Obtaining fuzzy rules from interval-censored data with genetic algorithms and a random sets-based semantic of the linguistic labels

Fuzzy memberships can be understood as coverage functions of random sets. This interpretation makes sense in the context of fuzzy rule learning: a random-sets-based semantic of the linguistic labels is compatible with the use of fuzzy statistics for obtaining knowledge bases from data. In particular, in this paper we formulate the learning of a fuzzy-rule-based classifier as a problem of statistical inference. We propose to learn rules by maximizing the likelihood of the classifier. Furthermore, we have extended this methodology to interval-censored data, and propose to use upper and lower bounds of the likelihood to evolve rule bases. Combining descent algorithms and a co-evolutionary scheme, we are able to obtain rule-based classifiers from imprecise data sets, and can also identify the conflictive instances in the training set: those that contribute the most to the indetermination of the likelihood of the model.     

Extracting linguistic rules from data sets using fuzzy logic and genetic algorithms

Linguistic rules in natural language are useful and consistent with human way of thinking. They are very important in multi-criteria decision making due to their interpretability. In this paper, our discussions concentrate on extracting linguistic rules from data sets. In the end, we firstly analyze how to extract complex linguistic data summaries based on fuzzy logic. Then, we formalize linguistic rules based on complex linguistic data summaries, in which, the degree of confidence of linguistic rules from a data set can be explained by linguistic quantifiers and its linguistic truth from the fuzzy logical point of view. In order to obtain a linguistic rule with a higher degree of linguistic truth, a genetic algorithm is used to optimize the number and parameters of membership functions of linguistic values. Computational results show that the proposed method is an alternative method for extracting linguistic rules with linguistic truth from data sets.     

On the influence of using fuzzy extensions in linguistic fuzzy rule-based regression systems

Fuzzy Rule-Based Systems, FRBSs, are powerful tools to address regression problems. They can model the relationship between inputs and outputs by linguistic concepts. However, those FRBSs which are based on the conventional Type-1 fuzzy sets may not be able to handle some difficulties of real-world applications. In such situations, using novel representations of fuzzy sets seems like a good idea. Different extensions of fuzzy sets usually help to provide more precise models in the real-world problems. In this study, the influence of using fuzzy extensions in improving the efficiency of linguistic fuzzy rule-based regression models is investigated. For this purpose, a conventional Type-1 Mamdani FRBS is adapted to the three extensions of fuzzy sets, namely Interval Type-2, Intuitionistic, and Interval Type-2 Intuitionistic fuzzy sets. A two-pass method is proposed to define membership (non-membership) functions of these fuzzy sets; this method is based on the 3-tuples representation of the standard Type-1 membership functions. Wang and Mendel’s rule learning method is adapted to extract fuzzy rules from regression data. In order to tune the membership functions up to different extents, three evolutionary extensions are also presented for each type of the proposed FRBSs. Individual, internal, and external comparisons of the proposed FRBSs were done using 22 real-world regression datasets and statistical tests. Experimental results confirm that all the three proposed FRBSs outperform the classical Type-1 framework; furthermore, the Interval Type-2 Intuitionistic FRBS is the superior system so that an appropriate tuning of its parameters makes it the most accurate model.     

Identification of fuzzy measures from sample data with genetic algorithms

In this paper, we introduce a method for the identification of fuzzy measures from sample data. It is implemented using genetic algorithms and is flexible enough to allow the use of different subfamilies of fuzzy measures for the learning, as k-additive or p-symmetric measures. The experiments performed to test the algorithm suggest that it is robust in situations where there exists noise in the considered data. We also explore some possibilities for the choice of the initial population, which lead to the study of the extremes of some subfamilies of fuzzy measures, as well as the proposal of a method for random generation of fuzzy measures.     

Unsatisfying functions and multiobjective fuzzy satisfaction design using genetic algorithms

This paper describes a new fuzzy satisfaction method using genetic algorithms (GA) for multiobjective problems. First, an unsatisfying function, which has a one-to-one correspondence with the membership function, is introduced for expressing "fuzziness". Next, the multiobjective design problem is transformed into a satisfaction problem of constraints by introducing an aspiration level for each objective. Here, in order to handle the fuzziness involved in aspiration levels and constraints, the unsatisfying function is used, and the problem is formulated as a multiobjective minimization problem of unsatisfaction ratings. Then, a GA is employed to solve the problem, and a new strategy is proposed to obtain a group of Pareto-optimal solutions in which the decision maker (DM) is interested. The DM can then seek a satisfaction solution by modifying parameters interactively according to preferences.     

