Weighted Fuzzy Interpolative Reasoning Based on Weighted Increment Transformation and Weighted Ratio Transformation Techniques

In this paper, we present a new weighted fuzzy interpolative reasoning method for sparse fuzzy rule-based systems. The proposed method uses weighted increment transformation and weighted ratio transformation techniques to handle weighted fuzzy interpolative reasoning in sparse fuzzy rule-based systems. It allows each variable that appears in the antecedent parts of fuzzy rules to associate with a weight between zero and one. Moreover, we also propose an algorithm that automatically tunes the optimal weights of the antecedent variables appearing in the antecedent parts of fuzzy rules. We also apply the proposed weighted fuzzy interpolative reasoning method to handle the truck backer-upper control problem. The proposed weighted fuzzy interpolative reasoning method performs better than the ones obtained by the traditional fuzzy inference system (2000), Huang and Shen's method (2008), and Chen and Ko's method (2008). The proposed method provides us with a useful way to deal with weighted fuzzy interpolative reasoning in sparse fuzzy rule-based systems.      

Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the slopes of fuzzy sets

In this paper, we present a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the slopes of fuzzy sets. The proposed method can deal with fuzzy rules interpolation involving complex polygonal fuzzy sets with the advantages of simplest calculation and get more reasonable fuzzy interpolative reasoning results. We also make some experiments to compare the fuzzy interpolative reasoning results of the proposed method with the ones of the existing methods. The experimental results show that the proposed method outperforms the existing fuzzy interpolative reasoning methods for sparse fuzzy rule-based systems.     

Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets

Fuzzy interpolative reasoning is an important research topic of sparse fuzzy rule-based systems. In recent years, some methods have been presented for dealing with fuzzy interpolative reasoning. However, the involving fuzzy sets appearing in the antecedents of fuzzy rules of the existing fuzzy interpolative reasoning methods must be normal and non-overlapping. Moreover, the reasoning conclusions of the existing fuzzy interpolative reasoning methods sometimes become abnormal fuzzy sets. In this paper, in order to overcome the drawbacks of the existing fuzzy interpolative reasoning methods, we present a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets. The proposed fuzzy interpolative reasoning method can handle the situation of non-normal and overlapping fuzzy sets appearing in the antecedents of fuzzy rules. It can overcome the drawbacks of the existing fuzzy interpolative reasoning methods in sparse fuzzy rule-based systems.     

Fuzzy Interpolative Reasoning for Sparse Fuzzy-Rule-Based Systems Based on the Areas of Fuzzy Sets

Fuzzy interpolative reasoning is an inference technique for dealing with the sparse rules problem in sparse fuzzy-rule-based systems. In this paper, we present a new fuzzy interpolative reasoning method for sparse fuzzy-rule-based systems based on the areas of fuzzy sets. The proposed method uses the weighted average method to infer the fuzzy interpolative reasoning results and has the following advantages: 1) it holds the normality and the convexity of the fuzzy interpolative reasoning result, 2) it can deal with fuzzy interpolative reasoning with complicated membership functions, 3) it can deal with fuzzy interpolative reasoning when the fuzzy sets of the antecedents and the consequents of the fuzzy rules have different kinds of membership functions, 4) it can handle fuzzy interpolative reasoning with multiple antecedent variables, 5) it can handle fuzzy interpolative reasoning with multiple fuzzy rules, and 6) it can handle fuzzy interpolative reasoning with logically consistent properties with respect to the ratios of fuzziness. We use some examples to compare the fuzzy interpolative reasoning results of the proposed method with those of the existing fuzzy interpolative reasoning methods. In terms of the six evaluation indices, the experimental results show that the proposed method performs more reasonably than the existing methods. The proposed method provides us a useful way to deal with fuzzy interpolative reasoning in sparse fuzzy-rule-based systems.      

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.     

A fuzzy reasoning approach for rule-based systems based on fuzzylogics

This paper presents a weighted fuzzy reasoning algorithm for rule-based systems based on weighted fuzzy logics. The proposed algorithm allows the truth values of the conditions appearing in the antecedent portions of the rules, the certainty factors of the rules, and the weights of the conditions appearing in the antecedent portions of the rules to be represented by trapezoidal fuzzy numbers. Given the fuzzy truth values of some conditions, the algorithm can perform weighted fuzzy reasoning to evaluate the fuzzy truth values of other conditions automatically.     

Fuzzy Interpolative Reasoning for Sparse Fuzzy Rule-Based Systems Based on ${bm alpha}$-Cuts and Transformations Techniques

In sparse fuzzy rule-based systems, the fuzzy rule bases are usually incomplete. In this situation, the system may not properly perform fuzzy reasoning to get reasonable consequences. In order to overcome the drawback of sparse fuzzy rule-based systems, there is an increasing demand to develop fuzzy interpolative reasoning techniques in sparse fuzzy rule-based systems. In this paper, we present a new fuzzy interpolative reasoning method via cutting and transformation techniques for sparse fuzzy rule-based systems. It can produce more reasonable results than the existing methods. The proposed method provides a useful way to deal with fuzzy interpolative reasoning in sparse fuzzy rule-based systems.      

