应用聚类和遗传算法获取模糊模型

针对在复杂系统的模糊建模中,模型的精确度和可解释性很难同时得到满足的问题,利用PNC2聚类算法和遗传算法各自的特点,提出了一种新的建模方法。利用PNC2聚类算法创建初始模型,然后对规则参数进行编码,借助实值遗传算法优化模型。PNC2是有指导的层次凝聚聚类算法,双重的合并测试使获得的初始模型达到局部最优解,具有很强的可解释性;遗传算法通过自适应优化来提高模型的精确度。通过运用Iris数据分类问题,验证了算法的可行性和有效性。     

Application of genetic algorithm for fuzzy rules optimization on semi expert judgment automation using Pittsburg approach

Genetic algorithm is well-known of its best heuristic search method. Fuzzy logic unveils the advantage of interpretability. Genetic fuzzy system exploits potential of optimization with ease of understanding that facilitates rules optimization. This paper presents the optimization of fourteen fuzzy rules for semi expert judgment automation of early activity based duration estimation in software project management. The goal of the optimization is to reduce linguistic terms complexity and improve estimation accuracy of the fuzzy rule set while at the same time maintaining a similar degree of interpretability. The optimized numbers of linguistic terms in fuzzy rules by 27.76% using simplistic binary encoding mechanism managed to improve accuracy by 14.29% and reduce optimization execution time by 6.95% without compromising on interpretability in addition to promote improvement of knowledge base in fuzzy rule based systems.     

Complexity reduction and interpretability improvement for fuzzy rule systems based on simple interpretability measures and indices by bi-objective evolutionary rule selection

The aim of this paper is to develop a general post-processing methodology to reduce the complexity of data-driven linguistic fuzzy models, in order to reach simpler fuzzy models preserving enough accuracy and better fuzzy linguistic performance with respect to their initial values. This post-processing approach is based on rule selection via the formulation of a bi-objective problem with one objective focusing on accuracy and the other on interpretability. The latter is defined via the aggregation of several interpretability measures, based on the concepts of similarity and complexity of fuzzy systems and rules. In this way, a measure of the fuzzy model interpretability is given. Two neuro-fuzzy systems for providing initial fuzzy models, Fuzzy Adaptive System ART based and Neuro-Fuzzy Function Approximation and several case studies, data sets from KEEL Project Repository, are used to check this approach. Both fuzzy and neuro-fuzzy systems generate Mamdani-type fuzzy rule-based systems, each with its own particularities and complexities from the point of view of the fuzzy sets and the rule generation. Based on these systems and data sets, several fuzzy models are generated to check the performance of the proposal under different restrictions of complexity and fuzziness.     

Takagi-Sugeno-Kang fuzzy system fusion: A survey at hierarchical,wide and stacked levels

With excellent global approximation performance and interpretability, Takagi-Sugeno-Kang (TSK) fuzzy systems have enjoyed a wide range of applications in various fields, such as smart control, medical, and finance. However, in handling high-dimensional complex data, the performance and interpretability of a single TSK fuzzy system are easily degraded by rule explosion due to the curse of dimensionality. Ensemble learning comes into play to deal with the problem by the fusion of multiple TSK fuzzy systems using appropriate ensemble learning strategies, which has shown to be effective in eliminating the issue of the curse of dimensionality curse problem and reducing the number of fuzzy rules, thereby maintaining the interpretability of fuzzy systems. To this end, this paper gives a comprehensive survey of TSK fuzzy system fusion to provide insights into further research development. First, we briefly review the fundamental concepts related to TSK fuzzy systems, including fuzzy rule structures, training methods, and interpretability, and discuss the three different development directions of TSK fuzzy systems. Next, along the direction of TSK fuzzy system fusion, we investigate in detail the current ensemble strategies for fusion at hierarchical, wide and stacked levels, and discuss their differences, merits and weaknesses from the aspects of time complexity, interpretability (model complexity) and classification performance. We then present some applications of TSK fuzzy systems in real-world scenarios. Finally, the challenges and future directions of TSK fuzzy system fusion are discussed to foster prospective research.     

