Optimal design of a fuzzy rule base

This paper considers the issues of forming a fuzzy database and ensuring its fullness and consistency. In addition, a certain method of designing the database structure using an hierarchical system of linguistic rules is suggested.     

基于子集融合与规则简约的磨矿过程模糊建模

针对选矿厂磨矿生产过程的模糊建模问题,本文提出一种基于模糊集融合和规则简约的模糊建模方法.该方法针对基于数据建立的磨矿过程Takagi-Sugeno模型,采用模糊C均值聚类方法对同一变量下的隶属度函数参数进行聚类,得到对不同工况具有代表性的融合后的隶属度函数,来降低过度拟合的影响.此外,本文根据规则库中的规则权值,对前件相同的冗余规则进行约简,形成最终的离线模糊规则库,有效提高了规则库的泛化能力.为验证本文方法的有效性,分别采用经典数据与实际工业数据进行了实验论证,从精度和泛化能力上体现了本文方法的优势.     

Similarity measures,penalty functions,and fuzzy entropy from new fuzzy subsethood measures

In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.     

模糊软集之间的相似性度量及应用

研究模糊软集的不确定度量问题,给出模糊软集的包含度、相似度公理化定义;基于模糊蕴含算子提出新的模糊软集包含度与相似度度量方法,该方法具有一定的普遍性,在某种程度上提供不同的模糊蕴含算子就可得到不同的包含度与相似度。基于新的相似度度量方法构造了一种决策方法并应用于金融企业流动性检测中。     

Reduction of fuzzy rule base via singular value decomposition

Introduces a singular value-based method for reducing a given fuzzy rule set. The method conducts singular value decomposition of the rule consequents and generates certain linear combinations of the original membership functions to form new ones for the reduced set. The present work characterizes membership functions by the conditions of sum normalization (SN), nonnegativeness (NN), and normality (NO). Algorithms to preserve the SN and NN conditions in the new membership functions are presented. Preservation of the NO condition relates to a high-dimensional convex hull problem and is not always feasible in which case a closed-to-NO solution may be sought. The proposed method is applicable regardless of the adopted inference paradigms. With product-sum-gravity inference and singleton support fuzzy rule base, output errors between the full and reduced fuzzy set are bounded by the sum of the discarded singular values. The work discusses three specific applications of fuzzy reduction: fuzzy rule base with singleton support, fuzzy rule base with nonsingleton support (which includes the case of missing rules), and the Takagi-Sugeno-Kang (TSK) model. Numerical examples are presented to illustrate the reduction process     

稀疏规则库条件下的模糊推理方法

当模糊规则库是稀疏型时,利用Kóczy线性插值推理方法不能保证推理结论的正规性和凸性,为了解决这一问题,石岩曾提出了插值推理方法的推理条件,当满足这些条件时利用Kóczy线性插值推理方法得到的推理结论也满足正规性和凸性;但是这些条件却限制了模糊推理系统的应用,而且如果多次推理中在同一输入点遇到稀疏情况,必须进行相同的计算才能得到正确的推理结果,这样增加了系统的计算量,降低了系统的速度和效率.因此提出了一种新的稀疏模糊推理方法,不仅能够简单的给出正确的推理结果,还能在相应的位置增加规则,提高规则库的紧密程度.     

Similarity measures for type-2 fuzzy sets and application in MCDM

Neural Computing and Applications - Type-2 fuzzy set theory is extensively used for decision making, pattern recognition and word computing due to exceptional expression of uncertain information....     

A neural network and fuzzy rule base hybrid for pattern classification

 This paper presents a novel hybrid of the two complimentary technologies of soft computing viz. neural networks and fuzzy logic to design a fuzzy rule based pattern classifier for problems with higher dimensional feature spaces. The neural network component of the hybrid, which acts as a pre-processor, is designed to take care of the all-important issue of feature selection. To circumvent the disadvantages of the popular back propagation algorithm to train the neural network, a meta-heuristic viz. threshold accepting (TA) has been used instead. Then, a fuzzy rule based classifier takes over the classification task with a reduced feature set. A combinatorial optimisation problem is formulated to minimise the number of rules in the classifier while guaranteeing high classification power. A modified threshold accepting algorithm proposed elsewhere by the authors (Ravi V, Zimmermann H.-J. (2000) Eur J Oper Res 123: 16–28) has been employed to solve this optimization problem. The proposed methodology has been demonstrated for (1) the wine classification problem having 13 features and (2) the Wisconsin breast cancer determination problem having 9 features. On the basis of these examples the results seem to be very interesting, as there is no reduction in the classification power in either of the problems, despite the fact that some of the original features have been completely eliminated from the study. On the contrary, the chosen features in both the problems yielded 100% classification power in some cases.     