Weighted fuzzy reasoning using weighted fuzzy Petri nets

This paper presents a Weighted Fuzzy Petri Net model (WFPN) and proposes a weighted fuzzy reasoning algorithm for rule-based systems based on Weighted Fuzzy Petri Nets. The fuzzy production rules in the knowledge base of a rule-based system are modeled by Weighted Fuzzy Petri Nets, where the truth values of the propositions appearing in the fuzzy production rules and the certainty factors of the rules are represented by fuzzy numbers. Furthermore, the weights of the propositions appearing in the rules are also represented by fuzzy numbers. The proposed weighted fuzzy reasoning algorithm can allow the rule-based systems to perform fuzzy reasoning in a more flexible and more intelligent manner     

Fuzzy interpolative reasoning via scale and move transformations

Interpolative reasoning does not only help reduce the complexity of fuzzy models but also makes inference in sparse rule-based systems possible. This paper presents an interpolative reasoning method by means of scale and move transformations. It can be used to interpolate fuzzy rules involving complex polygon, Gaussian or other bell-shaped fuzzy membership functions. The method works by first constructing a new inference rule via manipulating two given adjacent rules, and then by using scale and move transformations to convert the intermediate inference results into the final derived conclusions. This method has three advantages thanks to the proposed transformations: 1) it can handle interpolation of multiple antecedent variables with simple computation; 2) it guarantees the uniqueness as well as normality and convexity of the resulting interpolated fuzzy sets; and 3) it suggests a variety of definitions for representative values, providing a degree of freedom to meet different requirements. Comparative experimental studies are provided to demonstrate the potential of this method.     

基于OWA算子的区间值加权模糊推理

针对如何对区间值模糊产生式规则赋予合理权值的问题,将OWA算子引入到区间值模糊推理中。介绍一种基于OWA算子的区间值赋权方法,根据此方法给出区间值模糊集上的加权模糊产生式规则的推理算法。在采用该算法的过程中,为合理地计算输入事实与规则前件的匹配程度,引入基于OWA算子的区间值模糊匹配函数值和总体贴近度的计算方法。实例分析表明了所给出的区间值模糊推理算法的有效性和可行性。     

Effect of rule weights in fuzzy rule-based classification systems

This paper examines the effect of rule weights in fuzzy rule-based classification systems. Each fuzzy IF-THEN rule in our classification system has antecedent linguistic values and a single consequent class. We use a fuzzy reasoning method based on a single winner rule in the classification phase. The winner rule for a new pattern is the fuzzy IF-THEN rule that has the maximum compatibility grade with the new pattern. When we use fuzzy IF-THEN rules with certainty grades, the winner is determined as the rule with the maximum product of the compatibility grade and the certainty grade. In this paper, the effect of rule weights is illustrated by drawing classification boundaries using fuzzy IF-THEN rules with/without certainty grades. It is also shown that certainty grades play an important role when a fuzzy rule-based classification system is a mixture of general rules and specific rules. Through computer simulations, we show that comprehensible fuzzy rule-based systems with high classification performance can be designed without modifying the membership functions of antecedent linguistic values when we use fuzzy IF-THEN rules with certainty grades     

Fuzzy arithmetic-based interpolative reasoning for nonlinear dynamic fuzzy systems

FAIR (fuzzy arithmetic-based interpolative reasoning)—a fuzzy reasoning scheme based on fuzzy arithmetic, is presented here. Linguistic rules of the Mamdani type, with fuzzy numbers as consequents, are used in an inference mechanism similar to that of a Takagi–Sugeno model. The inference result is a weighted sum of fuzzy numbers, calculated by means of the extension principle. Both fuzzy and crisp inputs and outputs can be used, and the chaining of rule bases is supported without increasing the spread of the output fuzzy sets in each step. This provides a setting for modeling dynamic fuzzy systems using fuzzy recursion. The matching in the rule antecedents is done by means of a compatibility measure that can be selected to suit the application at hand. Different compatibility measures can be used for different antecedent variables, and reasoning with sparse rule bases is supported. The application of FAIR to the modeling of a nonlinear dynamic system based on a combination of knowledge-driven and data-driven approaches is presented as an example.     

Fuzzy Interpolation and Extrapolation: A Practical Approach

Fuzzy interpolation does not only help to reduce the complexity of fuzzy models, but also makes inference in sparse rule-based systems possible. It has been successfully applied to systems control, but limited work exists for its applications to tasks like prediction and classification. Almost all fuzzy interpolation techniques in the literature make strong assumptions that there are two closest adjacent rules available to the observation, and that such rules must flank the observation for each attribute. Also, some interpolation approaches cannot handle fuzzy sets whose membership functions involve vertical slopes. To avoid such limitations and develop a more practical approach, this paper extends the work of Huang and Shen. The result enables both interpolation and extrapolation which involve multiple fuzzy rules, with each rule consisting of multiple antecedents. Two realistic applications, namely truck backer-upper control and computer activity prediction, are provided in this paper to demonstrate the utility of the extended approach. Experiment-based comparisons to the most commonly used Mamdani fuzzy reasoning mechanism, and to other existing fuzzy interpolation techniques are given to show the significance and potential of this research.     