Designing fuzzy inference systems from data: Aninterpretability-oriented review

Fuzzy inference systems (FIS) are widely used for process simulation or control. They can be designed either from expert knowledge or from data. For complex systems, FIS based on expert knowledge only may suffer from a loss of accuracy. This is the main incentive for using fuzzy rules inferred from data. Designing a FIS from data can be decomposed into two main phases: automatic rule generation and system optimization. Rule generation leads to a basic system with a given space partitioning and the corresponding set of rules. System optimization can be done at various levels. Variable selection can be an overall selection or it can be managed rule by rule. Rule base optimization aims to select the most useful rules and to optimize rule conclusions. Space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters. Structure optimization is of a major importance: selecting variables, reducing the rule base and optimizing the number of fuzzy sets. Over the years, many methods have become available for designing FIS from data. Their efficiency is usually characterized by a numerical performance index. However, for human-computer cooperation another criterion is needed: the rule interpretability. An implicit assumption states that fuzzy rules are by nature easy to be interpreted. This could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts. The paper analyzes the main methods for automatic rule generation and structure optimization. They are grouped into several families and compared according to the rule interpretability criterion. For this purpose, three conditions for a set of rules to be interpretable are defined     

Fuzzy modeling of high-dimensional systems: complexity reductionand interpretability improvement

Fuzzy modeling of high-dimensional systems is a challenging topic. This paper proposes an effective approach to data-based fuzzy modeling of high-dimensional systems. An initial fuzzy rule system is generated based on the conclusion that optimal fuzzy rules cover extrema. Redundant rules are removed based on a fuzzy similarity measure. Then, the structure and parameters of the fuzzy system are optimized using a genetic algorithm and the gradient method. During optimization, rules that have a very low firing strength are deleted. Finally, interpretability of the fuzzy system is improved by fine training the fuzzy rules with regularization. The resulting fuzzy system generated by this method has the following distinct features: (1) the fuzzy system is quite simplified; (2) the fuzzy system is interpretable; and (3) the dependencies between the inputs and the output are clearly shown. This method has successfully been applied to a system that has 11 inputs and one output with 20 000 training data and 80 000 test data     

Model generation by domain refinement and rule reduction

The granularity and interpretability of a fuzzy model are influenced by the method used to construct the rule base. Models obtained by a heuristic assessment of the underlying system are generally highly granular with interpretable rules, while models algorithmically generated from an analysis of training data consist of a large number of rules with small granularity. This paper presents a method for increasing the granularity of rules while satisfying a prescribed precision bound on the training data. The model is generated by a two-stage process. The first step iteratively refines the partitions of the input domains until a rule base is generated that satisfies the precision bound. In this step, the antecedents of the rules are obtained from decomposable partitions of the input domains and the consequents are generated using proximity techniques. A greedy merging algorithm is then applied to increase the granularity of the rules while preserving the precision bound. To enhance the representational capabilities of a rule and reduce the number of rules required, the rules constructed by the merging procedure have multi-dimensional antecedents. A model defined with rules of this form incorporates advantageous features of both clustering and proximity methods for rule generation. Experimental results demonstrate the ability of the algorithm to reduce the number of rules in a fuzzy model with both precise and imprecise training information.     

A Proposal for the Genetic Lateral Tuning of Linguistic Fuzzy Systems and Its Interaction With Rule Selection

Linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings. However, since the accuracy and the interpretability of the obtained model are contradictory properties, the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled. To solve this problem, one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level. In this paper, a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions, with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability. To do so, we consider a new rule representation scheme base on the linguistic 2-tuples representation model which allows the lateral variation of the involved labels. Furthermore, the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper, presenting results on different real applications. The obtained results show the good performance of the proposed approach in high-dimensional problems and its ability to cooperate with methods to remove unnecessary rules.     

A new method for fuzzy information retrieval based on fuzzy hierarchical clustering and fuzzy inference techniques

In this paper, we extend the work of Kraft et al. to present a new method for fuzzy information retrieval based on fuzzy hierarchical clustering and fuzzy inference techniques. First, we present a fuzzy agglomerative hierarchical clustering algorithm for clustering documents and to get the document cluster centers of document clusters. Then, we present a method to construct fuzzy logic rules based on the document clusters and their document cluster centers. Finally, we apply the constructed fuzzy logic rules to modify the user's query for query expansion and to guide the information retrieval system to retrieve documents relevant to the user's request. The fuzzy logic rules can represent three kinds of fuzzy relationships (i.e., fuzzy positive association relationship, fuzzy specialization relationship and fuzzy generalization relationship) between index terms. The proposed fuzzy information retrieval method is more flexible and more intelligent than the existing methods due to the fact that it can expand users' queries for fuzzy information retrieval in a more effective manner.     

