Fuzzy approximation via grid point sampling and singular valuedecomposition

This paper introduces a new approach for fuzzy approximation of continuous function on a compact domain. The approach calls for sampling the function over a set of rectangular grid points and applying singular value decomposition to the sample matrix. The resulting quantities are then tailored to become rule consequences and membership functions via the conditions of sum normalization and non-negativeness. The inference paradigm of product-sum-gravity is apparent from the structure of the decomposition equation. All information are extracted directly from the function samples. The present approach yields a class of equivalent fuzzy approximator to a given function. A tight bounding technique to facilitate normal or close-to-normal membership functions is also formulated. The fuzzy output approximates the given function to within an error which is dependent on the sampling intervals and the singular values discarded from the approximation process. Trade-off between the number of membership functions and the desired approximation accuracy is also discussed.     

Rule Base Reduction and Genetic Tuning of Fuzzy Systems Based on the Linguistic 3-tuples Representation

Recently, a new linguistic rule representation model was presented to perform a genetic lateral tuning of membership functions. It is based on the linguistic 2-tuples representation model, that allows the symbolic translation of a label considering an unique parameter. It involves a reduction of the search space that eases the derivation of optimal models. This work presents a new symbolic representation with three values (s, α, β), respectively representing a label, the lateral displacement and the amplitude variation of the support of this label. Based on this new representation we propose a new method for fine tuning of membership functions that is combined with a rule base reduction method in order to extract the most useful tuned rules. This approach makes use of a modified inference system that consider non-covered inputs in order to improve the final fuzzy model generalization ability, specially in highly non-linear problems with noise points. Additionally, we analyze the proposed approach showing its behavior in two real-world applications. Supported by the Spanish Ministry of Science and Technology under Projects TIC-2002-04036-C05-01 and TIN-2005-08386-C05-01.     

Influential rule search scheme (IRSS) - a new fuzzy pattern classifier

Automatic generation of fuzzy rule base and membership functions from an input-output data set, for reliable construction of an adaptive fuzzy inference system, has become an important area of research interest. We propose a new robust, fast acting adaptive fuzzy pattern classification scheme, named influential rule search scheme (IRSS). In IRSS, rules which are most influential in contributing to the error produced by the adaptive fuzzy system are identified at the end of each epoch and subsequently modified for satisfactory performance. This fuzzy rule base adjustment scheme is accompanied by an output membership function adaptation scheme for fine tuning the fuzzy system architecture. This iterative method has shown a relatively high speed of convergence. Performance of the proposed IRSS is compared with other existing pattern classification schemes by implementing it for Fisher's iris data problem and Wisconsin breast cancer data problems.     

Type-2 fuzzy modeling for desulphurization of steel process

This paper presents a new type-2 fuzzy logic system model for desulphurization process of a real steel industry in Canada. The type-2 fuzzy logic system permits us to model rule uncertainties where every membership value of an element has a second order membership value of its own. In this paper, we propose an indirect method to create second order membership grades that are amplitudes of type-2 secondary membership functions, where the primary memberships are extracted by implementation of fuzzy clustering approach. In this research, Gaussian Mixture Model (GMM) is used for the creation of second order membership grades. Furthermore, a reduction scheme is implemented which results in type-1 membership grades. In turn, this leads to a reduction of the complexity of the system. Two methods are used for the estimation of the membership functions: indirect and direct methods. In the indirect method, the system uses an interpolation scheme for the estimation of the most appropriate membership functions. In the direct method, the system is tuned by an inference algorithm for the optimization of the main parametric system. In this case, the parameters are: Schweizer and Sklar t-norm and s-norm, combination of FATI and FITA inference approaches, and Yager defuzzification. Finally, the system model is applied to the desulphurization process of a real steel industry in Canada. It is shown that the proposed type-2 fuzzy logic system is superior in comparison to multiple regression and type-1 fuzzy logic systems in terms of robustness, and error reduction.     

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.     

