Performance evaluation of fuzzy classifier systems formultidimensional pattern classification problems

We examine the performance of a fuzzy genetics-based machine learning method for multidimensional pattern classification problems with continuous attributes. In our method, each fuzzy if-then rule is handled as an individual, and a fitness value is assigned to each rule. Thus, our method can be viewed as a classifier system. In this paper, we first describe fuzzy if-then rules and fuzzy reasoning for pattern classification problems. Then we explain a genetics-based machine learning method that automatically generates fuzzy if-then rules for pattern classification problems from numerical data. Because our method uses linguistic values with fixed membership functions as antecedent fuzzy sets, a linguistic interpretation of each fuzzy if-then rule is easily obtained. The fixed membership functions also lead to a simple implementation of our method as a computer program. The simplicity of implementation and the linguistic interpretation of the generated fuzzy if-then rules are the main characteristic features of our method. The performance of our method is evaluated by computer simulations on some well-known test problems. While our method involves no tuning mechanism of membership functions, it works very well in comparison with other classification methods such as nonfuzzy machine learning techniques and neural networks.     

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.     

A Strategy of Dynamic Reasoning in Knowledge-Based System with Fuzzy Production Rules

Fuzzy production rules have been successfully applied to represent uncertainty in a knowledge-based system. The knowledge organized as a knowledge base is static. On the other hand, a real system such as the stock market is dynamic in nature. Therefore we need a strategy to reflect the dynamic nature of a system when we make reasoning with a knowledge-based system.This paper proposes a strategy of dynamic reasoning that can be used to takes account the dynamic behavior of decision-making with the knowledge-based system consisted of fuzzy rules. A degree of match (DM) between actual input information and antecedent of a rule is represented by a value in interval [0, 1]. Weights of relative importance of attributes in a rule are obtained by the AHP (Analytic Hierarchy Process) method. Then these weights are applied as exponents for the DM, and the DMs in a rule are combined, with the Min operator, into a single DM for the rule. In this way, the importance of attributes of a rule, which can be changed from time to time, can be reflected to reasoning in knowledge-based system with fuzzy rules.With the proposed reasoning procedure, a decision maker can take his judgment on the given decision environment into a static knowledge base with fuzzy rules when he makes decision with the knowledge base. This procedure can be automated as a pre-processing system for fuzzy expert systems. Thereby the quality of decisions could be enhanced.     

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.     

Belief rule-base inference methodology using the evidential reasoning Approach-RIMER

In this paper, a generic rule-base inference methodology using the evidential reasoning (RIMER) approach is proposed. Existing knowledge-base structures are first examined, and knowledge representation schemes under uncertainty are then briefly analyzed. Based on this analysis, a new knowledge representation scheme in a rule base is proposed using a belief structure. In this scheme, a rule base is designed with belief degrees embedded in all possible consequents of a rule. Such a rule base is capable of capturing vagueness, incompleteness, and nonlinear causal relationships, while traditional if-then rules can be represented as a special case. Other knowledge representation parameters such as the weights of both attributes and rules are also investigated in the scheme. In an established rule base, an input to an antecedent attribute is transformed into a belief distribution. Subsequently, inference in such a rule base is implemented using the evidential reasoning (ER) approach. The scheme is further extended to inference in hierarchical rule bases. A numerical study is provided to illustrate the potential applications of the proposed methodology.     

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。     

Fuzzy knowledge representation and reasoning using a generalized fuzzy petri net and a similarity measure

In the study of weighted fuzzy production rules (WFPRs) reasoning, we often need to consider those rules whose consequences are represented by two or more propositions connected by “AND” or “OR”. To enhance the representation capability of those rules, this paper proposes two types of knowledge representation parameters, namely, the input weight and the output weight, for a rule. A Generalized Fuzzy Petri Net (GFPN) is also presented for WFPR reasoning. Furthermore, this paper gives a similarity measure to improve the evaluation method of WFPRs and the multilevel fuzzy reasoning in which the consequences and their certainty factors are deduced synchronously by using a GFPN.     

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.     

