Learning in the framework of fuzzy lattices

A basis for rigorous versatile learning is introduced theoretically, that is the framework of fuzzy lattices or FL-framework for short, which proposes a synergetic combination of fuzzy set theory and lattice theory. A fuzzy lattice emanates from a conventional mathematical lattice by fuzzifying the inclusion order relation. Learning in the FL-framework can be effected by handling families of intervals, where an interval is treated as a single entity/block the way explained here. Illustrations are provided in a lattice defined on the unit-hypercube where a lattice interval corresponds to a conventional hyperbox. A specific scheme for learning by clustering is presented, namely σ-fuzzy lattice learning scheme or σ-FLL (scheme) for short, inspired from adaptive resonance theory (ART). Learning by the σ-FLL is driven by an inclusion measure σ of the corresponding Cartesian product to be introduced here. We delineate a comparison of the σ-FLL scheme with various neural-fuzzy and other models. Applications are shown to one medical data set and two benchmark data sets, where σ-FLL's capacity for treating efficiently real numbers as well as lattice-ordered symbols separately or jointly is demonstrated. Due to its efficiency and wide scope of applicability the σ-FLL scheme emerges as a promising learning scheme     

Morphological perceptrons with competitive learning: Lattice-theoretical framework and constructive learning algorithm

A morphological neural network is generally defined as a type of artificial neural network that performs an elementary operation of mathematical morphology at every node, possibly followed by the application of an activation function. The underlying framework of mathematical morphology can be found in lattice theory.With the advent of granular computing, lattice-based neurocomputing models such as morphological neural networks and fuzzy lattice neurocomputing models are becoming increasingly important since many information granules such as fuzzy sets and their extensions, intervals, and rough sets are lattice ordered. In this paper, we present the lattice-theoretical background and the learning algorithms for morphological perceptrons with competitive learning which arise by incorporating a winner-take-all output layer into the original morphological perceptron model. Several well-known classification problems that are available on the internet are used to compare our new model with a range of classifiers such as conventional multi-layer perceptrons, fuzzy lattice neurocomputing models, k-nearest neighbors, and decision trees.     

On neurobiological, neuro-fuzzy, machine learning, and statisticalpattern recognition techniques

In this paper, we propose two new neuro-fuzzy schemes, one for classification and one for clustering problems. The classification scheme is based on Simpson's fuzzy min-max method (1992, 1993) and relaxes some assumptions he makes. This enables our scheme to handle mutually nonexclusive classes. The neuro-fuzzy clustering scheme is a multiresolution algorithm that is modeled after the mechanics of human pattern recognition. We also present data from an exhaustive comparison of these techniques with neural, statistical, machine learning, and other traditional approaches to pattern recognition applications. The data sets used for comparisons include those from the machine learning repository at the University of California, Irvine. We find that our proposed schemes compare quite well with the existing techniques, and in addition offer the advantages of one-pass learning and online adaptation.     

Rule inducing by fuzzy lattice reasoning classifier based on metric distances (FLRC-MD)

Recently much more attention has been paid to the applications of lattice theory in different fields. Fuzzy lattice reasoning (FLR) was described lately as a lattice data domain extension of fuzzy-ARTMAP based on a lattice inclusion measure function. In this work, we develop a fuzzy lattice reasoning classifier using various distance metrics. As a consequence, the new algorithm named FLRC-MD shows better classification results and more generalization and it will lead to generate fewer induced rules. To assess the effectiveness of the proposed model, twenty benchmark data sets are tested. The results are compared favorably with those from a number of state-of-the-art machine learning techniques published in the literature. Results obtained confirm the effectiveness of the proposed method.     

