Fuzzy Polynomial Neuron-Based Self-Organizing Neural Networks

We propose a new category of neurofuzzy networks—self-organizing neural networks (SONN) with fuzzy polynomial neurons (FPNs) and discuss a comprehensive design methodology supporting their development. Two kinds of SONN architectures, namely a basic SONN and a modified SONN architecture are discussed. Each of them comes with two topologies such as a generic and advanced type. Especially in the advanced type, the number of nodes in each layer of the SONN architecture can be modified with new nodes added, if necessary. SONN dwells on the ideas of fuzzy rule-based computing and neural networks. The architecture of the SONN is not fixed in advance as it usually takes place in the case of conventional neural networks, but becomes organized dynamically through a growth process. Simulation involves a series of synthetic as well as real-world data used across various neurofuzzy systems. A comparative analysis shows that the proposed SONN are models exhibiting higher accuracy than some other fuzzy models.     

Evolutionary design of hybrid self-organizing fuzzy polynomial neural networks with the aid of information granulation

We introduce a new architecture of information granulation-based and genetically optimized Hybrid Self-Organizing Fuzzy Polynomial Neural Networks (HSOFPNN). Such networks are based on genetically optimized multi-layer perceptrons. We develop their comprehensive design methodology involving mechanisms of genetic optimization and information granulation. The architecture of the resulting HSOFPNN combines fuzzy polynomial neurons (FPNs) that are located at the first layer of the network with polynomial neurons (PNs) forming the remaining layers of the network. The augmented version of the HSOFPNN, “IG_gHSOFPNN”, for brief, embraces the concept of information granulation and subsequently exhibits higher level of flexibility and leads to simpler architectures and rapid convergence speed to optimal structure in comparison with the HSOFPNNs and SOFPNNs.

The GA-based design procedure being applied at each layer of HSOFPNN leads to the selection of preferred nodes of the network (FPNs or PNs) whose local characteristics (such as the number of input variables, the order of the polynomial, a collection of the specific subset of input variables, the number of membership functions for each input variable, and the type of membership function) can be easily adjusted. In the sequel, two general optimization mechanisms are explored. The structural optimization is realized via GAs whereas the ensuing detailed parametric optimization is afterwards carried out in the setting of a standard least square method-based learning. The obtained results demonstrate a superiority of the proposed networks over the existing fuzzy and neural models.     

正则模糊神经网络是模糊值函数的泛逼近器

通过分析多元模糊值Bernstein多项式的近似特性，证明了4层前向正则模糊神经网络（FNN）的逼近性能，该类网络构成了模糊值函数的一类泛逼近器，即在欧氏空间的任何紧集上，任意连续模糊值函数能被这类FNN逼近到任意精度，最后通过实例给出了实现这种近似的具体步骤。     

Subsystem inference representation for fuzzy systems based uponproduct-sum-gravity rule

A new representation which expresses a product-sum-gravity (PSG) inference in terms of additive and multiplicative subsystem inferences of single variable is proposed. The representation yields additional insight into the structure of a fuzzy system and produces an approximate functional characterization of its inferred output. The form of the approximating function is dictated by the choice or polynomial, sinusoidal, or other designs of subsystem inferences. With polynomial inferences, the inferred output approximates a polynomial function the order of which is dependent on the numbers of input membership functions. Explicit expressions for the function and corresponding error of approximation are readily obtained for analysis. Subsystem inferences emulating sinusoidal functions are also discussed. With proper scaling, they produce a set of orthonormal subsystem inferences. The orthonormal set points to a possible “modal” analysis of fuzzy inference and yields solution to an additive decomposable approximation problem. This work also shows that, as the numbers of input membership functions become large, a fuzzy system with PSG inference would converge toward polynomial or Fourier series expansions. The result suggests a new framework to consider fuzzy systems as universal approximators     

Time series prediction with single multiplicative neuron model

Single neuron models are typical functional replica of the biological neuron that are derived using their individual and group responses in networks. In recent past, a lot of work in this area has produced advanced neuron models for both analog and binary data patterns. Popular among these are the higher-order neurons, fuzzy neurons and other polynomial neurons. In this paper, we propose a new neuron model based on a polynomial architecture. Instead of considering all the higher-order terms, a simple aggregation function is used. The aggregation function is considered as a product of linear functions in different dimensions of the space. The functional mapping capability of the proposed neuron model is demonstrated through some well known time series prediction problems and is compared with the standard multilayer neural network.     

