A reduction approach for fuzzy rule bases of fuzzy controllers

In this paper, a new approach to reducing the number of rules in a given fuzzy rule base of a fuzzy controller is presented. The fuzzy mechanism of the fuzzy controller under consideration consists of the product-sum inference, singleton output consequents and centroid defuzzification. The output consequents in the cells of the rule table are collected and represented as an output consequent matrix. The feature of the output consequent matrix is extracted by the singular values of the matrix. The output consequent matrix is reasonably approximated with a dominant consequent matrix. Also, the elements of the dominant consequent matrix is determined to minimize the approximation error function. Then the size of the dominant consequent matrix (the size of the fuzzy rule base) is reduced through the rule combination approach. The scaling factors for the fuzzy controller with the reduced rule table are adjusted to have the control system satisfy the performance indices. The effectiveness of the proposed approach is shown using simulation and experimental results.     

A generalized concept for fuzzy rule interpolation

The concept of fuzzy rule interpolation in sparse rule bases was introduced in 1993. It has become a widely researched topic in recent years because of its unique merits in the topic of fuzzy rule base complexity reduction. The first implemented technique of fuzzy rule interpolation was termed as /spl alpha/-cut distance based fuzzy rule base interpolation. Despite its advantageous properties in various approximation aspects and in complexity reduction, it was shown that it has some essential deficiencies, for instance, it does not always result in immediately interpretable fuzzy membership functions. This fact inspired researchers to develop various kinds of fuzzy rule interpolation techniques in order to alleviate these deficiencies. This paper is an attempt into this direction. It proposes an interpolation methodology, whose key idea is based on the interpolation of relations instead of interpolating /spl alpha/-cut distances, and which offers a way to derive a family of interpolation methods capable of eliminating some typical deficiencies of fuzzy rule interpolation techniques. The proposed concept of interpolating relations is elaborated here using fuzzy- and semantic-relations. This paper presents numerical examples, in comparison with former approaches, to show the effectiveness of the proposed interpolation methodology.     

Multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques

In this paper, we present a new method for multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques. First, the proposed method constructs training samples based on the variation rates of the training data set and then uses the training samples to construct fuzzy rules by making use of the fuzzy C-means clustering algorithm, where each fuzzy rule corresponds to a given cluster. Then, we determine the weight of each fuzzy rule with respect to the input observations and use such weights to determine the predicted output, based on the multiple fuzzy rules interpolation scheme. We apply the proposed method to the temperature prediction problem and the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) data. The experimental results show that the proposed method produces better forecasting results than several existing methods.     

Comprehensive analysis of a new fuzzy rule interpolation method

The first published result in fuzzy rule interpolation was the α-cut based fuzzy rule interpolation, termed as KH fuzzy rule interpolation, originally devoted for complexity reduction. A modified version of the KH approach has been presented by Yam et al. (1999), which eliminates the subnormality problem while at the same time intending to maintain the advantageous computational properties of the original method. This paper presents a comprehensive analysis of the new method, which includes detailed comparison with the original KH fuzzy rule interpolation method concerning the explicit functions of the methods, preservation of piecewise linearity, and stability. The fuzziness of the conclusion with respect to the fuzziness of the observation is also investigated in comparison with several interpolation techniques. All these comparisons shows that the new method preserves the advantageous properties of the KH method and alleviates its most significant disadvantage, the problem of subnormality     

Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction

Tuning fuzzy rule-based systems for linguistic fuzzy modeling is an interesting and widely developed task. It involves adjusting some of the components of the knowledge base without completely redefining it. This contribution introduces a genetic tuning process for jointly fitting the fuzzy rule symbolic representations and the meaning of the involved membership functions. To adjust the former component, we propose the use of linguistic hedges to perform slight modifications keeping a good interpretability. To alter the latter component, two different approaches changing their basic parameters and using nonlinear scaling factors are proposed. As the accomplished experimental study shows, the good performance of our proposal mainly lies in the consideration of this tuning approach performed at two different levels of significance. The paper also analyzes the interaction of the proposed tuning method with a fuzzy rule set reduction process. A good interpretability-accuracy tradeoff is obtained combining both processes with a sequential scheme: first reducing the rule set and subsequently tuning the model.     

