Fuzzy feature selection

In fuzzy classifier systems the classification is obtained by a number of fuzzy If–Then rules including linguistic terms such as Low and High that fuzzify each feature. This paper presents a method by which a reduced linguistic (fuzzy) set of a labeled multi-dimensional data set can be identified automatically. After the projection of the original data set onto a fuzzy space, the optimal subset of fuzzy features is determined using conventional search techniques. The applicability of this method has been demonstrated by reducing the number of features used for the classification of four real-world data sets. This method can also be used to generate an initial rule set for a fuzzy neural network.     

基于线性判别分析的Choquet积分的符号模糊测度提取

在解决分类问题时,建立在Choquet积分上的分类器以其非线性和不可加性的特点,扮演着越来越重要的角色。由于Choquet积分中的符号模糊测度可以描述各特征对结果的影响,因此Choquet积分在解决数据分类及融合 问题方面具有显著的优势。但是,关于Choquet积分符号模糊测度值的求解,学术界一直缺乏有效的方法。目前最常用的方法是遗传算法,但是遗传算法在解决符号模糊测度值的优化问题时存在算法较为复杂、耗时较长等缺陷。由于符号模糊测度值在Choquet积分分类器中是决定性的重要参数,因此设计出一种有效的符号模糊测度提取方法十分必要。文中提出基于线性判别分析的Choquet积分符号模糊测度的提取方法,推导出在分类问题下Choquet积分的符号模糊测度值的解析式表达,其能够有效、快速地得出关键性参数。分别在人工数据集及基准实际数据集上进行测试与验证,实验结果表明所提方法能有效解决Choquet积分分类器中符号模糊测度的优化问题。     

Treating fuzziness in subjective evaluation data

This paper proposes a technique to deal with fuzziness in subjective evaluation data, and applies it to principal component analysis and correspondence analysis. In the existing method, or techniques developed directly from it, fuzzy sets are defined from some standpoint on a data space, and the fuzzy parameters of the statistical model are identified with linear programming or the method of least squares. In this paper, we try to map the variation in evaluation data into the parameter space while preserving information as much as possible, and thereby define fuzzy sets in the parameter space. Clearly, it is possible to use the obtained fuzzy model to derive things like the principal component scores from the extension principle. However, with a fuzzy model which uses the extension principle, the possibility distribution spreads out as the explanatory variable values increase. This does not necessarily make sense for subjective evaluations, such as a 5-level evaluation, for instance. Instead of doing so, we propose a method for explicitly expressing the vagueness of evaluation, using certain quantities related to the eigenvalues of a matrix which specifies the fuzzy parameter spread. As a numerical example, we present an analysis of subjective evaluation data on local environments.     

正态模糊集合——Fuzzy集理论的新拓展

直觉模糊集（intuitionistic fuzzy sets）、区间值模糊集（interval-valued fuzzy sets）以及Vague集对普通fuzzy集的扩展是给出了隶属度的上下限,把隶属度从[0,1]区间中的一个单值推广到了[0,1]的子区间。但是该子区间犹如一个黑洞,隶属度在其内部的分布情况我们无从知晓,即这个子区间中的每一个值是等可能地作为元素的隶属度还是区间中的某些值较另外的值有更大的可能性呢？为了清晰的刻画出元素的隶属度在[0,1]区间中的分布情况,本文通过对投票模型的分析及正态分布理论,提出了一种新的模糊集合——正态模糊集合,同时对正态模糊集合的交、并、补等基本运算性质进行了讨论,文章最后对正态模糊集与fuzzy集、直觉模糊集的相互关系也作出了详细阐述。正态模糊集合是模糊集合理论的进一步推广,为我们处理模糊信息提供了一种全新的思想方法。     

A novel method for ranking of vague sets for handling the risk analysis of compressor system

Vague sets were first proposed by Gau and Buehrer [11] as an extension of fuzzy sets which encompass fuzzy sets, inter-valued fuzzy sets, as special cases. Vague sets consist of two parts, that is, the membership function and nonmembership function. Therefore, in accordance with practical demand these sets are more flexible than the existing fuzzy sets and provide much more information about the situation. In this paper, a new approach for the ranking of trapezoidal vague sets is introduced. Shortcomings in some existing ranking approaches have been pointed out. Validation of the proposed ranking method has been established through the reasonable properties of the fuzzy quantities. Further, the proposed ranking approach is applied to develop a new method for dealing with vague risk analysis problems to find the probability of failure, of each component of compressor system, which could be used for managerial decision making and future system maintenance strategy. Also, the proposed method provides a useful way for handling vague risk analysis problems.     