基于熵的模糊信息测度研究

模糊信息测度(Fuzzy Information Measures,FIM)是度量两个模糊集之间相似性大小的一种量度,在模式识别、机器学习、聚类分析等研究中,起着重要的作用.文中对模糊测度进行了分析,研究了基于熵的模糊信息测度理论:首先,概述了模糊测度理论,指出了其优缺点;其次,基于信息熵理论,研究了模糊熵理论,建立了模糊熵公理化体系,讨论了各种模糊熵,在此基础上,提出了模糊绝对熵测度、模糊相对熵测度等模糊熵测度;最后,基于交互熵理论,建立了模糊交互熵理论,进而提出了模糊交互熵测度.这些测度理论,不仅丰富与发展了 FIM理论,而且为模式识别、机器学习、聚类分析等理论与应用研究提供了新的研究方法.     

A design and analysis of objective function-based unsupervised neural networks for fuzzy clustering

Fuzzy clustering has played an important role in solving many problems. In this paper, we design an unsupervised neural network model based on a fuzzy objective function, called OFUNN. The learning rule for the OFUNN model is a result of the formal derivation by the gradient descent method of a fuzzy objective function. The performance of the cluster analysis algorithm is often evaluated by counting the number of crisp clustering errors. However, the number of clustering errors alone is not a reliable and consistent measure for the performance of clustering, especially in the case of input data with fuzzy boundaries. We introduce two measures to evaluate the performance of the fuzzy clustering algorithm. The clustering results on three data sets, Iris data and two artificial data sets, are analyzed using the proposed measures. They show that OFUNN is very competitive in terms of speed and accuracy compared to the fuzzy c-means algorithm.     

一种基于密度峰值的针对模糊混合数据的聚类算法

将CFSFDP算法拓展到连续型模糊集和离散型模糊集上,提出了一种针对模糊混合数据的拓展型CFSFDP算法,将其命名为FMD-CFSFDP算法。FMD-CFSFDP算法将样本涵盖的经典信息拓展到了模糊集上,利用寻找密度峰值的方法对模糊样本进行聚类,这是一种建立在模糊集上针对模糊混合数据的基于密度的聚类算法。首先简单介绍了CFSFDP算法及其改进,给出了"模糊混合数据"的数学概念;然后结合传统模糊欧氏距离的概念,分别提出了误差更小的针对连续型模糊集与离散型模糊集的改进型欧氏距离,在此基础上,依托权值构建了针对混合型模糊数据的整体距离。参考CFSFDP算法的聚类步骤给出了FMD-CFSFDP算法的聚类步骤。随后,在不同样本量、不同指标数量、不同簇数、不同取数规则的条件下,对算法进行了随机模拟实验并对聚类结果进行了分析。最后分别总结了FMD-CFSFDP算法的优缺点,并在此基础上提出了改进方案,为今后深入研究提供了参考。     

A Functional-Link-Based Neurofuzzy Network for Nonlinear System Control

This study presents a functional-link-based neurofuzzy network (FLNFN) structure for nonlinear system control. The proposed FLNFN model uses a functional link neural network (FLNN) to the consequent part of the fuzzy rules. This study uses orthogonal polynomials and linearly independent functions in a functional expansion of the FLNN. Thus, the consequent part of the proposed FLNFN model is a nonlinear combination of input variables. An online learning algorithm, which consists of structure learning and parameter learning, is also presented. The structure learning depends on the entropy measure to determine the number of fuzzy rules. The parameter learning, based on the gradient descent method, can adjust the shape of the membership function and the corresponding weights of the FLNN. Furthermore, results for the universal approximator and a convergence analysis of the FLNFN model are proven. Finally, the FLNFN model is applied in various simulations. Results of this study demonstrate the effectiveness of the proposed FLNFN model.      