Automatic generation of a fuzzy rule base for constant turning force

An automatic fuzzy rule base generation method is proposed to control nonlinear and timevarying turning processes with constant cutting forces. Based on this study, the optimum fuzzy rule base for the control of turning processes can be self-organized without the need for experienced manufacturing engineers. A fuzzy logic controller based on these fuzzy rules can adjust feed rate on line to achieve an optimal production rate in turning operations.     

广义犹豫模糊软集的相似度量及其应用

在广义模糊软集和犹豫模糊软集的基础上给出广义犹豫模糊软集的概念,并研究广义犹豫模糊软集的相似度量。首先利用三种犹豫模糊集合的包含度,构造犹豫模糊集间的相似度量公式。然后在犹豫模糊集相似度基础上给出广义犹豫模糊软集相似度量的公理化定义,并构造广义犹豫模糊软集的相似度量公式,这些公式可以计算参数集不同时两个广义犹豫模糊软集间的相似度。最后利用广义犹豫模糊软集相似度量方法构造了一种决策方法,并将这个决策方法应用于环境治理问题中。通过实例验证了所提出方法的可行性和有效性。     

Evolutionary design of fuzzy rule base for nonlinear systemmodeling and control

In designing fuzzy models and controllers, we encounter a major difficulty in the identification of an optimized fuzzy rule base, which is traditionally achieved by a tedious trial-and-error process. The paper presents an approach to the evolutionary design of an optimal fuzzy rule base for modeling and control. Evolutionary programming is used to simultaneously evolve the structure and the parameter of fuzzy rule base for a given task. To check the effectiveness of the suggested approach, four numerical examples are examined. The performance of the identified fuzzy rule bases is demonstrated     

Similarity measures

With complex multimedia data, we see the emergence of database systems in which the fundamental operation is similarity assessment. Before database issues can be addressed, it is necessary to give a definition of similarity as an operation. We develop a similarity measure, based on fuzzy logic, that exhibits several features that match experimental findings in humans. The model is dubbed fuzzy feature contrast (FFC) and is an extension to a more general domain of the feature contrast model due to Tversky (1977). We show how the FFC model can be used to model similarity assessment from fuzzy judgment of properties, and we address the use of fuzzy measures to deal with dependencies among the properties     

Real time traffic delay optimization using shadowed type-2 fuzzy rule base

Traffic networks are getting big and complex day by day with a rapid traffic growth. Existing Type-2 (T2) fuzzy logic works well in optimizing the waiting time of traffic at a big junction, but the rule base of T2 fuzzy logic is heavily dependent on previous traffic data, rather than real-time data. Moreover, it fails in changing and updating the waiting time in any junction with a high rate of traffic. In addition, very big junctions contain dynamic traffic data that is characterized by a high level of uncertainty, which is difficult to be handled by type-2 fuzzy logic. To cope with this situation, Shadowed Type-2 (ST2) fuzzy logic is proposed as it works well in the domain having very clumsy and uncertain data. It increases the uncertainty of a fuzzy set by partitioning it into different region. Thus, based on ST2 fuzzy rule base, a ST2 fuzzy waiting time simulator is created, whose output is implemented in a proposed real-time traffic-based Time Optimized Shortest Path (TOSP) model. It helps in structuring the optimized time path from one location to another. This can be done by taking real time traffic data from the upcoming junction, calculating the waiting time using ST2 fuzzy rule base, and finally directing the vehicle to take its optimized path, which results in a reduction in the overall waiting time of each junction. To demonstrate the superiority of the proposed model, a case study of a multi-directional (six directional) junction is presented. Success of this model easies the process of proposing it as a mobile application, which can help in reducing the waiting time in junctions of metropolitan areas.     

Similarity measures of picture fuzzy sets based on entropy and their application in MCDM

Pattern Analysis and Applications - An extension of fuzzy sets (Zadeh in Inf Control 8:338–353, 1965) and intuitionistic fuzzy sets (Atanassov in Fuzzy Sets Syst 20(1):87–96, 1986) is...     