基于聚类和遗传算法的解释性模糊模型设计

提出了一种基于模糊聚类和遗传算法构建解释性模糊模型的设计方法。定义了模糊模型的精确性指标,给出了模糊模型解释性的必要条件。然后利用模糊聚类算法和最小二乘法辨识初始的模糊模型;采用多目标遗传算法优化模糊模型;为提高模型的解释性,在遗传算法中利用基于相似性的模糊集合和模糊规则的简化方法对模型进行约简。采用该方法对Mackey-Glass系统进行建模,仿真结果验证了该方法的有效性。     

基于协同进化算法的高维模糊分类系统的设计

基于协同进化算法,提出一种高维模糊分类系统的设计方法．首先定义系统的精确性指标,给出解释性的必要条件,利用聚类算法辨识初始模型．相互协作的3类种群分别代表系统的特征变量、规则前件和模型隶属函数的参数,适应度函数采用3类种群合作计算的策略,在算法运行中利用基于相似性的模型简化技术约简模糊系统,最后利用该方法对Wine问题进行研究．仿真结果表明该方法能够对高维分类问题的特征变量进行选择,同时利用较少规则和模糊集合数达到较高的识别率．     

Interpretability of linguistic fuzzy rule-based systems: An overview of interpretability measures

Linguistic fuzzy modelling, developed by linguistic fuzzy rule-based systems, allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings. Linguistic fuzzy modelling comes with two contradictory requirements: interpretability and accuracy. In recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown.Whereas the measures of accuracy are straightforward and well-known, interpretability measures are difficult to define since interpretability depends on several factors; mainly the model structure, the number of rules, the number of features, the number of linguistic terms, the shape of the fuzzy sets, etc. Moreover, due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem.In this paper, we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule-based systems. To this end, we will propose a taxonomy based on a double axis: “Complexity versus semantic interpretability” considering the two main kinds of measures; and “rule base versus fuzzy partitions” considering the different components of the knowledge base to which both kinds of measures can be applied. The main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works.     

AGFS: Adaptive Genetic Fuzzy System for medical data classification

A Genetic Fuzzy System (GFS) is basically a fuzzy system augmented by a learning process based on a genetic algorithm (GA). Fuzzy systems have demonstrated their ability to solve different kinds of problems in various application domains. Currently, there is an increasing interest to augment fuzzy systems with learning and adaptation capabilities. Two of the most successful approaches to hybridize fuzzy systems with learning and adaptation methods have been made in the realm of soft computing. The GA can be merged with Fuzzy system for different purposes like rule selection, membership function optimization, rule generation, co-efficient optimization, for data classification. Here we propose an Adaptive Genetic Fuzzy System (AGFS) for optimizing rules and membership functions for medical data classification process. The primary intension of the research is 1) Generating rules from data as well as for the optimized rules selection, adapting of genetic algorithm is done and to explain the exploration problem in genetic algorithm, introduction of new operator, called systematic addition is done, 2) Proposing a simple technique for scheming of membership function and Discretization, and 3) Designing a fitness function by allowing the frequency of occurrence of the rules in the training data. Finally, to establish the efficiency of the proposed classifier the presentation of the anticipated genetic-fuzzy classifier is evaluated with quantitative, qualitative and comparative analysis. From the outcome, AGFS obtained better accuracy when compared to the existing systems.     

基于演进向量量化聚类的增量模糊关联分类方法

为了提高动态数据集上模糊关联分类器（FAC）的建模效率,提出了一种基于演进向量量化（eVQ）聚类的增量模糊关联分类方法。首先,采用eVQ聚类算法增量更新数量属性上的高斯隶属度函数参数;然后,扩展早剪枝更新（UWEP）算法,使之适用于增量挖掘模糊频繁项;最后,以模糊相关度（FCORR）和分类规则前件长度为度量方式裁剪并更新模糊关联分类规则库。在4个UCI标准数据集上的实验结果表明,与批量模糊关联分类建模方法相比,所提方法能够在保证分类精度和解释性的前提下,减少模糊关联分类器的训练时间;基于eVQ的高斯隶属度函数的增量更新有助于提高动态数据集上模糊关联分类器的分类精度。     

Fuzzy clustering with semantic interpretation

In the framework of Axiomatic Fuzzy Set (AFS) theory, we propose a new approach to data clustering. The objective of this clustering is to adhere to some principles of grouping exercised by humans when determining a structure in data. Compared with other clustering approaches, the proposed approach offers more detailed insight into the cluster's structure and the underlying decision making process. This contributes to the enhanced interpretability of the results via the representation capabilities of AFS theory. The effectiveness of the proposed approach is demonstrated by using real-world data, and the obtained results show that the performance of the clustering is comparable with other fuzzy rule-based clustering methods, and benchmark fuzzy clustering methods FCM and K-means. Experimental studies have shown that the proposed fuzzy clustering method can discover the clusters in the data and help specify them in terms of some comprehensive fuzzy rules.     