模糊系统万能逼近理论研究综述

模糊系统通用逼近理论是模糊理论研究的一个重要方向．目前，对模糊系统通用逼近性的研究已经取得了很大的进展．对模糊系统的通用逼近性、模糊系统作为通用逼近器的充分条件和必要条件以及模糊系统的逼近精度等方面的研究进行了较为详尽的综述，分析了各种分析方法的主要成果及其特点（包括优点和局限性），并指出了今后模糊系统通用逼近理论研究中有待解决的许多问题．     

模糊逻辑神经元研究进展

模糊逻辑神经元作为模糊神经网络的重要组成部分,一直以来备受关注,从模型设计到算法研究,成果颇多。以往的研究多集中于模糊逻辑神经元在某一方面的应用模型、性质和算法,具有突出的针对性和特殊性;而今的研究多立足于构造一种能够包容各种逻辑形态的通用神经元,以体现思维的灵活性和多样性,进而提高神经元的推广和应用价值。针对当前模糊逻辑神经元的研究进展做了综述,分析了前后两类不同模型的典型结构及特点,并明确了未来的发展方向。     

IG-based genetically optimized fuzzy polynomial neural networks with fuzzy set-based polynomial neurons

In this study, we introduce and investigate a new topology of fuzzy-neural networks—fuzzy polynomial neural networks (FPNN) that is based on a genetically optimized multiplayer perceptron with fuzzy set-based polynomial neurons (FSPNs). We also develop a comprehensive design methodology involving mechanisms of genetic optimization and information granulation. In the sequel, the genetically optimized FPNN (gFPNN) is formed with the use of fuzzy set-based polynomial neurons (FSPNs) composed of fuzzy set-based rules through the process of information granulation. This granulation is realized with the aid of the C-means clustering (C-Means). The design procedure applied in the construction of each layer of an FPNN deals with its structural optimization involving the selection of the most suitable nodes (or FSPNs) with specific local characteristics (such as the number of input variable, the order of the polynomial, the number of membership functions, and a collection of specific subset of input variables) and address main aspects of parametric optimization. Along this line, two general optimization mechanisms are explored. The structural optimization is realized via genetic algorithms (GAs) and HCM method whereas in case of the parametric optimization we proceed with a standard least square estimation (learning). Through the consecutive process of structural and parametric optimization, a flexible neural network is generated in a dynamic fashion. The performance of the designed networks is quantified through experimentation where we use two modeling benchmarks already commonly utilized within the area of fuzzy or neurofuzzy modeling.     

Sufficient conditions on uniform approximation of multivariatefunctions by general Takagi-Sugeno fuzzy systems with linear ruleconsequent

We have constructively proved a general class of multi-input single-output Takagi-Sugeno (TS) fuzzy systems to be universal approximators. The systems use any types of continuous fuzzy sets, fuzzy logic AND, fuzzy rules with linear rule consequent and the generalized defuzzifier. We first prove that the TS fuzzy systems can uniformly approximate any multivariate polynomial arbitrarily well, and then prove they can also uniformly approximate any multivariate continuous function arbitrarily well. We have derived a formula for computing the minimal upper bounds on the number of fuzzy sets and fuzzy rules necessary to achieve the prespecified approximation accuracy for any given bivariate function. A numerical example is furnished. Our results provide a solid-theoretical basis for fuzzy system applications, particularly as fuzzy controllers and models     

Fault diagnosis and cause analysis using fuzzy evidential reasoning approach and dynamic adaptive fuzzy Petri nets

Fault diagnosis is of great importance to all kinds of industries in the competitive global market today. However, as a promising fault diagnosis tool, fuzzy Petri nets (FPNs) still suffer a couple of deficiencies. First, traditional FPN-based fault diagnosis methods are insufficient to take into account incomplete and unknown information in diagnosis process. Second, most of the fault diagnosis methods using FPNs are only concerned with forward fault diagnosis, and no or less consider backward cause analysis. In this paper, we present a novel fault diagnosis and cause analysis (FDCA) model using fuzzy evidential reasoning (FER) approach and dynamic adaptive fuzzy Petri nets (DAFPNs) to address the problems mentioned above. The FER is employed to capture all types of abnormal event information which can be provided by experts, and processed by DAFPNs to identify the root causes and determine the consequences of the identified abnormal events. Finally, a practical fault diagnosis example is provided to demonstrate the feasibility and efficacy of the proposed model.     