Learning concurrently data and rule bases of Mamdani fuzzy rule-based systems by exploiting a novel interpretability index

Interpretability of Mamdani fuzzy rule-based systems (MFRBSs) has been widely discussed in the last years, especially in the framework of multi-objective evolutionary fuzzy systems (MOEFSs). Here, multi-objective evolutionary algorithms (MOEAs) are applied to generate a set of MFRBSs with different trade-offs between interpretability and accuracy. In MOEFSs interpretability has often been measured in terms of complexity of the rule base and only recently partition integrity has also been considered. In this paper, we introduce a novel index for evaluating the interpretability of MFRBSs, which takes both the rule base complexity and the data base integrity into account. We discuss the use of this index in MOEFSs, which generate MFRBSs by concurrently learning the rule base, the linguistic partition granularities and the membership function parameters during the evolutionary process. The proposed approach has been experimented on six real world regression problems and the results have been compared with those obtained by applying the same MOEA, with only accuracy and complexity of the rule base as objectives. We show that our approach achieves the best trade-offs between interpretability and accuracy.     

Fuzzy rule interpolation based on the ratio of fuzziness of interval type-2 fuzzy sets

In recent years, some fuzzy rule interpolation methods have been presented for sparse fuzzy rule-based systems based on interval type-2 fuzzy sets. However, the existing methods have the drawbacks that they cannot guarantee the convexity of the fuzzy interpolated result and may generate the same fuzzy interpolated results with respect to different observations. Moreover, they also cannot deal with fuzzy rule interpolation with bell-shaped interval type-2 fuzzy sets. In this paper, we present a new method for fuzzy rule interpolation for sparse fuzzy rule-based systems based on the ratio of fuzziness of interval type-2 fuzzy sets. The proposed method can overcome the drawbacks of the existing methods. First, it calculates the weights of the closest fuzzy rules with respect to the observation to obtain an intermediate consequence fuzzy set. Then, it uses the ratio of fuzziness of interval type-2 fuzzy sets to infer the fuzzy interpolated result based on the intermediate consequence fuzzy set. We also use some examples to compare the fuzzy interpolated results of the proposed method with the results by the existing methods. The experimental results show that the proposed fuzzy rule interpolation method gets more reasonable results than the existing methods.     

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.     

Combinatorial rule explosion eliminated by a fuzzy ruleconfiguration

Conventional fuzzy inference methodology relates the relevant subsets of each input universal set to the subsets of the other system inputs through an intersection-rule configuration. This strategy yields an exponential growth in the number of rules as inputs are added to the system, quickly reducing performance to unacceptable levels. A novel rule configuration and matrix design are presented in this paper that do not rely on rule multiplication to insure that antecedent elements are effectively related to their consequent counterparts. This alternative formulation models the entire system problem space with a simplified structure that increases linearly as the inference engine grows, providing significant computational savings to a broad range of commercial and scientific applications     

Multi-level knowledge discovery from rule bases

The discovery of multi-level knowledge is important to allow queries at and across different levels of abstraction. While there are some similarities between our research and that of others in this area, the work reported in this paper does not directly involve databases and is differently motivated. Our research is interested in taking data in the form of rule-bases and finding multi-level knowledge. This paper describes our motivation, our preferred technique for acquiring the initial knowledge known as Ripple-Down Rules, the use of Formal Concept Analysis to develop an abstraction hierarchy, and our application of these ideas to knowledge bases from the domain of chemical pathology. We also provide an example of how the approach can be applied to other prepositional knowledge bases and suggest that it can be used as an additional phase to many existing data mining approaches.     

Noise reduction by fuzzy image filtering

A new fuzzy filter is presented for the noise reduction of images corrupted with additive noise. The filter consists of two stages. The first stage computes a fuzzy derivative for eight different directions. The second stage uses these fuzzy derivatives to perform fuzzy smoothing by weighting the contributions of neighboring pixel values. Both stages are based on fuzzy rules which make use of membership functions. The filter can be applied iteratively to effectively reduce heavy noise. In particular, the shape of the membership functions is adapted according to the remaining noise level after each iteration, making use of the distribution of the homogeneity in the image. A statistical model for the noise distribution can be incorporated to relate the homogeneity to the adaptation scheme of the membership functions. Experimental results are obtained to show the feasibility of the proposed approach. These results are also compared to other filters by numerical measures and visual inspection.     

Linear interpolation in a fuzzy information space

An approach to solving a linear interpolation problem in a fuzzy information space is proposed. Two different schemes of interpolation are outlined: a heuristic one, based on the geometrical interpretation of operations, and an optimization one, based on the expansion principle. The results obtained allow performing fuzzy linear prediction. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 55–68, March–April 2006.     