Comments on “Reduction of fuzzy rule base via singular valuedecomposition”

Several comments are presented for the reduction of the fuzzy rule base in the paper of Yam et al. (1999). In their paper, the approach to determine the number of singular values necessary for the reduction process to obtain the effective and most efficient fuzzy rule base is not provided. Although the output error of the fuzzy controller is bounded, the performance of the system output may not be satisfied. Moreover, the computation load is increased for each input of the fuzzy mechanism since the input membership functions are modified     

A unified neurofuzzy model for classification

This work proposes a unified neurofuzzy modelling scheme. To begin with, the initial fuzzy base construction method is based on fuzzy clustering utilising a Gaussian mixture model (GMM) combined with the analysis of covariance (ANOVA) decomposition in order to obtain more compact univariate and bivariate membership functions over the subspaces of the input features. The mean and covariance of the Gaussian membership functions are found by the expectation maximisation (EM) algorithm with the merit of revealing the underlying density distribution of system inputs. The resultant set of membership functions forms the basis of the generalised fuzzy model (GFM) inference engine. The model structure and parameters of this neurofuzzy model are identified via the supervised subspace orthogonal least square (OLS) learning. Finally, instead of providing deterministic class label as model output by convention, a logistic regression model is applied to present the classifier’s output, in which the sigmoid type of logistic transfer function scales the outputs of the neurofuzzy model to the class probability. Experimental validation results are presented to demonstrate the effectiveness of the proposed neurofuzzy modelling scheme.     

Genetic tuning of fuzzy inference within fuzzy classifier systems

Abstract: In generating a suitable fuzzy classifier system, significant effort is often placed on the determination and the fine tuning of the fuzzy sets. However, in such systems little thought is given to the way in which membership functions are combined within the fuzzy rules. Often traditional fuzzy inference strategies are used which consequently provide no control over how strongly or weakly the inference is applied within these rules. Furthermore such strategies will allow no interaction between grades of membership. A number of theoretical fuzzy inference operators have been proposed for both regression and classification problems but they have not been investigated in the context of real-world applications. In this paper we propose a novel genetic algorithm framework for optimizing the strength of fuzzy inference operators concurrently with the tuning of membership functions for a given fuzzy classifier system. Each fuzzy system is generated using two well-established decision tree algorithms: C4.5 and CHAID. This will enable both classification and regression problems to be addressed within the framework. Each solution generated by the genetic algorithm will produce a set of fuzzy membership functions and also determine how strongly the inference will be applied within each fuzzy rule. We investigate several theoretical proven fuzzy inference techniques (T-norms) in the context of both classification and regression problems. The methodology proposed is applied to a number of real-world data sets in order to determine the effects of the simultaneous tuning of membership functions and inference parameters on the accuracy and robustness of fuzzy classifiers.     

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.     

一种用模糊－神经技术建造专家系统的方法

本文提出了一种用模糊－神经技术建造专家系统的方法（ＦＮＴ方法）。从领域专家处获取的知识是以模糊规则和隶属函数的形式表示的。根据本文提出的方法，首先将模糊规则和隶属函数用神经网络表示出来（导入）；生成的神经网络用于实现模糊推理，然后利用修改的反传算法训练神经网络，从而提高系统的精度，修改隶属函数，求精模糊规则；最后从神经网络中提取隶属函数和模糊规则（导出），帮助解释神经网络的内部表示和操作。利用本文所提出的方法建造的系统可实现快速的无匹配模糊推理，并具有较强的学习能力。     

A Three-Dimensional Fuzzy Control Methodology for a Class of Distributed Parameter Systems