General SISO Takagi-Sugeno fuzzy systems with linear ruleconsequent are universal approximators

Takagi-Sugeno (TS) fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice. Virtually all the TS fuzzy systems use linear rule consequent. At present, there exist no results (qualitative or quantitative) to answer the fundamentally important question that is especially critical to TS fuzzy systems as fuzzy controllers and models, “Are TS fuzzy systems with linear rule consequent universal approximators?” If the answer is yes, then how can they be constructed to achieve prespecified approximation accuracy and what are the sufficient renditions on systems configuration? In this paper, we provide answers to these questions for a general class of single-input single-output (SISO) fuzzy systems that use any type of continuous input fuzzy sets, TS fuzzy rules with linear consequent and a generalized defuzzifier containing the widely used centroid defuzzifier as a special case. We first constructively prove that this general class of SISO TS fuzzy systems can uniformly approximate any polynomial arbitrarily well and then prove, by utilizing the Weierstrass approximation theorem, that the general TS fuzzy systems can uniformly approximate any continuous function with arbitrarily high precision. Furthermore, we have derived a formula as part of sufficient conditions for the fuzzy approximation that can compute the minimal upper bound on the number of input fuzzy sets and rules needed for any given continuous function and prespecified approximation error bound, An illustrative numerical example is provided     

A comparative study on sufficient conditions for Takagi-Sugenofuzzy systems as universal approximators

Universal approximation is the basis of theoretical research and practical applications of fuzzy systems. Studies on the universal approximation capability of fuzzy systems have achieved great progress in recent years. In this paper, linear Takagi-Sugeno (TS) fuzzy systems that use linear functions of input variables as rule consequent and their special case, named simplified fuzzy systems that use fuzzy singletons as rule consequent, are investigated. On condition that overlapped fuzzy sets are employed, new sufficient conditions for simplified fuzzy systems and linear TS fuzzy systems as universal approximators are given, respectively. Then, a comparative study on existing sufficient conditions is carried out with numeric examples     

Approximation theory of fuzzy systems-MIMO case

In this paper, the approximation properties of MIMO fuzzy systems generated by the product inference are discussed. We first give an analysis of fuzzy basic functions (FBF's) and present several properties of FBF's. Based on these properties of FBF's, we obtain several basic approximation properties of fuzzy systems: 1) basic approximation property which reveals the basic approximation mechanism of fuzzy systems; 2) uniform approximation bounds which give the uniform approximation bounds between the desired (control or decision) functions and fuzzy systems; 3) uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions; and 4) universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect. The similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds     

一类非线性T-S模糊系统的通用逼近性

作为模糊系统理论研究和实际应用的基础，对线性T-S模糊系统通用逼近性的研究已经取得了很大的进展，但对非线性T-S模糊系统的研究却很少。本文将对一类非线性T-S模糊系统的非线性逼近能力进行研究，证明这种模糊系统在输入模糊子集为高斯型隶属函数的情况下，具有通用逼近性。     

Genetically optimized fuzzy polynomial neural networks with fuzzy set-based polynomial neurons