基于直觉模糊包含度的聚类有效性分析

针对直觉模糊集合数据的聚类有效性问题,提出了一种基于直觉模糊包含度的聚类有效性分析方法。该方法采用直觉模糊包含度和直觉模糊划分熵来评价直觉模糊聚类的有效性。其中,直觉模糊包含度通过增加非隶属度参数对模糊包含度进行直觉化扩展,用于评价类与类间包含的程度;而直觉模糊划分熵用于检验分类结果的可靠性。最后通过典型实例验证了该方法的有效性。     

区间二型模糊相似度与包含度

相似度与包含度是模糊集合理论中的两个重要概念,但对于二型模糊集合的研究还较为少见．鉴于此,提出了新的区间二型模糊相似度与包含度．首先选择了二者的公理化定义;然后基于公理化定义提出了新的计算公式,并讨论了二者的相互转换关系;最后通过实例来验证二者的性能,并将区间二型模糊相似度与Yang-Shih聚类方法相结合,用于高斯区间二型模糊集合的聚类分析,得到了合理的层次聚类树．仿真实例表明新测度具有一定的实用价值．     

用HCM聚类和遗传算法实现多级模糊神经网络

模糊集理论适用于一些实验数据中不确定性和模糊性的建模问题,而模糊推理系统拥有模糊IF-THEN格式的结构化知识表示,但缺少适应性。神经网络本身具有对外部很强的适应性和从过去数据中学习的机制,但基于线性推理的模糊神经网络(FNN)模型作为模糊推理方法不能得到存在于参数间的最终关系,也不能影响接着发生的模糊集合。因此,我们提出了一个多级模糊神经网络(Multi-FNN),使用硬C均值聚类和进化模糊颗粒,利用处理为近似推理的一个线性推理,获得信息微粒和模糊集之间的关系。     

适合大规模数据集且基于LLM的0阶TSK模糊分类器

针对传统分类器的泛化性能差、可解释性及学习效率低等问题, 提出0阶TSK-FC模糊分类器.为了将该分类器 应用到大规模数据的分类中, 提出增量式0阶TSK-IFC模糊分类器, 采用增量式模糊聚类算 法(IFCM($c+p$))训练模糊规则参数并通过适当的矩阵变换提升参数学习效率.仿真实验表明, 与FCPM-IRLS模糊分类器、径向基函数神经网 络相比, 所提出的模糊分类器在不同规模数据集中均能保持很好的性能, 且TSK-IFC模糊分类器在大规模数据分类中尤为突出.     

General fuzzy min-max neural network for clustering andclassification

This paper describes a general fuzzy min-max (GFMM) neural network which is a generalization and extension of the fuzzy min-max clustering and classification algorithms of Simpson (1992, 1993). The GFMM method combines supervised and unsupervised learning in a single training algorithm. The fusion of clustering and classification resulted in an algorithm that can be used as pure clustering, pure classification, or hybrid clustering classification. It exhibits a property of finding decision boundaries between classes while clustering patterns that cannot be said to belong to any of existing classes. Similarly to the original algorithms, the hyperbox fuzzy sets are used as a representation of clusters and classes. Learning is usually completed in a few passes and consists of placing and adjusting the hyperboxes in the pattern space; this is an expansion-contraction process. The classification results can be crisp or fuzzy. New data can be included without the need for retraining. While retaining all the interesting features of the original algorithms, a number of modifications to their definition have been made in order to accommodate fuzzy input patterns in the form of lower and upper bounds, combine the supervised and unsupervised learning, and improve the effectiveness of operations. A detailed account of the GFMM neural network, its comparison with the Simpson's fuzzy min-max neural networks, a set of examples, and an application to the leakage detection and identification in water distribution systems are given     

区间值模糊软集的信息测度及其聚类算法

针对区间值模糊软集信息测度难以精确定义的问题,提出了区间值模糊软集的距离测度、相似度、熵、包含度、子集度的公理化定义,给出了区间值模糊软集的信息测度公式,并讨论了它们的转换关系。然后提出了一个基于相似度的聚类算法,该算法结合区间值模糊软集的特性,着重对给出评价对象的具有相似知识水平的专家进行聚类,同时讨论了算法的计算复杂度。最后通过实例说明该算法能有效地处理专家聚类问题。     