The traditional fuzzy set is two-dimensional (2-D) with one dimension for the universe of discourse of the variable and the other for its membership degree. This 2-D fuzzy set is not able to handle the spatial information. The traditional fuzzy logic controller (FLC) developed from this 2-D fuzzy set should not be able to control the distributed parameter system that has the tempo-spatial nature. A three-dimensional (3-D) fuzzy set is defined to be made of a traditional fuzzy set and an extra dimension for spatial information. Based on concept of the 3-D fuzzy set, a new fuzzy control methodology is proposed to control the distributed parameter system. Similar to the traditional FLC, it still consists of fuzzification, rule inference, and defuzzification operations. Different to the traditional FLC, it uses multiple sensors to provide 3-D fuzzy inputs and possesses the inference mechanism with 3-D nature that can fuse these inputs into a so called ldquospatial membership function.rdquo Thus, a simple 2-D rule base can still be used for two obvious advantages. One is that rules will not increase as sensors increase for the spatial measurement; the other is that computation of this 3-D fuzzy inference can be significantly reduced for real world applications. Using only a few more sensors, the proposed FLC is able to process the distributed parameter system with little complexity increased from the traditional FLC. The 3-D FLC is successfully applied to a catalytic packed-bed reactor and compared with the traditional FLC. The results demonstrate its effectiveness to the nonlinear unknown distributed parameter process and its potential to a wide range of engineering applications.     

A reduction approach for fuzzy rule bases of fuzzy controllers

In this paper, a new approach to reducing the number of rules in a given fuzzy rule base of a fuzzy controller is presented. The fuzzy mechanism of the fuzzy controller under consideration consists of the product-sum inference, singleton output consequents and centroid defuzzification. The output consequents in the cells of the rule table are collected and represented as an output consequent matrix. The feature of the output consequent matrix is extracted by the singular values of the matrix. The output consequent matrix is reasonably approximated with a dominant consequent matrix. Also, the elements of the dominant consequent matrix is determined to minimize the approximation error function. Then the size of the dominant consequent matrix (the size of the fuzzy rule base) is reduced through the rule combination approach. The scaling factors for the fuzzy controller with the reduced rule table are adjusted to have the control system satisfy the performance indices. The effectiveness of the proposed approach is shown using simulation and experimental results.     

Fuzzy logic-based forecasting model

In this paper a fuzzy logic-based software tool, fuzzy logic advisory tool (FLAT), for demand forecasting of signal transmission products is presented. The FLAT was developed for the prediction of demand of about 1000 different products in order to aid materials purchasing process of about 14,000 different components in the electronics manufacturing processes of Nokia Network Systems's Haukipudas factory. The prediction values of different products are inferred by starting from a set of eight input values. Each input value is fuzzied by the FLAT. Thereafter, fuzzy results are inferred in three sequential phases. In each phase the number of variables is split due to hierarchical structure of the inference module. A data base and a rule base are divided accordingly into three hierarchical levels. Rules are represented by linguistic relations changed into matrix equations form in order to apply linguistic equations framework technique (LE). Fuzzy membership functions for input values are determined on-line from earlier input values of the products. Fuzzy rules were inferred by analyzing behavior of the products together with market experts and product experts of the company. The model is able to produce more accurate decision-making support than more traditional approaches. This is probably due to the model-based approach and systematic data management.     

Fuzzy classifications using fuzzy inference networks

In this paper, fuzzy inference models for pattern classifications have been developed and fuzzy inference networks based on these models are proposed. Most of the existing fuzzy rule-based systems have difficulties in deriving inference rules and membership functions directly from training data. Rules and membership functions are obtained from experts. Some approaches use backpropagation (BP) type learning algorithms to learn the parameters of membership functions from training data. However, BP algorithms take a long time to converge and they require an advanced setting of the number of inference rules. The work to determine the number of inference rules demands lots of experiences from the designer. In this paper, self-organizing learning algorithms are proposed for the fuzzy inference networks. In the proposed learning algorithms, the number of inference rules and the membership functions in the inference rules will be automatically determined during the training procedure. The learning speed is fast. The proposed fuzzy inference network (FIN) classifiers possess both the structure and the learning ability of neural networks, and the fuzzy classification ability of fuzzy algorithms. Simulation results on fuzzy classification of two-dimensional data are presented and compared with those of the fuzzy ARTMAP. The proposed fuzzy inference networks perform better than the fuzzy ARTMAP and need less training samples.     