Parameterized attribute reduction with Gaussian kernel based fuzzy rough sets

Fuzzy rough sets are considered as an effective tool to deal with uncertainty in data analysis, and fuzzy similarity relations are used in fuzzy rough sets to calculate similarity between objects. On the other hand in kernel tricks, a kernel maps data into a higher dimensional feature space where the resulting structure of the learning task is linearly separable, while the kernel is the inner product of this feature space and can also be viewed as a similarity function. It has been reported there is an overlap between family of kernels and collection of fuzzy similarity relations. This fact motivates the idea in this paper to use some kernels as fuzzy similarity relations and develop kernel based fuzzy rough sets. First, we consider Gaussian kernel and propose Gaussian kernel based fuzzy rough sets. Second we introduce parameterized attribute reduction with the derived model of fuzzy rough sets. Structures of attribute reduction are investigated and an algorithm with discernibility matrix to find all reducts is developed. Finally, a heuristic algorithm is designed to compute reducts with Gaussian kernel fuzzy rough sets. Several experiments are provided to demonstrate the effectiveness of the idea.     

一种基于模糊神经网络的模拟电路故障诊断方法

提出了一种采用小波分析与遗传算法相结合的模糊神经网络对模拟电路进行故障诊断的新方法。该方法采用基于小波分析的主成分分析方法对网络的训练样本进行预处理,提取优化向量后利用遗传算法对模糊神经网络进行训练。对两个模拟电路的诊断实例表明该方法故障覆盖率高,并能有效诊断出同类方法误诊的故障类型。     

红外图像的一种Type-Ⅱ模糊增强算法

提出了一种基于Type-Ⅱ模糊集的红外图像增强算法,该算法首先根据像素的邻域相关性对图像进行预处理,然后以Ostu分割阈值为基础,构造了红外图像的Type-Ⅱ模糊特征平面;然后,采用不同的变换规则对图像进行模糊增强,并将结果进行融合;最后,通过Type reduction和去模糊化操作得到增强后的输出图像。对几幅典型的红外图像的增强实验表明,提出的方法能够有效地提高红外图像的对比度。     

A new time invariant fuzzy time series forecasting method based on particle swarm optimization

In the analysis of time invariant fuzzy time series, fuzzy logic group relationships tables have been generally preferred for determination of fuzzy logic relationships. The reason of this is that it is not need to perform complex matrix operations when these tables are used. On the other hand, when fuzzy logic group relationships tables are exploited, membership values of fuzzy sets are ignored. Thus, in defiance of fuzzy set theory, fuzzy sets’ elements with the highest membership value are only considered. This situation causes information loss and decrease in the explanation power of the model. To deal with these problems, a novel time invariant fuzzy time series forecasting approach is proposed in this study. In the proposed method, membership values in the fuzzy relationship matrix are computed by using particle swarm optimization technique. The method suggested in this study is the first method proposed in the literature in which particle swarm optimization algorithm is used to determine fuzzy relations. In addition, in order to increase forecasting accuracy and make the proposed approach more systematic, the fuzzy c-means clustering method is used for fuzzification of time series in the proposed method. The proposed method is applied to well-known time series to show the forecasting performance of the method. These time series are also analyzed by using some other forecasting methods available in the literature. Then, the results obtained from the proposed method are compared to those produced by the other methods. It is observed that the proposed method gives the most accurate forecasts.     

Ranking efficient decision-making units in data envelopment analysis using fuzzy concept

This paper introduces a new mathematical method for improving the discrimination power of data envelopment analysis and to completely rank the efficient decision-making units (DMUs). Fuzzy concept is utilised. For this purpose, first all DMUs are evaluated with the CCR model. Thereafter, the resulted weights for each output are considered as fuzzy sets and are then converted to fuzzy numbers. The introduced model is a multi-objective linear model, endpoints of which are the highest and lowest of the weighted values. An added advantage of the model is its ability to handle the infeasibility situation sometimes faced by previously introduced models.     

