A fuzzy coherent rule mining algorithm

In real-world applications, transactions usually consist of quantitative values. Many fuzzy data mining approaches have thus been proposed for finding fuzzy association rules with the predefined minimum support from the give quantitative transactions. However, the common problems of those approaches are that an appropriate minimum support is hard to set, and the derived rules usually expose common-sense knowledge which may not be interesting in business point of view. In this paper, an algorithm for mining fuzzy coherent rules is proposed for overcoming those problems with the properties of propositional logic. It first transforms quantitative transactions into fuzzy sets. Then, those generated fuzzy sets are collected to generate candidate fuzzy coherent rules. Finally, contingency tables are calculated and used for checking those candidate fuzzy coherent rules satisfy the four criteria or not. If yes, it is a fuzzy coherent rule. Experiments on the foodmart dataset are also made to show the effectiveness of the proposed algorithm.     

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.     

一种基于多精度模糊划分提取模式识别规则的方法

对于具有多模糊特征变量的分类问题，提出一种自动提取适当的模糊模式识别规则集的方法．首先通过多精度划分模糊空间产生多个模糊规则表，然后采用人工免疫原理的克隆选择算法，得出一个优化的模糊分类规则集用于模式识别．实验表明该方法所提取的规则集规则数目少、分类正确率较高．     

Online tuning of a supervisory fuzzy controller for low-energy building system using reinforcement learning

This paper proposes a model-free method using reinforcement learning scheme to tune a supervisory controller for a low-energy building system online. The training time and computational demands are reduced by basing the supervisor on sets of fuzzy rules generated by off-line optimisation and by learning the optimal values of only one parameter, which selects the most appropriate set of rules. By carefully choosing the tuning targets, discretizing the state space, parameterizing the fuzzy rule base, using fuzzy trace-back, the proposed method can complete the training process in one season.     

Simultaneous design of membership functions and rule sets for fuzzycontrollers using genetic algorithms

This paper examines the applicability of genetic algorithms (GA's) in the simultaneous design of membership functions and rule sets for fuzzy logic controllers. Previous work using genetic algorithms has focused on the development of rule sets or high performance membership functions; however, the interdependence between these two components suggests a simultaneous design procedure would be a more appropriate methodology. When GA's have been used to develop both, it has been done serially, e.g., design the membership functions and then use them in the design of the rule set. This, however, means that the membership functions were optimized for the initial rule set and not the rule set designed subsequently. GA's are fully capable of creating complete fuzzy controllers given the equations of motion of the system, eliminating the need for human input in the design loop. This new method has been applied to two problems, a cart controller and a truck controller. Beyond the development of these controllers, we also examine the design of a robust controller for the cart problem and its ability to overcome faulty rules     

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.     

Rule base reduction for knowledge-based fuzzy controllers with application to a vacuum cleaner

A rule base reduction and tuning algorithm is proposed as a design tool for the knowledge-based fuzzy control of a vacuum cleaner. Given a set of expert-based control rules in a fuzzy rule base structure, proposed algorithm computes the inconsistencies and redundancies in the overall rule set based on a newly proposed measure of equality of the individual fuzzy sets. An inconsistency and redundancy measure is proposed and computed for each rule in the rule base. Then the rules with high inconsistency and redundancy levels are removed from the fuzzy rule base without affecting the overall performance of the controller. The algorithm is successfully tested experimentally for the control of a commercial household vacuum cleaner. Experimental results demonstrate the effective use of the proposed algorithm.     

