Checking the coherence and redundancy of fuzzy knowledge bases

Checking the coherence of a set of rules is an important step in knowledge base validation. Coherence is also needed in the field of fuzzy systems. Indeed, rules are often used regardless of their semantics, and it sometimes leads to sets of rules that make no sense. Avoiding redundancy is also of interest in real-time systems for which the inference engine is time consuming. A knowledge base is potentially inconsistent or incoherent if there exists a piece of input data that respects integrity constraints and that leads to logical inconsistency when added to the knowledge base. We more particularly consider knowledge bases composed of parallel fuzzy rules. Then, coherence means that the projection on the input variables of the conjunctive combination of the possibility distributions representing the fuzzy rules leaves these variables completely unrestricted (i.e., any value for these variables is possible) or, at least, not more restrictive than integrity constraints. Fuzzy rule representations can be implication-based or conjunction-based; we show that only implication-based models may lead to coherence problems. However, unlike conjunction-based models, they allow to design coherence checking processes. Some conditions that a set of parallel rules has to satisfy in order to avoid inconsistency problems are given for certainty or gradual rules. The problem of redundancy, which is also of interest for fuzzy knowledge bases validation, is addressed for these two kinds of rules     

Modeling Connectionist Neuro-Fuzzy network and Applications

The main objective of this paper is to propose a Neuro-Fuzzy network, which can model a system from input–output data by automatically dividing the input–output space and extracting fuzzy if-then rules from numerical data. The structure of the network is simple with input, rule and output layers only. The connections between input and rule layer is used to derive the membership functions of the fuzzified part. Kohonens self-organizing learning algorithm is applied to partition the pattern space. Using this algorithm, similar rules are mapped close by and extraction of if-then rules is made easy. It can also adapt to a number of rules automatically. The proposed network is verified for three benchmark applications. Experimental results show that the adaptive method discussed in this paper not only trains in a few hundred iterations but also provides better performance measures when compared to conventional methods.     

Designing fuzzy inference systems from data: Aninterpretability-oriented review

Fuzzy inference systems (FIS) are widely used for process simulation or control. They can be designed either from expert knowledge or from data. For complex systems, FIS based on expert knowledge only may suffer from a loss of accuracy. This is the main incentive for using fuzzy rules inferred from data. Designing a FIS from data can be decomposed into two main phases: automatic rule generation and system optimization. Rule generation leads to a basic system with a given space partitioning and the corresponding set of rules. System optimization can be done at various levels. Variable selection can be an overall selection or it can be managed rule by rule. Rule base optimization aims to select the most useful rules and to optimize rule conclusions. Space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters. Structure optimization is of a major importance: selecting variables, reducing the rule base and optimizing the number of fuzzy sets. Over the years, many methods have become available for designing FIS from data. Their efficiency is usually characterized by a numerical performance index. However, for human-computer cooperation another criterion is needed: the rule interpretability. An implicit assumption states that fuzzy rules are by nature easy to be interpreted. This could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts. The paper analyzes the main methods for automatic rule generation and structure optimization. They are grouped into several families and compared according to the rule interpretability criterion. For this purpose, three conditions for a set of rules to be interpretable are defined     

Simplification of a fuzzy mechanism with rule combination

 In this paper, a systematic approach to reduce the complexity of a fuzzy controller with the rule combination of a fuzzy rule base is presented. The complexity of a fuzzy controller is defined to be the computation load in this work. The proposed rule combination approach can be applied to the fuzzy mechanisms with product–sum and min–max inferences. With the input membership functions indexed in sequence for each input variable, the n-dimensional fuzzy rule table is represented as vectors so that the combination of the fuzzy rule base is realizable. Then the adjacent fuzzy rules with the same output consequent are combined to have smaller size of fuzzy rule base. The fuzzy mechanism with the combined rule table is shown to have the same output with the original fuzzy mechanism (without rule combination). Thus, in many applications, the rule combination approach presented in this paper can be used to reduce the complexity of the fuzzy mechanism without degrading the performances. Moreover, the Don't Care fuzzy rules are defined and it is indicated that the number of the necessary fuzzy rules might be decreased when the Don't Care fuzzy rules are taken into consideration. Further, the properties of the simplification approach for the fuzzy rule base of the fuzzy mechanism are discussed.     

