics are naturally interpreted in the language of this theory. First, probabilistic inequalites (like Chebychev's) offer a natural setting for devising possibility distributions from poor probabilistic information. Moreover, likelihood functions obey the laws of possibility theory when no prior probability is available. Possibility distributions also generalize the notion of confidence or prediction intervals, shedding some light on the role of the mode of asymmetric probability densities in the derivation of maximally informative interval substitutes of probabilistic information. Finally, the simulation of fuzzy sets comes down to selecting a probabilistic representation of a possibility distribution, which coincides with the Shapley value of the corresponding consonant capacity. This selection process is in agreement with Laplace indifference principle and is closely connected with the mean interval of a fuzzy interval. It sheds light on the “defuzzification” process in fuzzy set theory and provides a natural definition of a subjective possibility distribution that sticks to the Bayesian framework of exchangeable bets. Potential applications to risk assessment are pointed out.     

FUZZY MEASURES AND FUZZY INTEGRALS: AN OVERVIEW

This paper provides an overview of fuzzy measures, fuzzy integration theories and Choquet's capacity theory. Belief, plausibility, and possibility measures are characterized as Choquet capacities and as fuzzy measures. The relationship between possibility measures, fuzzy sets, and approximate reasoning is established. Recent results on extensions of fuzzy measures, structural characteristics of fuzzy measures, and convergence of function sequences on fuzzy measure spaces are presented. Fuzzy measure integration concepts due to Sugeno and Choquet and their applications are discussed. An extensive list of references to the literature of fuzzy measures, Sugeno and Choquet integrals, fuzzy probabilities, fuzzy random variables, probabilistic sets, and random sets is provided. Applicalions discussed or referenced include information fusion, information retrieval, approximate reasoning, artificial intelligence, uncertainty theory, and control and decision theory.     

BAYESIAN INFERENCE OF FUZZY PROBABILITIES

Abstract

In this paper, we propose a new method to generalize Bayesian methods both for fuzzy likelihood and fuzzy prior probabilities. Based on interval Bayesian inference developed by Pan and Klir [1996], the proposed method overcomes the difficulty in developing a normalized fuzzy Bayesian inference recognized in the literature [Friihwirth-Schnatter, 1993].     

Extension of Petri Nets for Representing and Reasoning with Tasks with Imprecise Durations

This paper presents an extension of Petri net framework with imprecise temporal properties. We use possibility theory to represent imprecise time by time-stamping tokens and assigning durations to firing of the transitions. A method for approximation of an arbitrary temporal distribution with a set of possibilistic intervals is used to introduce the composition operation for two possibilistic temporal distributions. We developed a method to determining an effective enabling time of a transition with incoming tokens with possibilistic distributions. The utility of the proposed theory is illustrated using an example of an automated manufacturing system. The proposed approach is novel and has a broad utility beyond a timed Petri network and its applications.     

FUZZY SETS IN COMMUNICATION SYSTEMS AND IN CONSENSUS FORMATION SYSTEMS

Abstract

In this paper we shall examine some applications of fuzzy set theory in the area of communications and information systems. With the advent of computers, the telephone and the television, there are potentialities of all kinds of communication services into the home. Fuzzy problems are very likely to occur with user groups requesting methods to understand fuzzy needs.

Thus problems in fuzzy communication, fuzzy problem-solving, fuzzy information retrieval, and fuzzy teleconferencing have direct applications of fuzzy set theory, which are considered.     

FUZZINESS VS. PROBABILITY

Fuzziness is explored as an alternative to randomness for describing uncertainty. The new sets-as-points geometric view of fuzzy sets is developed. This view identifies a fuzzy set with a point in a unit hypercube and a nonfuzzy set with a vertex of the cube. Paradoxes of two-valued logic and set theory, such as Russell's paradox, correspond to the midpoint of the fuzzy cube. The fundamental questions of fuzzy theory—How fuzzy is a fuzzy set? How much is one fuzzy set a subset of another?—are answered geometrically with the Fuzzy Entropy Theorem, the Fuzzy Subsethood Theorem, and the Entropy-Subsethood Theorem. A new geometric proof of the Subsethood Theorem is given, a corollary of which is that the apparently probabilistic relative frequency n_A /N turns out to be the deterministic subsethood S(X, A), the degree to which the sample space X is contained in its subset A. So the frequency of successful trials is viewed as the degree to which all trials are successful. Recent Bayesian polemics against fuzzy theory are examined in light of the new sets-as-points theorems.     

一种新的模糊时间序列模型的预测方法

针对目前在模糊时间序列模型中论域划分及数据模糊化方法存在的问题,首先提出了基于模糊聚类算法(FCM)的具有可调参数的模糊时间序列论域的非等分划分方法;然后,在数据模糊化时通过距离客观地定义了模糊集,并利用最小标准误差(RMSE)确定最优的预测结果和聚类数;最后,通过 Alabama 大学注册人数的预测表明了所提出算法的有效性.     