A multi-objective genetic optimization of interpretability-oriented fuzzy rule-based classifiers

The paper presents a multi-objective genetic approach to design interpretability-oriented fuzzy rule-based classifiers from data. The proposed approach allows us to obtain systems with various levels of compromise between their accuracy and interpretability. During the learning process, parameters of the membership functions, as well as the structure of the classifier's fuzzy rule base (i.e., the number of rules, the number of rule antecedents, etc.) evolve simultaneously using a Pittsburgh-type genetic approach. Since there is no particular coding of fuzzy rule structures in a chromosome (it reduces computational complexity of the algorithm), original crossover and mutation operators, as well as chromosome-repairing technique to directly transform the rules are also proposed. To evaluate both the accuracy and interpretability of the system, two measures are used. The first one – an accuracy measure – is based on the root mean square error of the system's response. The second one – an interpretability measure – is based on the arithmetic mean of three components: (a) the average length of rules (the average number of antecedents used in the rules), (b) the number of active fuzzy sets and (c) the number of active inputs of the system (an active fuzzy set or input means a set or input used by at least one fuzzy rule). Both measures are used as objectives in multi-objective (2-objective in our case) genetic optimization approaches such as well-known SPEA2 and NSGA-II algorithms. Moreover, for the purpose of comparison with several alternative approaches, the experiments are carried out both considering the so-called strong fuzzy partitions (SFPs) of attribute domains and without them. SFPs provide more semantically meaningful solutions, usually at the expense of their accuracy. The operation of the proposed technique in various classification problems is tested with the use of 20 benchmark data sets and compared to 11 alternative classification techniques. The experiments show that the proposed approach generates classifiers of significantly improved interpretability, while still characterized by competitive accuracy.     

A review of the construction of hierarchical fuzzy systems

Fuzzy rule‐based systems are nowadays one of the most successful applications of fuzzy sets and fuzzy logic. Most applications use a flat set of fuzzy rules. However, in complex applications with a large set of variables, it is not appropriate to define the system with a flat set of rules because, among other problems, the number of rules increases exponentially with the number of variables. Hierarchical fuzzy systems are one of the alternatives presented in the literature to overcome this problem. In this article we review the latest results related with this type of fuzzy system. © 2002 Wiley Periodicals, Inc.     

Self-organized fuzzy system generation from training examples

In the synthesis of a fuzzy system two steps are generally employed: the identification of a structure and the optimization of the parameters defining it. The paper presents a methodology to automatically perform these two steps in conjunction using a three-phase approach to construct a fuzzy system from numerical data. Phase 1 outlines the membership functions and system rules for a specific structure, starting from a very simple initial topology. Phase 2 decides a new and more suitable topology with the information received from the previous step; it determines for which variable the number of fuzzy sets used to discretize the domain must be increased and where these new fuzzy sets should be located. This, in turn, decides in a dynamic way in which part of the input space the number of fuzzy rules should be increased. Phase 3 selects from the different structures obtained to construct a fuzzy system the one providing the best compromise between the accuracy of the approximation and the complexity of the rule set. The accuracy and complexity of the fuzzy system derived by the proposed self-organized fuzzy rule generation procedure (SOFRG) are studied for the problem of function approximation. Simulation results are compared with other methodologies such as artificial neural networks, neuro-fuzzy systems, and genetic algorithms     

MULTITUNING FUZZY CONTROL SYSTEMS DESIGN

A multituning fuzzy control system structure that involves two simple, but effective tuning mechanisms, is proposed: one is called fuzzy control rule tuning mechanism (FCRTM); the other is called dynamic scalar tuning mechanism (DSTM). In FCRTM, it is used to generate the necessary control rules with a center extension method. In DSTM, it contains three fuzzy IF-THEN rules for determining the appropriate scaling factors for the fuzzy control system. In this paper, a method based on the genetic algorithm (GA) is proposed to simultaneously choose the appropriate parameters in FCRTM and DSTM. That is, the proposed GA-based method can automatically generate the required rule base of fuzzy controller and efficiently determine the appropriate map for building the dynamic scalars of fuzzy controller. A multiobjective fitness function is proposed to determine an appropriate parameter set such that not only the selected fuzzy control structure has fewer fuzzy rules, but also the controlled system has a good control performance. Finally, an inverted pendulum control problem is given to illustrate the effectiveness of the proposed control scheme.     