Trace-driven knowledge acquisition (TDKA) for rule-based real time scheduling systems

Expert scheduling systems, which develop the schedule automatically on a real time basis, are able to respond to the changes of product demand in Flexible Manufacturing Systems (FMS). While developing an expert scheduling system, the most time-consuming and difficult step is knowledge acquisition, the process that elicits the knowledge from experts and transfers it into the knowledge base. A trace-driven knowledge acquisition (TDKA) method is proposed to extract the expertise from the schedules produced by expert schedulers. Three phases are involved in the TDKA process: data collection, data analysis, and rule evaluation. In data collection, the expert schedulers are identified and decisions made during the scheduling process are recorded as a trace. In data analysis, a set of scheduling rules is developed based on the trace. The rules are then evaluated in the last phase. If the resulting rules do not perform as well as the expert schedulers, the process returns to phase two and refines the rules. The whole process stops whenever the resulting rules perform at least as well as the expert schedulers. A circuit board production line is used to demonstrate the feasibility of the TDKA methodology. The scheduling rules perform much better than the expert schedulers from whom the rules are extracted.     

Self-spawning neuro-fuzzy system for rule extraction

In this paper we propose self-spawning neuro-fuzzy system (SSNFS), a new neuro-fuzzy system to derive fuzzy rules from data. The SSNFS model is based on a generic definition of incremental perceptron and a new learning algorithm that is capable of both structural (rule) learning and parametric learning. It constructs the fuzzy system by detecting a suitable number of rule patches and their positions and shapes in the input space. Initially the rule base consists of one single fuzzy rule; during the iterative learning process the rule base expands according to a supervised spawning validity measure. The rule induction process terminates when a given stop criterion is satisfied. SSNFS is very general since it does not require the prior knowledge about the input space and can be used in any application based on the scatter-partitioning fuzzy system. To demonstrate the effectiveness and applicability of our algorithm, we present a synthetic example and real-world modelling problems.

Applying the JBOS reduction method for relevant knowledge extraction

This work presents results from an experiment used to assess the JBOS (junction based on objects similarity) reduction method. Two reductions were made of a formal context about patients having symptoms in a tuberculosis data base. The first reduction used the knowledge expressed in the original formal context and the second used the knowledge expressed in expert rules. The assessment was made, in the first case, by comparison of the performances of the sets of extracted rules (stem bases) before and after the reduction, and in the second case, by comparison of the performances of the set of extracted rules after reduction with that of the expert rules. The performance in the first case was exactly the same as before reduction. In the second case the performance even improved, showing that the weighting process, besides incorporating the expert knowledge, resulted in rules well adjusted to the knowledge expressed in the original formal context. So, both reductions resulted in rule sets absolutely consistent with the original ones. The expert rules, FCA rules and both set of rules obtained after reduction were used also to classify patients of a validation set. In this case, the results have shown that the performance was the same before and after reduction. Therefore, it was shown that by means of an appropriate attributes weight assignment it is possible, by the JBOS method, to achieve a suitable level of performance in a specific task after reduction.     

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.     

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.     

Integrating fuzzy knowledge by genetic algorithms

We propose a genetic algorithm-based fuzzy knowledge integration framework that can simultaneously integrate multiple fuzzy rule sets and their membership function sets. The proposed approach consists of two phases: fuzzy knowledge encoding and fuzzy knowledge integration. In the encoding phase, each fuzzy rule set with its associated membership functions is first transformed into an intermediary representation and then further encoded as a string. The combined strings form an initial knowledge population, which is then ready for integration. In the knowledge-integration phase, a genetic algorithm is used to generate an optimal or nearly optimal set of fuzzy rules and membership functions from the initial knowledge population. Two application domains, the hepatitis diagnosis and the sugarcane breeding prediction, were used to show the performance of the proposed knowledge-integration approach. Results show that the fuzzy knowledge base derived using our approach performs better than every individual knowledge base     

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.     