Representing time intervals in a two-dimensional space: An empirical study

ObjectiveInstead of the linear model (LM), time intervals can be represented by a two-dimensional (2D) model, which is called the triangular model (TM). Although the TM has been introduced for decades and applied in some areas, there still a lack of empirical studies on its usability. To fill this gap, this study aims to evaluate how people perform when using the TM to answer questions on time intervals, in comparison with using the traditional LM.MethodAround 250 novice participants took part in the experiment, which consisted of a video training, a pretest and posttest. The video training introduced the basic knowledge of temporal relations and the two representations. The pretest allowed participants to practice the knowledge they have learned and receive feedbacks of the answers. In the posttest, participants' accuracy and speed when answering the questions were recorded for analysis. The results of using the TM and the LM were compared in pairs. The null hypothesis is that the participants produce equal results with the two models.ResultThe results showed that the participants scored better and spent less time when answering questions with the TM, which rejected the null hypothesis. Moreover, the score and speed when they used the TM did decline in the questions containing a larger number of intervals. In contrast, the score and accuracy when they used the LM declined when questions containing a large number of intervals.Conclusion

• •The TM is easy to learn. After a 20-min training, novice participants can use it to solve questions and produce satisfactory result.
• •The TM is easy and efficient for visual queries of time intervals.
• •The TM is easy to use for handling a large number of intervals.

Implication

• •The TM can be widely applied in analysing time intervals and linear data.
• •Tools implementing the TM can be learned and used by novice users.

     

Theory of Fuzzy Generalized Morphisms and Relational Inequalities

We present here the theory of generalized morphisms and fuzzy relational inequalities and discuss some of these applications. Results presented in this paper extend the work of Bandler and Kohout (1986) ["On the general theory of relational morphisms", International Journal of General Systems, 13, pp. 47-66] published in this journal previously, to relational systems based on residuated t-norm fuzzy logics.     

Fuzzy set theory applications in production management research: a literature survey

Fuzzy set theory has been used to model systems that are hard to define precisely. As a methodology, fuzzy set theory incorporates imprecision and subjectivity into the model formulation and solution process. Fuzzy set theory represents an attractive tool to aid research in production management when the dynamics of the production environment limit the specification of model objectives, constraints and the precise measurement of model parameters. This paper provides a survey of the application of fuzzy set theory in production management research. The literature review that we compiled consists of 73 journal articles and nine books. A classification scheme for fuzzy applications in production management research is defined. We also identify selected bibliographies on fuzzy sets and applications.     

FRACTAL GEOMETRY: A TOOL FOR FUZZY REASONING

Abstract

A new procedure for construction of a fuzzy number is given, based on the covering of points by consequently contracting intervals. The approach leads to the membership function in the form of Devil's staircase, a borderline fractal finding many applications in the modern physics. The alternative definition of subsethood based on the self-similarity of the sets is also given.     

Novel intuitionistic fuzzy soft multiple-attribute decision-making methods

An intuitionistic fuzzy soft set plays a significant role as a mathematical tool for mathematical modeling, system analysis and decision making. This mathematical tool gives more precision, flexibility and compatibility to the system when compared to systems that are designed using fuzzy graphs and fuzzy soft graphs. In this paper, we use intuitionistic fuzzy soft graphs and possibility intuitionistic fuzzy soft graphs for parameterized representation of a system involving some uncertainty. We present novel multiple-attribute decision-making methods based on an intuitionistic fuzzy soft graph and possibility intuitionistic fuzzy soft graph. We also present our methods as algorithms that are used in our applications.

     

基于模糊测度和证据理论的模糊聚类集成方法

针对现有集成方法在处理模糊聚类时存在的不足,提出一种基于证据理论的模糊聚类集成方法.以各聚类成员作为证据元,以样本点间的类别关系作为焦元,通过证据积累构造互相关矩阵.考虑到模糊聚类对于各样本点的聚类有效性,提出一种结合点模糊度和模糊贴近度的类别关系表示方法,并以此作为各证据元的基本概率赋值函数.最后基于互相关矩阵构造样本点间相似性关系,并利用谱聚类算法对其聚类. 实验中通过与多种已有聚类集成方法的对比表明,该方法具有较高的聚类性能.     

模糊控制理论的发展应用与展望

介绍了模糊控制理论的发展概况及在许多领域应用进化过程。在此基础上,论述了模糊控制理论应用的一般特点。指出了模糊控制理论应用中需要解决的问题,单纯模糊控制在某些情况下,难以适应不同的运行状态,影响了控制效果。实际应用中需要将模糊控制或模糊推理思想与其他成熟的控制理论与方法结合起来应用,以达到更理想的控制效果。展望了模糊控制理论的发展方向。由于模糊控制技术是当今比较先进的控制技术,因此,它有着广阔的应用前景。     

Rough set theory for the interval-valued fuzzy information systems

The notion of a rough set was originally proposed by Pawlak [Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (5) (1982) 341-356]. Later on, Dubois and Prade [D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System 17 (2-3) (1990) 191-209] introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets. This paper deals with an interval-valued fuzzy information system by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory and discusses the basic rough set theory for the interval-valued fuzzy information systems. In this paper we firstly define the rough approximation of an interval-valued fuzzy set on the universe U in the classical Pawlak approximation space and the generalized approximation space respectively, i.e., the space on which the interval-valued rough fuzzy set model is built. Secondly several interesting properties of the approximation operators are examined, and the interrelationships of the interval-valued rough fuzzy set models in the classical Pawlak approximation space and the generalized approximation space are investigated. Thirdly we discuss the attribute reduction of the interval-valued fuzzy information systems. Finally, the methods of the knowledge discovery for the interval-valued fuzzy information systems are presented with an example.     