A neural fuzzy system with fuzzy supervised learning

A neural fuzzy system learning with fuzzy training data (fuzzy if-then rules) is proposed in this paper. This system is able to process and learn numerical information as well as linguistic information. At first, we propose a five-layered neural network for the connectionist realization of a fuzzy inference system. The connectionist structure can house fuzzy logic rules and membership functions for fuzzy inference. We use alpha-level sets of fuzzy numbers to represent linguistic information. The inputs, outputs, and weights of the proposed network can be fuzzy numbers of any shape. Furthermore, they can be hybrid of fuzzy numbers and numerical numbers through the use of fuzzy singletons. Based on interval arithmetics, a fuzzy supervised learning algorithm is developed for the proposed system. It extends the normal supervised learning techniques to the learning problems where only linguistic teaching signals are available. The fuzzy supervised learning scheme can train the proposed system with desired fuzzy input-output pairs which are fuzzy numbers instead of the normal numerical values. With fuzzy supervised learning, the proposed system can be used for rule base concentration to reduce the number of rules in a fuzzy rule base. Simulation results are presented to illustrate the performance and applicability of the proposed system.     

A multistage genetic fuzzy classifier for land cover classification from satellite imagery

A linguistic boosted genetic fuzzy classifier (LiBGFC) is proposed in this paper for land cover classification from multispectral images. The LiBGFC is a three-stage process, aiming at effectively tackling the interpretability versus accuracy tradeoff problem. The first stage iteratively generates fuzzy rules, as directed by a boosting algorithm that localizes new rules in uncovered subspaces of the feature space, implicitly preserving the cooperation with previously derived ones. Each rule is able to select the required features, further improving the interpretability of the obtained model. Special provision is taken in the formulation of the fitness function to avoid the creation of redundant rules. A simplification stage follows the first one aiming at further improving the interpretability of the initial rule base, providing a more compact and interpretable solution. Finally, a genetic tuning stage fine tunes the fuzzy sets database improving the classification performance of the obtained model. The LiBGFC is tested using an IKONOS multispectral VHR image, in a lake-wetland ecosystem of international importance. The results indicate the effectiveness of the proposed system in handling multidimensional feature spaces, producing easily understandable fuzzy models.     

Additive Fuzzy Systems: From Generalized Mixtures to Rule Continua

A generalized probability mixture density governs an additive fuzzy system. The fuzzy system's if‐then rules correspond to the mixed probability densities. An additive fuzzy system computes an output by adding its fired rules and then averaging the result. The mixture's convex structure yields Bayes theorems that give the probability of which rules fired or which combined fuzzy systems fired for a given input and output. The convex structure also results in new moment theorems and learning laws and new ways to both approximate functions and exactly represent them. The additive fuzzy system itself is just the first conditional moment of the generalized mixture density. The output is a convex combination of the centroids of the fired then‐part sets. The mixture's second moment defines the fuzzy system's conditional variance. It describes the inherent uncertainty in the fuzzy system's output due to rule interpolation. The mixture structure gives a natural way to combine fuzzy systems because mixing mixtures yields a new mixture. A separation theorem shows how fuzzy approximators combine with exact Watkins‐based two‐rule function representations in a higher‐level convex sum of the combined systems. Two mixed Gaussian densities with appropriate Watkins coefficients define a generalized mixture density such that the fuzzy system's output equals any given real‐valued function if the function is bounded and not constant. Statistical hill‐climbing algorithms can learn the generalized mixture from sample data. The mixture structure also extends finite rule bases to continuum‐many rules. Finite fuzzy systems suffer from exponential rule explosion because each input fires all their graph‐cover rules. The continuum system fires only a special random sample of rules based on Monte Carlo sampling from the system's mixture. Users can program the system by changing its wave‐like meta‐rules based on the location and shape of the mixed densities in the mixture. Such meta‐rules can help mitigate rule explosion. The meta‐rules grow only linearly with the number of mixed densities even though the underlying fuzzy if‐then rules can have high‐dimensional if‐part and then‐part fuzzy sets.     