弧焊过程神经网络模糊控制

提出一种将FLC与神经网络技术相结合的方法对钨极氩弧焊（GTAW）过程进行控制,它克服了模糊规则产生对专家的依赖及模糊集非自适应性的问题。隶属函数的自适应及模糊规则的自组织通过神经网络的自学习和竞争获得。该方法实现了弧焊过程中模糊规则的自动确定和隶属度函数在线调度。 以GTAW过程焊缝几何参数调节为对象,验证了算法的有效性。计算机仿真表明,采用该方法的系统性能有较大的提高。     

Knowledge modeling based on interval-valued fuzzy rough set and similarity inference: prediction of welding distortion

Knowledge-based modeling is a trend in complex system modeling technology. To extract the process knowledge from an information system, an approach of knowledge modeling based on interval-valued fuzzy rough set is presented in this paper, in which attribute reduction is a key to obtain the simplified knowledge model. Through defining dependency and inclusion functions, algorithms for attribute reduction and rule extraction are obtained. The approximation inference plays an important role in the development of the fuzzy system. To improve the inference mechanism, we provide a method of similaritybased inference in an interval-valued fuzzy environment. larity based approximate reasoning, an inference result is Combining the conventional compositional rule of inference with simideduced via rule translation, similarity matching, relation modification, and projection operation. This approach is applied to the problem of predicting welding distortion in marine structures, and the experimental results validate the effectiveness of the proposed methods of knowledge modeling and similarity-based inference.     

A trainable transparent universal approximator for defuzzificationin Mamdani-type neuro-fuzzy controllers

A novel technique of designing application specific defuzzification strategies with neural learning is presented. The proposed neural architecture considered as a universal defuzzification approximator is validated by showing the convergence when approximating several existing defuzzification strategies. The method is successfully tested with fuzzy controlled reverse driving of a model truck. The transparent structure of the universal defuzzification approximator allows us to analyze the generated customized defuzzification method using the existing theories of defuzzification. The integration of universal defuzzification approximator instead of traditional methods in Mamdani-type fuzzy controllers can also be considered as an addition of trainable nonlinear noise to the output of the fuzzy rule inference before calculating the defuzzified crisp output. Therefore, nonlinear noise trained specifically for a given application shows a grade of confidence on the rule base, providing an additional opportunity to measure the quality of the fuzzy rule base. The possibility of modeling a Mamdani-type fuzzy controller as a feedforward neural network with the ability of gradient descent training of the universal defuzzification approximator and antecedent membership functions fulfil the requirement known from multilayer preceptrons in finding solutions to nonlinear separable problems     

自调整模糊控制器的软件仿真系统设计

该文提出一种具有修正因子的自调整模糊控制器软件仿真系统RMDFCSS。它包含有编辑器、调试器、模糊推理机和代码生成器功能,用编辑器定义模糊规则、隶属函数、特定的推理方法和反模糊化方法;用调试器能检查整个推理过程的每一个步骤,完成模糊控制算法模型的确立、论证和优化操作。     

Fast inference in SAM fuzzy systems using transition matrices

Fast inference using transition matrices (FITM) is a new fast algorithm for performing inferences in fuzzy systems. It is based on the assumption that fuzzy inputs can be expressed as a linear composition of the fuzzy sets used in the rule base. This representation let us interpret a fuzzy set as a vector, so we can just work with the coordinates of it instead of working with the whole set. The inference is made using transition matrices. The key of the method is the fact that a lot of operations can be precomputed offline to obtain the transition matrices, so actual inferences are reduced to a few online matrix additions and multiplications. The algorithm is designed for the standard additive model using the sum-product inference composition.     

Fuzzy functions with support vector machines

A new fuzzy system modeling (FSM) approach that identifies the fuzzy functions using support vector machines (SVM) is proposed. This new approach is structurally different from the fuzzy rule base approaches and fuzzy regression methods. It is a new alternate version of the earlier FSM with fuzzy functions approaches. SVM is applied to determine the support vectors for each fuzzy cluster obtained by fuzzy c-means (FCM) clustering algorithm. Original input variables, the membership values obtained from the FCM together with their transformations form a new augmented set of input variables. The performance of the proposed system modeling approach is compared to previous fuzzy functions approaches, standard SVM, LSE methods using an artificial sparse dataset and a real-life non-sparse dataset. The results indicate that the proposed fuzzy functions with support vector machines approach is a feasible and stable method for regression problems and results in higher performances than the classical statistical methods.