Design of Mamdani fuzzy logic controllers with rule base minimisation using genetic algorithm

This paper presents a design procedure for Mamdani fuzzy logic controller including rule base minimisation. The rules are modelled with binary weights on which constraints are imposed in order to ensure consistency. A genetic algorithm is used for finding stabilising controllers that minimise the number of rules. The cost function includes a stability/performance coefficient which insures that stable, performance satisfying controllers are given the highest possible fitness. The number of fuzzy sets for the input and the control variables are set by the user and the design procedure is concerned only with the rule base and the distribution of the fuzzy sets in the universes of discourses. Two examples were studied: the control of the pole and cart system and the control of the concentration in CSTR. In both cases, the fuzzy sets were isosceles triangles evenly distributed, in the universe of discourses.     

Gene selection for designing optimal fuzzy rule base classifier by estimating missing value

DNA microarray technology, a high throughput technology evaluates the expression of thousands of genes simultaneously under different experimental conditions. Analysis of the gene expression data reveals that not all but few important genes are responsible for the diseases. However, the DNA microarray data set usually contain multiple missing value and therefore, selection of important genes using the incomplete data set may be erroneous, resulting misclassification in disease prediction. In the paper we propose an integrated framework, which first imputes the missing value and then in order to achieve maximum accuracy in classifying the patients a classifier has been designed to select the genes using the complete microarray data set.Here functionally similar genes are employed to estimate the missing value unlike the existing gene expression value based distance similarity measure. However, the functionally similar genes may differ in their protein production capacity and so the degree of similarity between the genes varies from gene to gene. The problem has been dealt by proposing a novel method to impute the missing value using the concept of fuzzy similarity. After imputing the missing value, the continuous gene expression matrix is discretized using fuzzy sets to distinguish the activation levels of different genes. The proposed fuzzy importance factor (FIf) of each gene represents its activation level or protein production capacity both in the disease and normal class. The importance of each gene is evaluated while optimizing the number of rules in the fuzzy classifier depending on the FIf. The methodology we propose has been demonstrated using nine different cancer data sets and compared with the state of the art methods. Analysis of experimental results reveals that the proposed framework able to classify the diseased and normal patients with improved accuracy.     

A fuzzy rule base for the improved control of a pressurized waternuclear reactor

In France, nuclear energy provides about 80% of the whole electricity production. A modulation of the nuclear power plants must be able to respond to the demand on the network. The pressurized water nuclear reactor has to yield correctly a load set point. Fundamentally, two parameters are concerned in leading this task to a successful conclusion: the power axial-offset and the control rods position. The focus of this study is the automation of the control of the power axial-offset by adding soluble boron and by minimizing the volume flows through the water pump. It is also important to take into consideration the liquid waste volume. Water or boron is injected into the reactor primary circuit. At the present time this task is still performed manually by an operator, for all previous attempts to automate it failed. A nonfuzzy device, earlier developed by Electricite de France, was intensively tested at Cruas, France, power plant and allowed us to prove the feasibility of automating the boration-dilution function. But it could not be definitely adopted because it was too difficult to be tuned for industrial purposes. That device, sketchily described in the paper, gave rise to the development of a real-time fuzzy controller for the power axial-offset and the control rods insertion in a pressurized water reactor (PWR). The fuzzy controller, which is the main subject of the paper, expresses more naturally the human expertise, thus avoiding the previous issue of empirical tunings. It was implemented in simulation using Matlab-Simulink on a Sun workstation. Two realistic tests discussed show that the fuzzy controller runs as efficiently as an expert operator does     

Comments on “Reduction of fuzzy rule base via singular valuedecomposition”

Several comments are presented for the reduction of the fuzzy rule base in the paper of Yam et al. (1999). In their paper, the approach to determine the number of singular values necessary for the reduction process to obtain the effective and most efficient fuzzy rule base is not provided. Although the output error of the fuzzy controller is bounded, the performance of the system output may not be satisfied. Moreover, the computation load is increased for each input of the fuzzy mechanism since the input membership functions are modified     

Similarity measures in computer vision

A similarity measure is needed in many Computer Vision problems. Although Euclidean distance has traditionally been used, median distance was recently proposed as an alternative, mostly due to its robustness properties. In this paper, a parametric class of distances is presented which allow to introduce a notion of similarity depending on the problem being considered.     

Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications

In this paper, we presented 10 similarity measures between Pythagorean fuzzy sets (PFSs) based on the cosine function by considering the degree of membership, degree of nonmembership and degree of hesitation in PFSs. Then, we applied these similarity measures and weighted similarity measures between PFSs to pattern recognition and medical diagnosis. Finally, two illustrative examples are given to demonstrate the efficiency of the similarity measures for pattern recognition and medical diagnosis.