A Self-Evolving Interval Type-2 Fuzzy Neural Network With Online Structure and Parameter Learning

This paper proposes a self-evolving interval type-2 fuzzy neural network (SEIT2FNN) with online structure and parameter learning. The antecedent parts in each fuzzy rule of the SEIT2FNN are interval type-2 fuzzy sets and the fuzzy rules are of the Takagi–Sugeno–Kang (TSK) type. The initial rule base in the SEIT2FNN is empty, and the online clustering method is proposed to generate fuzzy rules that flexibly partition the input space. To avoid generating highly overlapping fuzzy sets in each input variable, an efficient fuzzy set reduction method is also proposed. This method independently determines whether a corresponding fuzzy set should be generated in each input variable when a new fuzzy rule is generated. For parameter learning, the consequent part parameters are tuned by the rule-ordered Kalman filter algorithm for high-accuracy learning performance. Detailed learning equations on applying the rule-ordered Kalman filter algorithm to the SEIT2FNN consequent part learning, with rules being generated online, are derived. The antecedent part parameters are learned by gradient descent algorithms. The SEIT2FNN is applied to simulations on nonlinear plant modeling, adaptive noise cancellation, and chaotic signal prediction. Comparisons with other type-1 and type-2 fuzzy systems in these examples verify the performance of the SEIT2FNN.      

GenSoFNN: a generic self-organizing fuzzy neural network

Existing neural fuzzy (neuro-fuzzy) networks proposed in the literature can be broadly classified into two groups. The first group is essentially fuzzy systems with self-tuning capabilities and requires an initial rule base to be specified prior to training. The second group of neural fuzzy networks, on the other hand, is able to automatically formulate the fuzzy rules from the numerical training data. No initial rule base needs to be specified prior to training. A cluster analysis is first performed on the training data and the fuzzy rules are subsequently derived through the proper connections of these computed clusters. However, most existing neural fuzzy systems (whether they belong to the first or second group) encountered one or more of the following major problems. They are (1) inconsistent rule-base; (2) heuristically defined node operations; (3) susceptibility to noisy training data and the stability-plasticity dilemma; and (4) needs for prior knowledge such as the number of clusters to be computed. Hence, a novel neural fuzzy system that is immune to the above-mentioned deficiencies is proposed in this paper. This new neural fuzzy system is named the generic self-organizing fuzzy neural network (GenSoFNN). The GenSoFNN network has strong noise tolerance capability by employing a new clustering technique known as discrete incremental clustering (DIC). The fuzzy rule base of the GenSoFNN network is consistent and compact as GenSoFNN has built-in mechanisms to identify and prune redundant and/or obsolete rules. Extensive simulations were conducted using the proposed GenSoFNN network and its performance is encouraging when benchmarked against other neural and neural fuzzy systems.     

基于RPCL的模糊关联规则挖掘

在模糊关联规则的挖掘过程中,很难预先知道每个属性合适的模糊集。针对该问题,提出基于次胜者受罚竞争学习的模糊关联规则挖掘算法,无需先验知识,即可根据每个属性的性质找出对应的模糊集,并确定模糊集的数目。实验结果表明,与同类算法相比,该算法可以挖掘出更多有趣的关联规则。     

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     

模糊控制规则优化方法研究

模糊控制规则的选择是模糊控制器设计的关键问题之一,在现有应用遗传算法优化模糊控制规则的方法进行研究的基础上,以模糊控制规则的完整性和一致性为出发点,提出了一种用遗传算法来优化模糊控制规则的改进算法,具体给出了遗传算法设计中的各种函数和算子的确定,并将优化过的规则用于设计模糊控制器,进行仿真研究,取得了令人满意的效果。