Fuzzy rough granular neural networks, fuzzy granules, and classification

We introduce a fuzzy rough granular neural network (FRGNN) model based on the multilayer perceptron using a back-propagation algorithm for the fuzzy classification of patterns. We provide the development strategy of the network mainly based upon the input vector, initial connection weights determined by fuzzy rough set theoretic concepts, and the target vector. While the input vector is described in terms of fuzzy granules, the target vector is defined in terms of fuzzy class membership values and zeros. Crude domain knowledge about the initial data is represented in the form of a decision table, which is divided into subtables corresponding to different classes. The data in each decision table is converted into granular form. The syntax of these decision tables automatically determines the appropriate number of hidden nodes, while the dependency factors from all the decision tables are used as initial weights. The dependency factor of each attribute and the average degree of the dependency factor of all the attributes with respect to decision classes are considered as initial connection weights between the nodes of the input layer and the hidden layer, and the hidden layer and the output layer, respectively. The effectiveness of the proposed FRGNN is demonstrated on several real-life data sets.     

Fuzzy integral-based perceptron for two-class pattern classification problems

The single-layer perceptron with single output node is a well-known neural network for two-class classification problems. Furthermore, the sigmoid or logistic function is usually used as the activation function in the output neuron. A critical step is to compute the sum of the products of the connection weights with the corresponding inputs, which indicates the assumption of additivity among individual variables. Unfortunately, because the input variables are not always independent of each other, an assumption of additivity may not be reasonable enough. In this paper, the inner product can be replaced with an aggregation value obtained by a useful fuzzy integral by viewing each of the connection weights as a value of a λ-fuzzy measure for the corresponding variable. A genetic algorithm is then employed to obtain connection weights by maximizing the number of correctly classified training patterns and minimizing the errors between the actual and desired outputs of individual training patterns. The experimental results further demonstrate that the proposed method outperforms the traditional single-layer perceptron and performs well in comparison with other fuzzy or non-fuzzy classification methods.     

Classification performance evaluation of single-layer perceptron with Choquet integral-based TOPSIS

This paper interprets an output of the single-layer perceptron as the synthetic evaluation of the similarity between the input pattern and two classes for two-class pattern classification problems. For this, an appropriate tool for multiple attribute decision making, TOPSIS, can be an appropriate tool. A novel single-layer perceptron based on the TOPSIS is proposed by measuring the grades of similarity between an input pattern and the typical patterns of the respective classes. To obtain the relative similarity in terms of certain distance measure, since the interaction among attributes cannot be ignored, the nonlinear Choquet integral with respect to the fuzzy measure is incorporated into the activation function. Moreover, a genetic algorithm-based method is designed to determine the connection weights, which are interpreted as the degrees of importance of the respective input signals. The experimental results further demonstrate that classification results obtained by the single-layer perceptron with the Choquet integral-based TOPSIS are comparable to those obtained by the other fuzzy or non-fuzzy classification methods.     

Fuzzy declustering-based vector quantization

Vector quantization is a useful approach for multi-dimensional data compression and pattern classification. One of the most popular techniques for vector quantization design is the LBG (Linde, Buzo, Gray) algorithm. To address the problem of producing poor estimate of vector centroids which are subjected to biased data in vector quantization; we propose a fuzzy declustering strategy for the LBG algorithm. The proposed technique calculates appropriate declustering weights to adjust the global data distribution. Using the result of fuzzy declustering-based vector quantization design, we incorporate the notion of fuzzy partition entropy into the distortion measures that can be useful for classification of spectral features. Experimental results obtained from simulated and real data sets demonstrate the effective performance of the proposed approach.     

一种基于模糊规则的神经网络结构及其学习算法研究

文章提出了一种基于模糊规则的神经网络结构,并用形式化语言进行描述。基于模糊规则的神经网络由输入层、规则层和输出层三层网络结构组成,以隶属度函数（语义值）作为网络权值,输入值沿权值的传播即进行隶属度计算。在充分分析三角形函数特征的基础上,应用启发式方法,导出了FRBNN网络的学习算法。最后应用FRBNN评价船舶碰撞危险度,表明FRBNN兼备神经网络和模糊推理系统的优点。     

A fuzzy set based solution method for multiobjective optimal design problem of mechanical and structural systems using functional-link net

The principle of solving multiobjective optimization problems with fuzzy sets theory is studied. Membership function is the key to introduce the fuzzy sets theory to multiobjective optimization. However, it is difficult to determine membership functions in engineering applications. On the basis of rapid quadratic optimization in the learning of weights, simplification in hardware as well as in computational procedures of functional-link net, discrete membership functions are used as sample training data. When the network converges, the continuous membership functions implemented with the network. Membership functions based on functional-link net have been used in multiobjective optimization. An example is given to illustrate the method.     