Performance evaluation of fuzzy rule-based classification systems obtained by multi-objective genetic algorithms

In this paper, we examine the classification performance of fuzzy if-then rules selected by a GA-based multi-objective rule selection method. This rule selection method can be applied to high-dimensional pattern classification problems with many continuous attributes by restricting the number of antecedent conditions of each candidate fuzzy if-then rule. As candidate rules, we only use fuzzy if-then rules with a small number of antecedent conditions. Thus it is easy for human users to understand each rule selected by our method. Our rule selection method has two objectives: to minimize the number of selected fuzzy if-then rules and to maximize the number of correctly classified patterns. In our multi-objective fuzzy rule selection problem, there exist several solutions (i.e., several rule sets) called “non-dominated solutions” because two conflicting objectives are considered. In this paper, we examine the performance of our GA-based rule selection method by computer simulations on a real-world pattern classification problem with many continuous attributes. First we examine the classification performance of our method for training patterns by computer simulations. Next we examine the generalization ability for test patterns. We show that a fuzzy rule-based classification system with an appropriate number of rules has high generalization ability.     

Identification of evolving fuzzy rule-based models

An approach to identification of evolving fuzzy rule-based (eR) models is proposed. eR models implement a method for the noniterative update of both the rule-base structure and parameters by incremental unsupervised learning. The rule-base evolves by adding more informative rules than those that previously formed the model. In addition, existing rules can be replaced with new rules based on ranking using the informative potential of the data. In this way, the rule-base structure is inherited and updated when new informative data become available, rather than being completely retrained. The adaptive nature of these evolving rule-based models, in combination with the highly transparent and compact form of fuzzy rules, makes them a promising candidate for modeling and control of complex processes, competitive to neural networks. The approach has been tested on a benchmark problem and on an air-conditioning component modeling application using data from an installation serving a real building. The results illustrate the viability and efficiency of the approach.     

Progressive design methodology for complex engineering systems based on multiobjective genetic algorithms and linguistic decision making

This work focuses on a design methodology that aids in design and development of complex engineering systems. This design methodology consists of simulation, optimization and decision making. Within this work a framework is presented in which modelling, multi-objective optimization and multi criteria decision making techniques are used to design an engineering system. Due to the complexity of the designed system a three-step design process is suggested. In the first step multi-objective optimization using genetic algorithm is used. In the second step a multi attribute decision making process based on linguistic variables is suggested in order to facilitate the designer to express the preferences. In the last step the fine tuning of selected few variants are performed. This methodology is named as progressive design methodology. The method is applied as a case study to design a permanent magnet brushless DC motor drive and the results are compared with experimental values.     

Simplifying fuzzy rule-based models using orthogonal transformationmethods

An important issue in fuzzy-rule-based modeling is how to select a set of important fuzzy rules from a given rule base. Even though it is conceivable that removal of redundant or less important fuzzy rules from the rule base can result in a compact fuzzy model with better generalizing ability, the decision as to which rules are redundant or less important is not an easy exercise. In this paper, we introduce several orthogonal transformation-based methods that provide new or alternative tools for rule selection. These methods include an orthogonal least squares (OLS) method, an eigenvalue decomposition (ED) method, a singular value decomposition and QR with column pivoting (SVD-QR) method, a total least squares (TLS) method, and a direct singular value decomposition (D-SVD) method. A common attribute of these methods is that they all work on a firing strength matrix and employ some measure index to detect the rules that should be retained and eliminated. We show the performance of these methods by applying them to solving a nonlinear plant modeling problem. Our conclusions based on analysis and simulation can be used as a guideline for choosing a proper rule selection method for a specific application.     

Selection of the optimum promotion mix by integrating a fuzzy linguistic decision model with genetic algorithms

Integrated marketing communication (IMC) is an important process by which a company can influence a target market, improve the position of that company’s product/service in the target market, and effectively build up its brand image. Sales promotion is an important communication channel designed to influence the customer’s purchasing behavior in the target market. There are many promotion tools available. Variations in business objectives and budgetary limits make it impossible for a company to employ all these promotion tools to convey sales messages to the customers. The selection of the best mix of promotion tools involves subjective information processing, instead of a numerically expressed objective decision-making process. In this research, we integrate a fuzzy linguistic decision model with a genetic algorithm (GA) to extract the optimum promotion mix of a variety of tools under satisfying expected marketing performance and budget limitations. The proposed methodology shows satisfactory results in an empirical study in terms of estimating the degree of satisfaction for achieving the business objectives, determining the optimum promotion mix, and minimizing expenditure on sales promotion activities.     

Reconstruct street network from imprecise excavation data using fuzzy Hough transforms

This paper proposes an approach for handling multivariate data in an archaeological Geographical Information System (GIS), providing a new tool to archaeologists and historians. Our method extracts potential objects of known shapes in a geographical database (GDB) devoted to archaeological excavations. In this work, archaeological information is organized according to three components: location, date and a shape parameter, in a context where data are imprecise and lacunar. To manage these aspects, a three-step methodology was developed using fuzzy sets modeling and adapting the fuzzy Hough transform. This methodology is applied in order to define the appropriate tool for a GDB of Roman street remains in Reims, France. The defined queries return an estimation of the possible presence of streets during a fuzzy time interval given by experts on the Roman period in Reims.     