Classification by nonlinear integral projections

A new method based on nonlinear integral projections for classification is presented. The contribution rate of each combination of the feature attributes, including each singleton, toward the classification is represented by a fuzzy measure. The nonadditivity of the fuzzy measure reflects the interactions among the feature attributes. The weighted Choquet integral with respect to the fuzzy measure serves as an aggregation tool to project the feature space onto a real axis optimally according to an error criterion, and the classifying attribute is properly numerical analysed on the axis simultaneously making the classification simple. To implement the classification, we need to determine the unknown parameters, the values of fuzzy measure and the weight function. This can be done by running an adaptive genetic algorithm on the given training data. The new classifier is tested by recovering the preset parameters from a set of artificial training data generated from these parameters. It also performs well on several real-world data sets. Beyond discriminating classes, this method can also learn the scaling requirements and the respective importance indexes of the feature attributes as well as the relationships among them. A comprehensive discussion on the semantic and geometric meanings of the parameters is given. Moreover, we show how these parameters' values can be used for short-listing important feature attributes to reduce the complexity (dimensions) of the classification problem. Our new method also compares favorably with other methods on some well-known real-world benchmarks.     

红外图像的一种Type-II模糊增强算法

提出了一种基于Type-II模糊集的红外图像增强算法,该算法首先根据像素的邻域相关性对图像进行预处理,然后以Ostu分割阈值为基础,构造了红外图像的Type-II模糊特征平面;然后,采用不同的变换规则对图像进行模糊增强,并将结果进行融合;最后,通过Type reduction和去模糊化操作得到增强后的输出图像。对几幅典型的红外图像的增强实验表明,提出的方法能够有效地提高红外图像的对比度。     

Adaptive fuzzy identification and predictive control for industrial processes

This paper proposes a method for adaptive identification and control for industrial applications. The learning of a T–S fuzzy model is performed from input/output data to approximate unknown nonlinear processes by a hierarchical genetic algorithm (HGA). The HGA approach is composed by five hierarchical levels where the following parameters of the T–S fuzzy system are learned: input variables and their respective time delays, antecedent fuzzy sets, consequent parameters, and fuzzy rules. In order to reduce the computational cost and increase the algorithm’s performance an initialization method is applied on HGA. To deal with nonlinear plants and time-varying processes, the T–S fuzzy model is adapted online to maintain the quality of the identification/control. The identification methodology is proposed for two application problems: (1) the design of data-driven soft sensors, and (2) the learning of a model for the Generalized predictive control (GPC) algorithm. The integration of the proposed adaptive identification method with the GPC results in an effective adaptive predictive fuzzy control methodology. To validate and demonstrate the performance and effectiveness of the proposed methodologies, they are applied on identification of a model for the estimation of the flour concentration in the effluent of a real-world wastewater treatment system; and on control of a simulated continuous stirred tank reactor (CSTR) and on a real experimental setup composed of two coupled DC motors. The results are presented, showing that the developed evolving T–S fuzzy model can identify the nonlinear systems satisfactorily and it can be used successfully as a prediction model of the process for the GPC controller.     

Design of a two-stage fuzzy classification model

This paper proposes a novel two-stage fuzzy classification model established by the fuzzy feature extraction agent (FFEA) and the fuzzy classification unit (FCU). At first, we propose a FFEA to validly extraction the feature variables from the original database. And then, the FCU, which is the main determination of the classification result, is developed to generate the if–then rules automatically. In fact, both the FFEA and FCU are fuzzy models themselves. In order to obtain better classification results, we utilize the genetic algorithms (GAs) and adaptive grade mechanism (AGM) to tune the FFEA and FCU, respectively, to improve the performance of the proposed fuzzy classification model. In this model, GAs are used to determine the distribution of the fuzzy sets for each feature variable of the FFEA, and the AGM is developed to regulate the confidence grade of the principal if–then rule of the FCU. Finally, the well-known Iris, Wine, and Glass databases are exploited to test the performances. Computer simulation results demonstrate that the proposed fuzzy classification model can provide a sufficiently high classification rate in comparison with other models in the literature.     

Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images

Two measures are presented for comparing fuzzy sets. The method for constructing these measures is studied, starting from fuzzy DI-subsethood measures (see [H. Bustince, V. Mohedano, E. Barrenechea, M. Pagola, Definition and construction of fuzzy DI-subsethood measures, Information Sciences 176 (2006) 3190-3231]). We then analyze and compare the properties satisfied by the measures and those satisfied by other classical indices used in the literature on fuzzy sets. The minimal set of conditions are studied that, from our point of view, must be met by any given measure for comparing images. We also prove that only one of the measures identified fulfills such conditions. Finally, all the measures studied are applied to different images and the results are analyzed, indicating that measures constructed from fuzzy DI-subsethood measures provide the best results.     