Solving fully fuzzy polynomials using feed-back neural networks

Recently, there has been a considerable amount of interest and practice in solving many problems of several applied fields by fuzzy polynomials. In this paper, we have designed an artificial fuzzified feed-back neural network. With this design, we are able to find a solution of fully fuzzy polynomial with degree n. This neural network can get a fuzzy vector as an input, and calculates its corresponding fuzzy output. It is clear that the input–output relation for each unit of fuzzy neural network is defined by the extension principle of Zadeh. In this work, a cost function is also defined for the level sets of fuzzy output and fuzzy target. Next a learning algorithm based on the gradient descent method will be defined that can adjust the fuzzy connection weights. Finally, our approach is illustrated by computer simulations on numerical examples. It is worthwhile to mention that application of this method in fluid mechanics has been shown by an example.     

Recognition of control chart patterns using fuzzy SVM with a hybrid kernel function

Accurate control chart patterns recognition (CCPR) plays an essential role in the implementation of control charts. However, it is a challenging problem since nonrandom control chart patterns (CCPs) are normally distorted by “common process variations”. In this paper, a novel method of CCPR by integrating fuzzy support vector machine (SVM) with hybrid kernel function and genetic algorithm (GA) is proposed. Firstly, two shape features and two statistical features that do not depend on the distribution parameters and number of samples are presented to explicitly describe the characteristics of CCPs. Then, a novel multiclass method based on fuzzy SVM with a hybrid kernel function is proposed. In this method, the influence of outliers on classification accuracy of SVM-based classifiers is weakened by assigning a degree of membership for every training sample. Meanwhile, a hybrid kernel function combining Gaussian kernel and polynomial kernel is adopted to further enhance the generalization ability of the classifiers. To solve the issue of features selection and parameters optimization, GA is used to simultaneously optimize the input features subsets and parameters of fuzzy SVM-based classifier. Finally, several simulation experiments and a real example are addressed to validate the feasibility and effectiveness of the proposed methodology. And the results of simulation experiments demonstrate that it can achieve excellent performance for CCPR and outperforms other approaches, such as learning vector quantization network, multi-layer perceptron network, probability neural network, fuzzy clustering and SVM, in term of recognition accuracy. The results of the practical cases manifest that the proposed method has application potential for solving the problem of control chart interpretation in real-world.     

Support vector learning for fuzzy rule-based classification systems

To design a fuzzy rule-based classification system (fuzzy classifier) with good generalization ability in a high dimensional feature space has been an active research topic for a long time. As a powerful machine learning approach for pattern recognition problems, the support vector machine (SVM) is known to have good generalization ability. More importantly, an SVM can work very well on a high- (or even infinite) dimensional feature space. This paper investigates the connection between fuzzy classifiers and kernel machines, establishes a link between fuzzy rules and kernels, and proposes a learning algorithm for fuzzy classifiers. We first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function. Fuzzy inference on the IF-part of a fuzzy rule can be viewed as evaluating the kernel function. The kernel function is then proven to be a Mercer kernel if the reference functions meet a certain spectral requirement. The corresponding fuzzy classifier is named positive definite fuzzy classifier (PDFC). A PDFC can be built from the given training samples based on a support vector learning approach with the IF-part fuzzy rules given by the support vectors. Since the learning process minimizes an upper bound on the expected risk (expected prediction error) instead of the empirical risk (training error), the resulting PDFC usually has good generalization. Moreover, because of the sparsity properties of the SVMs, the number of fuzzy rules is irrelevant to the dimension of input space. In this sense, we avoid the "curse of dimensionality." Finally, PDFCs with different reference functions are constructed using the support vector learning approach. The performance of the PDFCs is illustrated by extensive experimental results. Comparisons with other methods are also provided.     

A novel fuzzy classifier with Choquet integral-based grey relational analysis for pattern classification problems

This paper aims to propose a fuzzy classifier, which is a one-class-in-one-network structure consisting of multiple novel single-layer perceptrons. Since the output value of each single-layer perceptron can be interpreted as the overall grade of the relationship between the input pattern and one class, the degree of relationship between an attribute of the input pattern and that of this class can be taken into account by establishing a representative pattern for each class. A feature of this paper is that it employs the grey relational analysis to compute the grades of relationship for individual attributes. In particular, instead of using the sigmoid function as the activation function, a non-additive technique, the Choquet integral, is used as an activation function to synthesize the performance values, since an assumption of noninteraction among attributes may not be reasonable. Thus, a single-layer perceptron in the proposed structure performs the synthetic evaluation of the Choquet integral-based grey relational analysis for a pattern. Each connection weight is interpreted as a degree of importance of an attribute and can be determined by a genetic algorithm-based method. The experimental results further demonstrate that the test results of the proposed fuzzy classifier are better than or comparable to those of other fuzzy or non-fuzzy classification methods.