Performance evaluation of evolutionary multiobjective optimization algorithms for multiobjective fuzzy genetics-based machine learning

Recently, evolutionary multiobjective optimization (EMO) algorithms have been utilized for the design of accurate and interpretable fuzzy rule-based systems. This research area is often referred to as multiobjective genetic fuzzy systems (MoGFS), where EMO algorithms are used to search for non-dominated fuzzy rule-based systems with respect to their accuracy and interpretability. In this paper, we examine the ability of EMO algorithms to efficiently search for Pareto optimal or near Pareto optimal fuzzy rule-based systems for classification problems. We use NSGA-II (elitist non-dominated sorting genetic algorithm), its variants, and MOEA/D (multiobjective evolutionary algorithm based on decomposition) in our multiobjective fuzzy genetics-based machine learning (MoFGBML) algorithm. Classification performance of obtained fuzzy rule-based systems by each EMO algorithm is evaluated for training data and test data under various settings of the available computation load and the granularity of fuzzy partitions. Experimental results in this paper suggest that reported classification performance of MoGFS in the literature can be further improved using more computation load, more efficient EMO algorithms, and/or more antecedent fuzzy sets from finer fuzzy partitions.     

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.     

Adaptive genetic algorithms applied to dynamic multiobjective problems

This paper describes an adaptive genetic algorithm (AGA) with dynamic fitness function for multiobjective problems (MOPs) in a dynamic environment. In order to see performance of the algorithm, AGA was applied to two kinds of MOPs. Firstly, the algorithm was used to find an optimal force allocation for a combat simulation. The paper discusses four objectives that need to be optimized and presents a fuzzy inference system that forms an aggregation of the four objectives. A second fuzzy inference system is used to control the crossover and mutation rates based on statistics of the aggregate fitness. In addition to dynamic force allocation optimization problem, a simple example of a dynamic multiobjective optimization problem taken from Farina et al. [M. Farina, K. Deb, P. Amato, Dynamic multiobjective optimization problems: test cases, approximations, and applications, IEEE Trans. Evol. Comput. 8 (5) (2004) 425–442] is presented and solved with the proposed algorithm. The results obtained here indicate that performance of the fuzzy-augmented GA is better than a standard GA method in terms of improvement of convergence to solutions of dynamic MOPs.     

On improving multiobjective genetic algorithms for design optimization

This paper presents some improvements to Multi-Objective Genetic Algorithms (MOGAs). MOGA modifies certain operators within the GA itself to produce a multiobjective optimization technique. The improvements are made to overcome some of the shortcomings in niche formation, stopping criteria and interaction with a design decision-maker. The technique involves filtering, mating restrictions, the idea of objective constraints, and detecting Pareto solutions in the non-convex region of the Pareto set. A step-by-step procedure for an improved MOGA has been developed and demonstrated via two multiobjective engineering design examples: (i) two-bar truss design, and (ii) vibrating platform design. The two-bar truss example has continuous variables while the vibrating platform example has mixed-discrete (combinatorial) variables. Both examples are solved by MOGA with and without the improvements. It is shown that MOGA with the improvements performs better for both examples in terms of the number of function evaluations.     

Estimating the uniaxial compressive strength of some clay-bearing rocks selected from Turkey by nonlinear multivariable regression and rule-based fuzzy models

Abstract: Although the use of predictive models in rock engineering and engineering geology is an important issue, some simple and multivariate regression techniques traditionally employed in these areas have recently been challenged by the use of fuzzy inference systems and artificial neural networks. The purpose of this study was to construct some predictive models to estimate the uniaxial compressive strength of some clay-bearing rocks, depending on examination of their slake durability indices and clay contents. For this purpose, the simple and nonlinear multivariable regression techniques and the Mamdani fuzzy algorithm are compared in terms of their accuracy. To increase the accuracy of the Mamdani fuzzy inference system, the weighted if–then rules are extracted. To compare the predictive performances of the models, the statistical performance indices (root mean square error and variance account for) are calculated and the results are discussed. The indices reveal that the fuzzy inference system has a slightly higher prediction capacity than the regression models. The basic reason for the higher performance of the fuzzy inference system is the flexibility of the fuzzy approach.     

Missing data imputation for fuzzy rule-based classification systems

Fuzzy rule-based classification systems (FRBCSs) are known due to their ability to treat with low quality data and obtain good results in these scenarios. However, their application in problems with missing data are uncommon while in real-life data, information is frequently incomplete in data mining, caused by the presence of missing values in attributes. Several schemes have been studied to overcome the drawbacks produced by missing values in data mining tasks; one of the most well known is based on preprocessing, formerly known as imputation. In this work, we focus on FRBCSs considering 14 different approaches to missing attribute values treatment that are presented and analyzed. The analysis involves three different methods, in which we distinguish between Mamdani and TSK models. From the obtained results, the convenience of using imputation methods for FRBCSs with missing values is stated. The analysis suggests that each type behaves differently while the use of determined missing values imputation methods could improve the accuracy obtained for these methods. Thus, the use of particular imputation methods conditioned to the type of FRBCSs is required.