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.     

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 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.     

Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems

In this paper, we present a weighted fuzzy interpolative reasoning method for sparse fuzzy rule-based systems, where the antecedent variables appearing in the fuzzy rules have different weights. We also present a weights-learning algorithm to automatically learn the optimal weights of the antecedent variables of the fuzzy rules for the proposed weighted fuzzy interpolative reasoning method. We also apply the proposed weighted fuzzy interpolative reasoning method and the proposed weights-learning algorithm to handle the truck backer-upper control problem. The experimental results show that the proposed fuzzy interpolative reasoning method using the optimally learned weights by the proposed weights-learning algorithm gets better truck backer-upper control results than the ones by the traditional fuzzy inference system and the existing fuzzy interpolative reasoning methods. The proposed method provides us with a useful way for fuzzy rules interpolation 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.     

稀疏规则条件下的相似插值推理研究

模糊推理本质上就是插值器。但在稀疏规则库的务件下，当输入的事实落入规则“空隙”时，采用传统的CRI方法是得不到任何推理结果的。而采用KH线性插值推理也存在着难以保证推理结果的凸性和正规性等问题。为了在稀疏规则条件下能有好的插值推理结果，提出了一种相似插值推理方法。谊方法能较好地保证推理结果隶属函数的凸性和正规性，这为智能系统中的模糊推理提供了一个十分有用的工具。     

稀疏规则条件下的线性插值推理研究

模糊推理本质上就是某种插值方法．但在稀疏规则库的条件下，当输入的事实落入规则“空隙”时，采用传统的CRI方法是得不到任何推理结果的．而采用KH线性插值推理也存在着难以保证推理结果的凸性和正规性等问题．在分析了Koczy和Hirota提出的线性插值推理方法的基础上，本文提出了一个新的线性插值推理的方法，该方法能很好地保证推理结果的凸性和正规性，这为智能系统中的模糊推理提供了一个十分有用的工具．     

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.     

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．     

多变量规则的线性插值推理方法

插值推理是稀疏规则条件下的一类重要的推理方法，单变量的情况已有较多研究，但针对多变量情况的研究还不多，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。多变量规则的插值推理是插值推理研究的重要方面，为了在多变量稀疏规则条件下能得到好的插值推理结果，本文对多变量规则的插值推理方法进行了研究，提出了一个多变量规则的线性插值推理方法。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。     

一种基于核集与相似性的模糊推理方法

在稀疏规则库条件下,当给定的输入落入规则"间隙"时,采用传统的模糊推理方法是得不到任何结论的.学者已经证明模糊推理本质上就是插值器.Koczy和Hirota首先提出了KH线性插值推理方法,然而推理结果存在着无法保证凸性和正规性等问题.为了能有一个较好的插值推理结果,本文提出了一种基于核集与相似性的模糊插值推理方法,并把此方法扩展到多维变量的情况,该方法不仅推理简单,推理结果较好,并且能很好地保证推理结果的凸性和正规性.这为智能系统中的模糊推理提供了一个非常有用的工具.     

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.     

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 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.     

A proposal for improving the accuracy of linguistic modeling

We propose accurate linguistic modeling, a methodology to design linguistic models that are accurate to a high degree and may be suitably interpreted. This approach is based on two main assumptions related to the interpolative reasoning developed by fuzzy rule-based systems: a small change in the structure of the linguistic model based on allowing the linguistic rule to have two consequents associated; and a different way to obtain the knowledge base based on generating a preliminary fuzzy rule set composed of a large number of rules and then selecting the subset of them best cooperating. Moreover, we introduce two variants of an automatic design method for these kinds of linguistic models based on two well-known inductive fuzzy rule generation processes and a genetic process for selecting rules. The accuracy of the proposed methods is compared with other linguistic modeling techniques with different characteristics when solving of three different applications     

Fuzzy if... then rule models and their transformation into oneanother

The two most essential fuzzy rule-based models used in the literature and in the industrial applications are briefly described. The way of reasoning in these models is shown. Interpolative reasoning for the case of sparse rule bases is also discussed. Rule base compression by eliminating redundant rules whose information can be reconstructed within a set accuracy interval from the remaining rules by using the previous interpolation method is shown. A general fuzzy model is discussed that contains the previous fuzzy models (as well as the nonfuzzy one) as special cases. Ways of transforming the different approximative models into each other via the general model and interpolation are presented     

Fuzzy rule base systems verification using high-level Petri nets

In this paper, we propose a Petri nets formalism for the verification of rule-based systems. Typical structural errors in a rule-based system are redundancy, inconsistency, incompleteness, and circularity. Since our verification is based on Petri nets and their incidence matrix, we need to transform rules into a Petri nets first, then derive an incidence matrix from the net. In order to let fuzzy rule-based systems detect above the structural errors, we are presenting a Petri-nets-based mechanism. This mechanism consists of three phases: rule normalization, rules transformation, and rule verification. Rules will be first normalized into Horn clauses, then transform the normalized rules into a high-level Petri net, and finally we verify these normalized rules. In addition, we are presenting our approach to simulate the truth conditions which still hold after a transition firing and negation in Petri nets for rule base modeling. In this paper, we refer to fuzzy rules as the rules with certainty factors, the degree of truth is computed in an algebraic form based on state equation which can be implemented in matrix computation in Petri nets. Therefore, the fuzzy reasoning problems can be transformed as the liner equation problems that can be solved in parallel. We have implemented a Petri nets tool to realize the mechanism presented fuzzy rules in this paper.     

基于优序图加权的多维稀疏模糊推理方法

在稀疏规则库条件下,多数的多维稀疏规则条件近似推理方法都难以保证推理结果的凸性和正规性,且没有考虑到多维变量对结论的影响权值。提出一种基于优序图加权的多维模糊推理方法,运用优序图确定权值,实验结果表明,该方法不仅减小推理结果的误差,而且能较好地保证推理结果的凸性和正规性。