Evolutionary design of a fuzzy classifier from data

Genetic algorithms show powerful capabilities for automatically designing fuzzy systems from data, but many proposed methods must be subjected to some minimal structure assumptions, such as rule base size. In this paper, we also address the design of fuzzy systems from data. A new evolutionary approach is proposed for deriving a compact fuzzy classification system directly from data without any a priori knowledge or assumptions on the distribution of the data. At the beginning of the algorithm, the fuzzy classifier is empty with no rules in the rule base and no membership functions assigned to fuzzy variables. Then, rules and membership functions are automatically created and optimized in an evolutionary process. To accomplish this, parameters of the variable input spread inference training (VISIT) algorithm are used to code fuzzy systems on the training data set. Therefore, we can derive each individual fuzzy system via the VISIT algorithm, and then search the best one via genetic operations. To evaluate the fuzzy classifier, a fuzzy expert system acts as the fitness function. This fuzzy expert system can effectively evaluate the accuracy and compactness at the same time. In the application section, we consider four benchmark classification problems: the iris data, wine data, Wisconsin breast cancer data, and Pima Indian diabetes data. Comparisons of our method with others in the literature show the effectiveness of the proposed method.     

SIGNIFICANT FUZZY RULES EXTRACTION BY AN SVD–QR-BASED APPROACH

ABSTRACT

In this article, an SVD–QR-based approach is proposed to extract the important fuzzy rules from a rule base with several fuzzy rule tables to design an appropriate fuzzy system directly from some input-output data of the identified system. A fuzzy system with fuzzy rule tables is defined to approach the input-output pairs of an identified system. In the rule base of the defined fuzzy system, each fuzzy rule table corresponds to a partition of an input space. In order to extract the important fuzzy rules from the rule base of the defined fuzzy system, a firing strength matrix determined by the membership functions of the premise fuzzy sets is constructed. According to the firing strength matrix, the number of important fuzzy rules is determined by the Singular Value Decomposition SVD, and the important fuzzy rules are selected by the SVD–QR-based method. Consequently, a reconstructed fuzzy rule base composed of significant fuzzy rules is determined by the firing strength matrix. Furthermore, the recursive least-squares method is applied to determine the consequent part of the reconstructed fuzzy system according to the gathered input-output data so that a fine fuzzy system is determined by the proposed method. Finally, three nonlinear systems illustrate the efficiency of the proposed method.     

SOGARG: A self-organized genetic algorithm-based rule generation scheme for fuzzy controllers

This paper presents a self-organized genetic algorithm-based rule generation (SOGARG) method for fuzzy logic controllers. It is a three-stage hierarchical scheme that does not require any expert knowledge and input-output data. The first stage selects rules required to control the system in the vicinity of the set point. The second stage extends this to the entire input space, giving a rulebase that can bring the system to its set point from almost all initial states. The third stage refines the rulebase and reduces the number of rules. The first two stages use the same fitness function whose aim is only to acquire the controllability, but the last stage uses a different one, which attempts to optimize both the settling time and number of rules. The effectiveness of SOGARG is demonstrated using an inverted pendulum and the truck reversing.     

一类提取模糊规则的新方法及其在干燥建模中的应用

基于系统的输入－输出数据,提出一种通过划分输入空间撮模糊规则的方法,并将该方法应用于建模非线性程度较高物干燥过程。建模过程中,首先利用训练样本将输入空间动态地划分为若干个子空间,然后针对第一个子空间了生一条模糊规则,再将所产生的模糊规则共成一个模糊规则库,模糊逻辑系统的最终输出从模糊规则库中产生。仿真结果表明,该方法可很好地预测干燥系统的物料降水率,而且简单实用,十分可靠。     

A characteristic-point-based fuzzy inference system aimed to minimize the number of fuzzy rules

This paper presents a characteristic-point-based fuzzy inference system (CPFIS) for fuzzy modeling from training data. The aim of the CPFIS is not only satisfactory precision performance, but also to employ as few purely linguistic fuzzy rules as possible by using a minimization-based systematic training method. Characteristic points (CPs) are defined as the few data points among the original training data which, when they are directly mapped to fuzzy rules and thus form the entire rule base, allow the underlying system to be effectively modeled. Three minimization-based algorithms in a sequence are proposed to train the CPFIS: a gradient-projection method, a Gauss-Jordan-elimination-based column elimination, and back-propagation. The CPs are determined by iterative computations of the first two minimization algorithms, after which the resulting fuzzy sets are further fine-tuned by the third algorithm. Experiments conducted on three benchmark problems showed that the CPFIS used one of the smallest number of fuzzy rules among the reported results for other methods. The Gaussian membership functions in both the input and output fuzzy sets and the small number of fuzzy rules make the rule interpretation of the CPFIS much easier than that of other methods, thus enhancing human-computer cooperation in knowledge discovery.     