Lipschitz聚合算子的模糊推理鲁棒性研究

模糊推理中,输入和推理规则发生摄动时,内部连接算子的选择是影响推理输出的主要因素。给出了模糊Lipschitz聚合算子的定义,论证了满足Lipschitz条件的三角模算子和蕴涵算子,研究了一类稳定的Lipschitz聚合算子对模糊推理的鲁棒性影响,指出了当系统发生输入摄动和规则摄动时,内部连接算子为1-Lipschitz算子,能有效地抑制模糊推理的输出摄动,特别是当内部连接算子既是1-k∞-Lipschitz又是quasi-copulas时,模糊推理输出更稳定安全可行,模糊推理的鲁棒性得到更好调控;另外,从实验结果看,规则摄动对推理输出影响较大。实验部分既是对文中所提理论的很好验证,同时也是该理论在图像处理和人脸联想方面的具体应用。     

Learning fuzzy rules for similarity assessment in case-based reasoning

Fundamental to case-based reasoning is the assumption that similar problems have similar solutions. The meaning of the concept of “similarity” can vary in different situations and remains an issue. This paper proposes a novel similarity model consisting of fuzzy rules to represent the semantics and evaluation criteria for similarity. We believe that fuzzy if-then rules present a more powerful and flexible means to capture domain knowledge for utility oriented similarity modeling than traditional similarity measures based on feature weighting. Fuzzy rule-based reasoning is utilized as a case matching mechanism to determine whether and to which extent a known case in the case library is similar to a given problem in query. Further, we explain that such fuzzy rules for similarity assessment can be learned from the case library using genetic algorithms. The key to this is pair-wise comparisons of cases with known solutions in the case library such that sufficient training samples can be derived for genetic-based fuzzy rule learning. The evaluations conducted have shown the superiority of the proposed method in similarity modeling over traditional schemes as well as the feasibility of learning fuzzy similarity rules from a rather small case base while still yielding competent system performance.     

Embedding defaults into terminological knowledge representation formalisms

We consider the problem of integrating Reiter's default logic into terminological representation systems. It turns out that such an integration is less straightforward than we expected, considering the fact that the terminological language is a decidable sublanguage of first-order logic. Semantically, one has the unpleasant effect that the consequences of a terminological default theory may be rather unintuitive, and may even vary with the syntactic structure of equivalent concept expressions. This is due to the unsatisfactory treatment of open defaults via Skolemization in Reiter's semantics. On the algorithmic side, we show that this treatment may lead to an undecidable default consequence relation, even though our base language is decidable, and we have only finitely many (open) defaults. Because of these problems, we then consider a restricted semantics for open defaults in our terminological default theories: default rules are applied only to individuals that are explicitly present in the knowledge base. In this semantics it is possible to compute all extensions of a finite terminological default theory, which means that this type of default reasoning is decidable. We describe an algorithm for computing extensions and show how the inference procedures of terminological systems can be modified to give optimal support to this algorithm.This is a revised and extended version of a paper presented at the3rd International Conference on Principles of Knowledge Representation and Reasoning, October 1992, Cambridge, MA.     

某武器系统电控设备的故障诊断专家系统

以Visual Basic6.0为开发环境，Access97为数据库结构形式，DAO为数据库访问技术，开发了某武器系统电控设备故障诊断专家系统，介绍了系统的功能组成和实现方法。研究了一般产生式规则与模糊产生式规则相结合的知识表示方法以及精确推理与模糊推理相结合、基于规则的推理和基于实例的推理相结合的推理机制。     

A fuzzy expert system shell using both exact and inexact reasoning

This paper presents a comprehensive expert system shell which can deal with both exact and inexact reasoning. A prototype of this proposed shell, code named as SYSTEM Z-IIe, has been implemented successfully. It is a rule-based system which employs fuzzy logic and numbers for its reasoning. Two basic inexact concepts, fuzziness and uncertainty, are both used and distinct from each other clearly in the system. Moreover, these two concepts have been built into two levels for inexact reasoning, i.e. the level of the rules and facts, and the level of the values of the objects of these rules and facts. Other features of Z-IIe include multiple fuzzy propositions in rules and dual fact input mechanisms. It also allows any combinations of fuzzy and normal terms and uncertainties. Fuzzy numeric comparison logic control is also available for the rules and facts. Its natural language interface which uses English with restricted syntax improves the efficiency of knowledge engineering. Z-IIe is also coupled to a Database Management System for supplying facts from existing databases if appropriate. All these features can be combined to build very powerful expert systems and are illustrated by an example.     

A fuzzy neural network with fuzzy impact grades

Fuzzy rule derivation is often difficult and time-consuming, and requires expert knowledge. This creates a common bottleneck in fuzzy system design. In order to solve this problem, many fuzzy systems that automatically generate fuzzy rules from numerical data have been proposed. In this paper, we propose a fuzzy neural network based on mutual subsethood (MSBFNN) and its fuzzy rule identification algorithms. In our approach, fuzzy rules are described by different fuzzy sets. For each fuzzy set representing a fuzzy rule, the universe of discourse is defined as the summation of weighted membership grades of input linguistic terms that associate with the given fuzzy rule. In this manner, MSBFNN fully considers the contribution of input variables to the joint firing strength of fuzzy rules. Afterwards, the proposed fuzzy neural network quantifies the impacts of fuzzy rules on the consequent parts by fuzzy connections based on mutual subsethood. Furthermore, to enhance the knowledge representation and interpretation of the rules, a linear transformation from consequent parts to output is incorporated into MSBFNN so that higher accuracy can be achieved. In the parameter identification phase, the backpropagation algorithm is employed, and proper linear transformation is also determined dynamically. To demonstrate the capability of the MSBFNN, simulations in different areas including classification, regression and time series prediction are conducted. The proposed MSBFNN shows encouraging performance when benchmarked against other models.     