Protecting data: a fuzzy approach

Privacy issues represent a longstanding problem nowadays. Measures such as k-anonymity, l-diversity and t-closeness are among the most used ways to protect released data. This work proposes to extend these three measures when the data are protected using fuzzy sets instead of intervals or representative elements. The proposed approach is then tested using Energy Information Authority data set and different fuzzy partition methods. Results shows an improvement in protecting data when data are encoded using fuzzy sets.     

On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods

In this paper, a kind of novel soft set model called a Z-soft fuzzy rough set is presented by means of three uncertain models: soft sets, rough sets and fuzzy sets, which is an important generalization of Z-soft rough fuzzy sets. As a novel Z-soft fuzzy rough set, its applications in the corresponding decision making problems are established. It is noteworthy that the underlying concepts keep the features of classical Pawlak rough sets. Moreover, this novel approach will involve fewer calculations when one applies this theory to algebraic structures. In particular, an approach for the method of decision making problem with respect to Z-soft fuzzy rough sets is proposed and the validity of the decision making methods is testified by a given example. At the same time, an overview of techniques based on some types of soft set models is investigated. Finally, the numerical experimentation algorithm is developed, in which the comparisons among three types of hybrid soft set models are analyzed.     

Modelling and optimization of grinding processes

The paper describes different methods for modelling and optimization of grinding processes. First the process and product quality characterizing quantities have to be measured. Afterwards different model types, e.g. physical–empirical basic grinding models as well as empirical process models based on neural networks, fuzzy set theory and standard multiple regression methods, are discussed for an off-line process conceptualization and optimization using a genetic algorithm. The assessment of grinding process results, which build the individuals in the genetic algorithm's population, is carried out using a target tree method. The methods presented are integrated into an existing grinding information system, which is part of a three control loop system for quality assurance.     

基于MIC架构稳健投资组合并行算法及实现

投资组合选择是数量化投资管理领域中的一项关键技术,目前其在应用中亟需高性能算法与实现研究。本论文针对现实投资场景下的稳健投资组合选择最优化模型,设计出高效的并行算法,利用并行计算技术多层级优化性能,实现对稳健投资组合计算的快速响应。该稳健投资组合将模糊集理论与投资组合理论相结合,建立基于可能性理论和机会测度的投资组合模型,用BP神经网络算法和遗传算法对模型进行求解,并在最新的高性能计算集成众核(Many Integrated Core,MIC)架构上实现并行。文章选取上证50指数成份股近两年的交易数据,对并行算法及其性能进行分析。结果显示,该算法计算得到的投资组合收益率优于经典模型收益率和上证50指数同期收益率,基于MIC架构的并行求解性能优于传统的CPU架构,平均并行效率达到80%。     

Medical diagnosis of cardiovascular diseases using an interval-valued fuzzy rule-based classification system

ObjectiveTo develop a classifier that tackles the problem of determining the risk of a patient of suffering from a cardiovascular disease within the next 10 years. The system has to provide both a diagnosis and an interpretable model explaining the decision. In this way, doctors are able to analyse the usefulness of the information given by the system.MethodsLinguistic fuzzy rule-based classification systems are used, since they provide a good classification rate and a highly interpretable model. More specifically, a new methodology to combine fuzzy rule-based classification systems with interval-valued fuzzy sets is proposed, which is composed of three steps: (1) the modelling of the linguistic labels of the classifier using interval-valued fuzzy sets; (2) the use of the K_α operator in the inference process and (3) the application of a genetic tuning to find the best ignorance degree that each interval-valued fuzzy set represents as well as the best value for the parameter α of the K_α operator in each rule.ResultsThe suitability of the new proposal to deal with this medical diagnosis classification problem is shown by comparing its performance with respect to the one provided by two classical fuzzy classifiers and a previous interval-valued fuzzy rule-based classification system. The performance of the new method is statistically better than the ones obtained with the methods considered in the comparison. The new proposal enhances both the total number of correctly diagnosed patients, around 3% with respect the classical fuzzy classifiers and around 1% vs. the previous interval-valued fuzzy classifier, and the classifier ability to correctly differentiate patients of the different risk categories.ConclusionThe proposed methodology is a suitable tool to face the medical diagnosis of cardiovascular diseases, since it obtains a good classification rate and it also provides an interpretable model that can be easily understood by the doctors.