A Meteorological Fuzzy Expert System Incorporating Subjective User Input

We present a fuzzy expert system, MEDEX, for forecasting gale-force winds in the Mediterranean basin. The most successful local wind forecasting in this region is achieved by an expert human forecaster with access to numerical weather prediction products. That forecaster's knowledge is expressed as a set of ‘rules-of-thumb’. Fuzzy set methodologies have proved well suited for encoding the forecaster's knowledge, and for accommodating the uncertainty inherent in the specification of rules, as well as in subjective and objective input. MEDEX uses fuzzy set theory in two ways: as a fuzzy rule base in the expert system, and for fuzzy pattern matching to select dominant wind circulation patterns as one input to the expert system. The system was developed, tuned, and verified over a two-year period, during which the weather conditions from 539 days were individually analyzed. Evaluations of MEDEX performance for both the onset and cessation of winter and summer winds are presented, and demonstrate that MEDEX has forecasting skill competitive with the US Navy's regional forecasting center in Rota, Spain. Received 5 May 1999 / Revised 8 August 2000 / Accepted in revised form 23 April 2001     

The use of knowledge acquisition in instructional design

Expert systems and knowledge based systems have emerged from “esoteric” laboratory research in Artificial Intelligence (AI) to become an important tool for approaching real world problems. Expert systems are distinctive in that they are designed to address problems in a similar manner and with similar results as a human expert. The basic structure of an expert system is comprised of three functionally separate components: (a) knowledge base, which contains a representation of domain related facts; (b) means of knowledge base use to solve a problem, inference mechanism; and (c) working memory, which records the input data and progress for each problem. Given the complexity and cost of expert system construction, it is imperative that system developers and researchers attend to research issues which are critical to knowledge engineering. These questions can be categorized according to the parts of an expert system: (a) knowledge representation; (b) knowledge utilization; and (c) knowledge acquisition. A knowledge acquisition procedure is presented which displays the relationship between subject matter expert expertise consisting of declarative knowledge, procedural knowledge, heuristics, formal rules, and meta-rules. The knowledge engineer uses one or a combination of elicitation methods to gather relevant data to eventually build the components of an expert system. Further explained are the acquisition methods: (a) structured interview; (b) verbal reports; (c) teaching the subject matter; (d) observation; and (e) automated knowledge acquisition tools. The paper concludes with a discussion of the future research issues concerned with using knowledge mapping and task analysis vs. knowledge acquisition techniques.     

Fuzzy rule interpolation for multidimensional input spaces with applications: a case study

Fuzzy rule based systems have been very popular in many engineering applications. However, when generating fuzzy rules from the available information, this may result in a sparse fuzzy rule base. Fuzzy rule interpolation techniques have been established to solve the problems encountered in processing sparse fuzzy rule bases. In most engineering applications, the use of more than one input variable is common, however, the majority of the fuzzy rule interpolation techniques only present detailed analysis to one input variable case. This paper investigates characteristics of two selected fuzzy rule interpolation techniques for multidimensional input spaces and proposes an improved fuzzy rule interpolation technique to handle multidimensional input spaces. The three methods are compared by means of application examples in the field of petroleum engineering and mineral processing. The results show that the proposed fuzzy rule interpolation technique for multidimensional input spaces can be used in engineering applications.     

A graph-based approach for rule integrity and maintainability in expert system maintenance

Just as conventional software systems have maintenance costs far exceeding development costs, so too do rule-based expert systems. They are frequently developed by an incremental and iterative method, where knowledge and decision rules are extracted and added to the system in a piecemeal manner throughout system evolution. Thus, ensuring the correctness and consistency of the rule base (RB) becomes an important, though challenging task. However, most research work in expert systems has focused on building and validating rule bases, leaving the maintenance issue unexplored. We propose a graph-based approach, called the object classification model (OCM), as a methodology for RB maintenance. An experiment was conducted to compare the OCM with traditional RB maintenance methods. The results show that the OCM helps knowledge engineers retain rule-base integrity and, thus, increase rule-base maintainability.