In this paper, we propose and investigate a new category of neurofuzzy networks—fuzzy polynomial neural networks (FPNN) endowed with fuzzy set-based polynomial neurons (FSPNs) We develop a comprehensive design methodology involving mechanisms of genetic optimization, and genetic algorithms (GAs) in particular. The conventional FPNNs developed so far are based on the mechanisms of self-organization, fuzzy neurocomputing, and evolutionary optimization. The design of the network exploits the FSPNs as well as the extended group method of data handling (GMDH). Let us stress that in the previous development strategies some essential parameters of the networks (such as the number of input variables, the order of the polynomial, the number of membership functions, and a collection of the specific subset of input variables) being available within the network are provided by the designer in advance and kept fixed throughout the overall development process. This restriction may hamper a possibility of developing an optimal architecture of the model. The design proposed in this study addresses this issue. The augmented and genetically developed FPNN (gFPNN) results in a structurally optimized structure and comes with a higher level of flexibility in comparison to the one we encounter in the conventional FPNNs. The GA-based design procedure being applied at each layer of the FPNN leads to the selection of the most suitable nodes (or FSPNs) available within the FPNN. In the sequel, two general optimization mechanisms are explored. First, the structural optimization is realized via GAs whereas the ensuing detailed parametric optimization is carried out in the setting of a standard least square method-based learning. The performance of the gFPNN is quantified through experimentation in which we use a number of modeling benchmarks—synthetic and experimental data being commonly used in fuzzy or neurofuzzy modeling. The obtained results demonstrate the superiority of the proposed networks over the models existing in the references.     

基于模糊聚类的神经元识别方法

大脑是生物体内结构和功能最复杂的组织,其中包含上千亿个神经元。作为大脑构造的基本单位,神经元的结构和功能包含很多因素,其中神经元的几何形态特征就是一个重要方面。大脑中神经元的几何形态复杂多样,对其识别分类问题是一个难题。本文在模糊聚类的基础上根据神经元的几何形态建立了模糊集模型 ,并利用多数据库分类模型中的最优划分模型对模糊聚类分析法进行改进。将改进后的模糊聚类方法用于对神经元的识别分类,得到最优的分类结果。根据聚类的评价方法,与其他的聚类方法比较,证明了改进的模糊聚类方法能够得到更好的聚类效果。     

K-积分模意义下折线模糊神经网络的泛逼近性

为克服模糊数运算的复杂性引入折线模糊数的定义,利用折线模糊数的优良性质获得了两个重要不等式,并给出实例说明折线模糊数的逼近能力有效.其次,引进K-拟可加积分和K-积分模概念,在折线模糊数空间满足可分性的基础上,借助于模糊值简单函数和模糊值Bernstein多项式研究了若干函数空间的稠密性问题,获得了可积有界模糊值函数类依K-积分模构成完备可分的度量空间.最后,在K-积分模意义下讨论了四层正则折线模糊神经网络对模糊值简单函数的泛逼近性,进而得到该网络对可积有界函数类也具有泛逼近性.该结果表明正则折线模糊神经网络对连续模糊系统的逼近能力可以推广为对一般可积系统的逼近能力.     

A Note on the Universal Approximation Capability of Support Vector Machines

The approximation capability of support vector machines (SVMs) is investigated. We show the universal approximation capability of SVMs with various kernels, including Gaussian, several dot product, or polynomial kernels, based on the universal approximation capability of their standard feedforward neural network counterparts. Moreover, it is shown that an SVM with polynomial kernel of degree p − 1 which is trained on a training set of size p can approximate the p training points up to any accuracy. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximation capability of fuzzy systems using translations anddilations of one fixed function as membership functions

This paper reports on a related study on approximation theory of fuzzy systems. First, some basic principles are presented to construct membership functions. Then, an approach is proposed to form membership functions by using translations and dilations of one fixed function (called a basis function) which is very similar to that in wavelets analysis. The properties of this type of membership function reflect the advantages of the given approach. Finally, it is proved that fuzzy systems based on such membership functions are universal approximators under certain mild conditions on the basis function. This conclusion expands the family of fuzzy systems which can be universal approximators     

Sufficient and necessary conditions for Boolean fuzzy systems as universal approximators

A formula is first presented to compute the lower upper bounds on the number of fuzzy sets to achieve pre-specified approximation accuracy for an arbitrary multivariate continuous function. The necessary condition for Boolean fuzzy systems as universal approximators with minimal system configurations is then discussed. Two examples are provided to demonstrate how to design a Boolean fuzzy system in order to approximate a given continuous function with a required approximation accuracy.