基于模糊理论和遗传算法的导弹故障诊断方法研究

研究了一种将有向图、模糊理论和遗传算法相结合的智能故障诊断方法。用有向图来构造系统模型,用模糊集来解决模型中的不确定性问题,用遗传算法来对可能的故障传播路径进行搜索。当被诊断的系统含有不可测量节点时,该方法仍然可以很好地进行诊断。将该方法应用于某大型导弹武器装备的故障诊断系统中,实践证明,该方法行之有效并可以大大提高故障诊断效率。     

Intensity standardization simplifies brain MR image segmentation

Typically, brain MR images present significant intensity variation across patients and scanners. Consequently, training a classifier on a set of images and using it subsequently for brain segmentation may yield poor results. Adaptive iterative methods usually need to be employed to account for the variations of the particular scan. These methods are complicated, difficult to implement and often involve significant computational costs. In this paper, a simple, non-iterative method is proposed for brain MR image segmentation. Two preprocessing techniques, namely intensity-inhomogeneity-correction, and more importantly MR image intensity standardization, used prior to segmentation, play a vital role in making the MR image intensities have a tissue-specific numeric meaning, which leads us to a very simple brain tissue segmentation strategy.Vectorial scale-based fuzzy connectedness and certain morphological operations are utilized first to generate the brain intracranial mask. The fuzzy membership value of each voxel within the intracranial mask for each brain tissue is then estimated. Finally, a maximum likelihood criterion with spatial constraints taken into account is utilized in classifying all voxels in the intracranial mask into different brain tissue groups. A set of inhomogeneity corrected and intensity standardized images is utilized as a training data set. We introduce two methods to estimate fuzzy membership values. In the first method, called SMG (for simple membership based on a gaussian model), the fuzzy membership value is estimated by fitting a multivariate Gaussian model to the intensity distribution of each brain tissue whose mean intensity vector and covariance matrix are estimated and fixed from the training data sets. The second method, called SMH (for simple membership based on a histogram), estimates fuzzy membership value directly via the intensity distribution of each brain tissue obtained from the training data sets. We present several studies to evaluate the performance of these two methods based on 10 clinical MR images of normal subjects and 10 clinical MR images of Multiple Sclerosis (MS) patients. A quantitative comparison indicates that both methods have overall better accuracy than the k-nearest neighbors (kNN) method, and have much better efficiency than the Finite Mixture (FM) model-based Expectation-Maximization (EM) method. Accuracy is similar for our methods and EM method for the normal subject data sets, but much better for our methods for the patient data sets.     

Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets

In this paper, we present a new method to handle fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. First, we present the arithmetic operations between interval type-2 fuzzy sets. Then, we present a fuzzy ranking method to calculate the ranking values of interval type-2 fuzzy sets. We also make a comparison of the ranking values of the proposed method with the existing methods. Based on the proposed fuzzy ranking method and the proposed arithmetic operations between interval type-2 fuzzy sets, we present a new method to handle fuzzy multiple attributes group decision-making problems. The proposed method provides us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 fuzzy sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of attributes.     

Towards optimal use of incomplete classification data

Data for classification are often incomplete. The multiple-values construction method (MVCM) can be used to include data with missing values for classification. In this study, the MVCM is implemented by using fuzzy sets theory in the context of classification with discrete data. By using the fuzzy sets based MVCM, data with missing values can add values to classification, but can also introduce excessive uncertainty. Furthermore, the computational cost for the use of incomplete data could be prohibitive if the scale of missing values is large. This paper discusses the association between classification performance and the use of incomplete data. It proposes an algorithm of near-optimal use of incomplete classification data. An experiment with real-world data demonstrates the usefulness of the algorithm.     

模糊决策的基础——模糊集比较与排序

研究模糊决策中模糊集的比较与排序问题。通过引入模糊极大集和模糊极小集为参照系统并以海明距离为计量工具，定义了两个模糊效用函数和一个模糊优先关系作为模糊集的排序指标。前者适合于多个模糊集的整体分析，后者适合于两两之间的比较判别。对于两个模糊集的排序问题，模糊效用函数自动退化为相应的模糊优先关系。系统分析了3种指标的性能及关系，并举例说明了它们的应用。