模糊推理Petri网及其在故障诊断中的应用

分析了推理Petri网与传统Petri网的共性和区别,给出了模糊产生式规则推理 Petri网模型.在此基础上,给出了有效的推理算法,并以极大代数矩阵算子进行了形式化表 示,此算法充分利用了Petri网的数学理论基础和描述并发系统的能力,具有并行推理能力, 可以同时得到推理后系统的全部状态值.最后举例说明了其在故障诊断中的应用.     

一种基于改进T-S模糊推理的模糊神经网络学习算法

针对模糊神经网络学习算法计算量过大,在预测模型设计中提出了基于改进T-S模糊推理的模糊神经网络学习算法。主要工作如下:首先,改进T-S模糊推理方法,定义基于偏移率的T-s模糊推理方法;然后,通过将此模糊推理方法与基于合成规则的模糊推理方法及距离型模糊推理方法相比较可以看出,所提方法有较少的计算量,且比较有效;最后,在此基础上改善了模糊神经网络学习算法,并将其应用于天气预测与安全态势预测。测试结果表明,该方法明显改善了学习效率,减少了预测模型设计中的学习次数与时间复杂度,并降低了学习误差。     

基于信息熵Petri网模型的知识发现研究

为描述命题和规则的可信度,定义了命题和规则的可信度信息熵。从熵的角度研究产生式规则中的不确定性推理,应用Petri网和可信度信息熵,建立了一类新的Information Entropy Petri网模型(IEPN),介绍了不确定性知识表示和推理步骤。同时分析IEPN推理对知识发现(KDD)的指导意义,并给出了IEPN推理过程及知识发现(KDK)方法。     

规则摄动时模糊蕴涵算子对模糊推理的鲁棒性的影响

列举了模糊规则发生摄动的常见情形,建立了一般性的模糊推理算法对规则摄动的鲁棒性的概念;就多重、链式和多维模糊推理情形,重点研究了一般性的模糊蕴涵算子对几个重要的模糊推理算法的这种鲁棒性的影响,并分别给出了相应的充分必要条件;初步尝试了通过一定的摄动制约来改善这种鲁棒性;同时指出了很多现有的模糊蕴涵算子使得所讨论的这些推理算法拥有好的鲁棒性,此时,即使规则中的隶属度有适度的粗糙或摄动,推理仍是可行的、安全的．文中工作对模糊系统的分析、模糊蕴涵算子的选择以及规则获取过程有一定的指导意义．     

Probabilistic Reasoning Under Coherence in System P

In this paper we apply a probabilistic reasoning under coherence to System P. We consider a notion of strict probabilistic consistency, we show its equivalence to Adams' probabilistic consistency, and we give a necessary and sufficient condition for probabilistic entailment. We consider the inference rules of System P in the framework of coherent imprecise probabilistic assessments. Exploiting our coherence-based approach, we propagate the lower and upper probability bounds associated with the conditional assertions of a given knowledge base, obtaining the precise probability bounds for the derived conclusions of the inference rules. This allows a more flexible and realistic use of System P in default reasoning and provides an exact illustration of the degradation of the inference rules when interpreted in probabilistic terms. We also examine the disjunctive Weak Rational Monotony rule of System P⁺ proposed by Adams in his extended probabilistic logic. Finally, we examine the propagation of lower bounds with real -values and, to illustrate our probabilistic reasoning, we consider an example.     

基于数值型数据的模糊规则快速挖掘方法

针对数值型数据能准确反应现实世界,但难以理解的问题,为了从数值型数据中挖掘出易于理解的知识,提出了基于数值型数据的模糊规则快速挖掘方法。该方法能从数值型数据中挖掘出一个零阶的Sugeno模糊规则,并采用一种启发式方法将这个零阶的Sugeno模糊规则的数值结论转变为两个带置信度的语言变量,并给出了规则库的存储结构。最后通过实例证明了这种快速模糊规则挖掘方法能避免复杂的数值型计算和能有效逼近非线性函数的优点．