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.     

Fuzzy backward reasoning using fuzzy Petri nets

Chen, Ke and Chang (1990) have presented a fuzzy forward reasoning algorithm for rule-based systems using fuzzy Petri nets. In this paper, we extend the work of Chen, Ke and Chang (1990) to present a fuzzy backward reasoning algorithm for rule-based systems using fuzzy Petri nets, where the fuzzy production rules of a rule-based system are represented by fuzzy Petri nets. The system can perform fuzzy backward reasoning automatically to evaluate the degree of truth of any proposition specified by the user. The fuzzy backward reasoning capability allows the computers to perform reasoning in a more flexible manner and to think more like people.     

Hybridization of fuzzy GBML approaches for pattern classification problems

We propose a hybrid algorithm of two fuzzy genetics-based machine learning approaches (i.e., Michigan and Pittsburgh) for designing fuzzy rule-based classification systems. First, we examine the search ability of each approach to efficiently find fuzzy rule-based systems with high classification accuracy. It is clearly demonstrated that each approach has its own advantages and disadvantages. Next, we combine these two approaches into a single hybrid algorithm. Our hybrid algorithm is based on the Pittsburgh approach where a set of fuzzy rules is handled as an individual. Genetic operations for generating new fuzzy rules in the Michigan approach are utilized as a kind of heuristic mutation for partially modifying each rule set. Then, we compare our hybrid algorithm with the Michigan and Pittsburgh approaches. Experimental results show that our hybrid algorithm has higher search ability. The necessity of a heuristic specification method of antecedent fuzzy sets is also demonstrated by computational experiments on high-dimensional problems. Finally, we examine the generalization ability of fuzzy rule-based classification systems designed by our hybrid algorithm.     

Learning fuzzy inference systems using an adaptive membershipfunction scheme

An adaptive membership function scheme for general additive fuzzy systems is proposed in this paper. The proposed scheme can adapt a proper membership function for any nonlinear input-output mapping, based upon a minimum number of rules and an initial approximate membership function. This parameter adjustment procedure is performed by computing the error between the actual and the desired decision surface. Using the proposed adaptive scheme for fuzzy system, the number of rules can be minimized. Nonlinear function approximation and truck backer-upper control system are employed to demonstrate the viability of the proposed method.     

一种新的多维稀疏规则模糊插值推理方法

经典的插值理论针对一维稀疏规则库的条件，提出了各种不同的插值方法，取得了很多很好的经验．但对多维稀疏规则条件的近似推理，研究很少，仅有的几种插值方法，存在着难以保证推理结果的凸性和正规性等问题．为了在多维稀疏规则条件下能得到好的插值推理结果。提出了一种基于几何相似的插值推理方法．该方法能较好地保证推理结果隶属函数的凸性和正规性，为智能系统中的模糊推理提供了一个十分有用的工具．     

A multilevel weighted fuzzy reasoning algorithm for expert systems

The applications of fuzzy production rules (FPR) are rather limited if the relative degree of importance of each proposition in the antecedent contributing to the consequent (i.e., the weight) is ignored or assumed to be equal. Unfortunately, this is the case for many existing FPR and most existing fuzzy expert system development shells or environments offer no such functionality for users to incorporate different weights in the antecedent of FPR. This paper proposes to assign a weight parameter to each proposition in the antecedent of a FPR and a new fuzzy production rule evaluation method (FPREM) which generalizes the traditional method by taking the weight factors into consideration is devised. Furthermore, a multilevel weighted fuzzy reasoning algorithm (MLWFRA) incorporating this new FPREM, which is based on the reachability and adjacent place characteristics of a fuzzy Petri net, is developed. The MLWFRA has the advantages that i) it offers multilevel reasoning capability; ii) it allows multiple conclusions to be drawn if they exist; iii) it offers a new fuzzy production rule evaluation method; and iv) it is capable of detecting cycle rules     

Generating weighted fuzzy rules from relational database systems for estimating values using genetic algorithms

In recent years, some methods have been proposed to estimate values in relational database systems. However, the estimated accuracy of the existing methods are not good enough. In this paper, we present a new method to generate weighted fuzzy rules from relational database systems for estimating values using genetic algorithms (GAs), where the attributes appearing in the antecedent part of generated fuzzy rules have different weights. After a predefined number of evolutions of the GA, the best chromosome contains the optimal weights of the attributes, and they can be translated into a set of rules to be used for estimating values. The proposed method can get a higher average estimated accuracy rate than the methods we presented in two previous papers.