A novel generic hebbian ordering-based fuzzy rule base reduction approach to mamdani neuro-fuzzy system

There are two important issues in neuro-fuzzy modeling: (1) interpretability--the ability to describe the behavior of the system in an interpretable way--and (2) accuracy--the ability to approximate the outcome of the system accurately. As these two objectives usually exert contradictory requirements on the neuro-fuzzy model, certain compromise has to be undertaken. This letter proposes a novel rule reduction algorithm, namely, Hebb rule reduction, and an iterative tuning process to balance interpretability and accuracy. The Hebb rule reduction algorithm uses Hebbian ordering, which represents the degree of coverage of the samples by the rule, as an importance measure of each rule to merge the membership functions and hence reduces the number of the rules. Similar membership functions (MFs) are merged by a specified similarity measure in an order of Hebbian importance, and the resultant equivalent rules are deleted from the rule base. The rule with a higher Hebbian importance will be retained among a set of rules. The MFs are tuned through the least mean square (LMS) algorithm to reduce the modeling error. The tuning of the MFs and the reduction of the rules proceed iteratively to achieve a balance between interpretability and accuracy. Three published data sets by Nakanishi (Nakanishi, Turksen, & Sugeno, 1993), the Pat synthetic data set (Pal, Mitra, & Mitra, 2003), and the traffic flow density prediction data set are used as benchmarks to demonstrate the effectiveness of the proposed method. Good interpretability, as well as high modeling accuracy, are derivable simultaneously and are suitably benchmarked against other well-established neuro-fuzzy models.     

Fuzzy rule interpolation based on principle membership functions and uncertainty grade functions of interval type-2 fuzzy sets

Fuzzy rule interpolation is an important research topic in sparse fuzzy rule-based systems. In this paper, we present a new method for dealing with fuzzy rule interpolation in sparse fuzzy rule-based systems based on the principle membership functions and uncertainty grade functions of interval type-2 fuzzy sets. The proposed method deals with fuzzy rule interpolation based on the principle membership functions and the uncertainty grade functions of interval type-2 fuzzy sets. It can deal with fuzzy rule interpolation with polygonal interval type-2 fuzzy sets and can handle fuzzy rule interpolation with multiple antecedent variables. We also use some examples to compare the fuzzy interpolative reasoning results of the proposed method with the ones of an existing method. The experimental result shows that the proposed method gets more reasonable results than the existing method for fuzzy rule interpolation based on interval type-2 fuzzy sets.     

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.     

Clustering rule bases using ontology-based similarity measures

Rules are increasingly becoming an important form of knowledge representation on the Semantic Web. There are currently few methods that can ensure that the acquisition and management of rules can scale to the size of the Web. We previously developed methods to help manage large rule bases using syntactical analyses of rules. This approach did not incorporate semantics. As a result, rule categorization based on syntactic features may not be effective. In this paper, we present a novel approach for grouping rules based on whether the rule elements share relationships within a domain ontology. We have developed our method for rules specified in the Semantic Web Rule Language (SWRL), which is based on the Web Ontology Language (OWL) and shares its formal underpinnings. Our method uses vector space modeling of rule atoms and an ontology-based semantic similarity measure. We apply a clustering method to detect rule relatedness, and we use a statistical model selection method to find the optimal number of clusters within a rule base. Using three different SWRL rule bases, we evaluated the results of our semantic clustering method against those of our syntactic approach. We have found that our new approach creates clusters that better match the rule bases’ logical structures. Semantic clustering of rule bases may help users to more rapidly comprehend, acquire, and manage the growing numbers of rules on the Semantic Web.     

Model construction, rule reduction, and robust compensation forgeneralized form of Takagi-Sugeno fuzzy systems

This paper presents a systematic procedure of fuzzy control system design that consists of fuzzy model construction, rule reduction, and robust compensation for nonlinear systems. The model construction part replaces the nonlinear dynamics of a system with a generalized form of Takagi-Sugeno fuzzy systems, which is newly developed by us. The generalized form has a decomposed structure for each element of A_i and B_i matrices in consequent parts. The key feature of this structure is that it is suitable for constructing IF-THEN rules and reducing the number of IF-THEN rules. The rule reduction part provides a successive procedure to reduce the number of IF-THEN rules. Furthermore, we convert the reduction error between reduced fuzzy models and a system to model uncertainties of reduced fuzzy models. The robust compensation part achieves the decay rate controller design guaranteeing robust stability for the model uncertainties. Finally, two examples demonstrate the utility of the systematic procedure developed