基于阴影集的粗糙模糊可能性C均值聚类算法

相对于硬聚类算法,软聚类算法可以更好地表示具有不精确边界的类簇。粗糙集和模糊集均是用于描述不确定数据的有效的数学工具,二者互为补充。研究人员已经将粗糙集和模糊集的概念相结合,并应用到聚类算法中,提出了粗糙模糊可能性C均值聚类算法。而文中通过引入阴影集,有效地解决了粗糙模糊可能性C均值聚类算法中的阈值选择问题。     

Iterative Bayesian fuzzy clustering toward flexible icon-based assistive software for the disabled

A novel fuzzy clustering technique, called iterative Bayesian fuzzy clustering (IBFC), is presented and applied for grouping and recommendation of icons associated with assistive software meant for the physically disabled. The algorithm incorporates a modified fuzzy competitive learning structure with a Bayesian decision rule. In order to ignore unintended behavior of the user, a Bayesian minimum risk classification rule with two loss coefficients is built into the algorithm. This provides a rational basis for outlier detection in noisy data. In addition, we show that the inclusion of a unique control parameter of IBFC allows for establishment of a strong relationship between learning region and cluster congestion. This interpretation leads to an agglomerative iterative Bayesian fuzzy clustering (AIBFC) framework capable of clustering data of complex structure. The proposed AIBFC framework is applied to design a flexible interface for the icon-based assistive software for the disabled. The latter is utilized in grouping and recommendation of icons. Additionally, the proposed algorithm is shown to outperform several well-known methods for both IRIS and Wisconsin benchmark data sets. Finally, it is shown, using a questionnaire survey of real end-users, that the software designed using AIBFC framework meets users’ needs.     

模糊超球神经网络在模式聚类中的应用

提出和实现了用于模式聚类的无监督模糊超球神经网络.模式集是一个具有超球核 的用隶属函数表示的模糊集,模式集又可以合并成模式类.模糊超球神经网络学习算法能在 几次循环学习中形成模式集,无需对已知模式集重新训练就可融合新样例和精炼已存在的模 式集.模式聚类的数值仿真解释了模糊超球聚类神经网络的优越性能.     

基于神经网络的模糊决策树改进算法

传统决策树通过对特征空间的递归划分寻找决策边界,给出特征空间的“硬”划分。但对于处理大数据和复杂模式问题时,这种精确决策边界降低了决策树的泛化能力。为了让决策树算法获得对不精确知识的自动获取,把模糊理论引进了决策树,并在建树过程中,引入神经网络作为决策树叶节点,提出了一种基于神经网络的模糊决策树改进算法。在神经网络模糊决策树中,分类器学习包含两个阶段：第一阶段采用不确定性降低的启发式算法对大数据进行划分,直到节点划分能力低于真实度阈值[ε]停止模糊决策树的增长;第二阶段对该模糊决策树叶节点利用神经网络做具有泛化能力的分类。实验结果表明,相较于传统的分类学习算法,该算法准确率高,对识别大数据和复杂模式的分类问题能够通过结构自适应确定决策树规模。     

A novel approach for distance-based semi-supervised clustering using functional link neural network

Semi-supervised clustering is gaining importance these days since neither supervised nor unsupervised learning methods in a stand-alone manner provide satisfactory results. Existing semi-supervised clustering techniques are mostly based on pair-wise constraints, which could be misleading. These semi-supervised clustering algorithms also fail to address the problem of dealing with attributes having different weights. In most of the real-life applications, all attributes do not have equal importance and hence same weights cannot be assigned for each attribute. In this paper, a novel distance-based semi-supervised clustering algorithm has been proposed, which uses functional link neural network (FLNN) for finding weights for attributes with small amount of labeled data for further use in parametric Minkowski’s model for clustering. In FLNN, the nonlinearity is captured by enhancing the input using orthonormal basis functions. The effectiveness of the approach has been illustrated over a number of datasets taken from UCI machine learning repository. Comparative performance evaluation demonstrates that the proposed approach outperforms the existing semi-supervised clustering algorithms. The proposed approach has also been successfully used to cluster the crime locations and to find crime hot spots in India on the data provided by National Crime Records Bureau (NCRB).