A genetic approach to modeling fuzzy systems based on information granulation and successive generation-based evolution method

In this paper, we introduce and investigate a new category of fuzzy inference systems based on information granulation and genetic optimization used to system identification. We show the applications of such systems to identification of nonlinear systems. The formal framework of information granulation and resulting information granules themselves become an important design facet of the fuzzy models. By embracing fuzzy sets, the model is geared towards capturing essential relationship between information granules rather than concentrating on plain numeric data. Information granulation realized with the use of the commonly exploited C-Means clustering helps determine the initial values of the parameters of the fuzzy models. This in particular concerns such essential components of the rules as the initial apexes of the membership functions standing in the premise part of the fuzzy rules and the points of the polynomial functions standing in the consequence part. The initial apexes (center points) of the membership functions based on C-Means algorithm are tuned with the aid of the genetic algorithm (GA), while the tuned apexes are also used to adjust the points of the consequent polynomials (conclusions) of the rules. In particular, the initial apexes of the membership functions and the initial points of the consequent polynomials are adjusted and updated every time through successive evolution process. The overall design methodology involves a hybrid structural and parametric optimization. Genetic algorithms and C-Means clustering are used to optimize the model with respect to its structure and parameters. To determine the structure and estimate the values of the parameters of the fuzzy model we consider the successive tuning method with generation-based evolution by means of genetic algorithms. The model is evaluated with the use of numerical experimentation and its quality is compared with respect to some other fuzzy models already encountered in the literature.     

An ensemble method for fuzzy rule-based classification systems

Fuzzy rule-based classification systems are very useful tools in the field of machine learning as they are able to build linguistic comprehensible models. However, these systems suffer from exponential rule explosion when the number of variables increases, degrading, therefore, the accuracy of these systems as well as their interpretability. In this article, we propose to improve the comprehensibility through a supervised learning method by automatic generation of fuzzy classification rules, designated SIFCO–PAF. Our method reduces the complexity by decreasing the number of rules and of antecedent conditions, making it thus adapted to the representation and the prediction of rather high-dimensional pattern classification problems. We perform, firstly, an ensemble methodology by combining a set of simple classification models. Subsequently, each model uses a subset of the initial attributes: In this case, we propose to regroup the attributes using linear correlation search among the training set elements. Secondly, we implement an optimal fuzzy partition thanks to supervised discretization followed by an automatic membership functions construction. The SIFCO–PAF method, analyzed experimentally on various data sets, guarantees an important reduction in the number of rules and of antecedents without deteriorating the classification rates, on the contrary accuracy is even improved.     

DENFIS: dynamic evolving neural-fuzzy inference system and itsapplication for time-series prediction

This paper introduces a new type of fuzzy inference systems, denoted as dynamic evolving neural-fuzzy inference system (DENFIS), for adaptive online and offline learning, and their application for dynamic time series prediction. DENFIS evolve through incremental, hybrid (supervised/unsupervised), learning, and accommodate new input data, including new features, new classes, etc., through local element tuning. New fuzzy rules are created and updated during the operation of the system. At each time moment, the output of DENFIS is calculated through a fuzzy inference system based on m-most activated fuzzy rules which are dynamically chosen from a fuzzy rule set. Two approaches are proposed: (1) dynamic creation of a first-order Takagi-Sugeno-type fuzzy rule set for a DENFIS online model; and (2) creation of a first-order Takagi-Sugeno-type fuzzy rule set, or an expanded high-order one, for a DENFIS offline model. A set of fuzzy rules can be inserted into DENFIS before or during its learning process. Fuzzy rules can also be extracted during or after the learning process. An evolving clustering method (ECM), which is employed in both online and offline DENFIS models, is also introduced. It is demonstrated that DENFIS can effectively learn complex temporal sequences in an adaptive way and outperform some well-known, existing models