Fuzzy rough set theory for the interval-valued fuzzy information systems

The concept of the rough set was originally proposed by Pawlak as a formal tool for modelling and processing incomplete information in information systems, then in 1990, Dubois and Prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets. The aim of this paper is to present a new extension of the rough set theory by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory, i.e., the interval-valued fuzzy rough set model is presented based on the interval-valued fuzzy information systems which is defined in this paper by a binary interval-valued fuzzy relations RF⁽ⁱ⁾(U×U) on the universe U. Several properties of the rough set model are given, and the relationships of this model and the others rough set models are also examined. Furthermore, we also discuss the knowledge reduction of the classical Pawlak information systems and the interval-valued fuzzy information systems respectively. Finally, the knowledge reduction theorems of the interval-valued fuzzy information systems are built.     

基于改进模糊决策粗糙集的最小化决策代价特征选择算法

模糊决策粗糙集是决策粗糙集理论在模糊集环境下的重要延伸,然而该模型对含噪声的数据不具有很好的容忍性。为此在传统的模糊相似关系中引入一个限定阈值,提出一种改进的模糊相似关系。在其基础上对原始的模糊决策粗糙集进行重构,提出一种改进的模糊决策粗糙集模型。根据不同的特征选择方式,利用所提出的改进模型设计出两种搜索策略的最小化决策代价特征选择算法。实验分析表明,该算法比传统算法具有更高的优越性。     

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.     

基于模糊邻域粗糙集的启发式属性约简算法

属性约简是机器学习等领域中常用的数据预处理方法。在基于粗糙集理论的属性约简算法中,大多是根据单一的方法来度量属性重要度。为了从多角度对属性达到更为优越的评估效果,首先在已有的模糊邻域粗糙集模型中定义属性依赖度度量,然后根据粒计算理论中知识粒度的概念,在模糊邻域粗糙集模型下提出了模糊邻域粒度度量。由于属性依赖度和知识粒度代表了不同视角的属性评估方法,因此将这两种方法结合起来用于信息系统的属性重要度评估,最后给出一种启发式属性约简算法。实验结果表明,所提出的算法具有较好的属性约简性能。     

基于模糊知识粒度的混合属性约简算法

现实世界中常常包含着海量的、不完整的、模糊及不精确的数据或对象,使得模糊信息粒化成为近年来研究趋势。利用论域上的模糊等价关系定义了模糊粒度世界的模糊知识粒度,给出了新的属性约简条件和核属性计算方法,以便更好地挖掘出潜在的、有利用价值的信息。针对粗糙集在对连续属性约简的过程中容易造成信息缺失和不能对模糊属性处理的现象,提出了一种基于模糊知识粒度对混合决策系统约简的启发式算法,省去了连续属性离散化过程,减少了计算量,为离散值域和混合值域约简提供了统一的方法。最后通过实例验证了其有效性。     

基于直觉模糊粗糙集的属性约简

针对Jensen下近似定义的局限性,提出一种新的等价类形式的近似算子表示,并将其推广到直觉模糊环境.在此基础上,将相对正域、相对约简、相对核等粗糙集的知识约简概念推广到直觉模糊环境,提出一种直觉模糊信息系统的启发式属性约筒算法.实例计算表明.该方法比Jensen的属性约简方法更为合理有效.     

Roughness based on fuzzy ideals

The theory of rough set, proposed by Pawlak and the theory of fuzzy set, proposed by Zadeh are complementary generalizations of classical set theory. Many sets are naturally endowed with two binary operations: addition and multiplication. One concept which does this is a ring. This paper concerns a relationship between rough sets, fuzzy sets and ring theory. It is a continuation of ideas presented by Kuroki and Wang [N. Kuroki, P.P. Wang, The lower and upper approximations in a fuzzy group, Inform. Sci. 90 (1996) 203-220]. We consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal. In fact, we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring. Some characterizations of the above approximations are made and some examples are presented.     

不完备模糊目标信息系统粗集模型与知识约简

在不完备信息系统和模糊目标信息系统的基础上,将不完备近似空间和模糊目标信息系统有机地结台起来,提出了不完备模糊目标信息系统的概念;同时还给出了不完备模糊目标信息系统的粗糙集模型,该模型是完备模糊目标信息系统和经典目标信息系统粗糙集模型的推广。还给出了系统知识约简的一些相关概念及其辨识矩阵的知识约简方法。     

A fuzzy genetic algorithm for the discovery of process parameter settings using knowledge representation

In this paper, we propose a fuzzy genetic algorithm (Fuzzy-GA) approach integrating fuzzy rule sets and their membership function sets, in a chromosome. The proposed approach consists of two processes: knowledge representation and knowledge assimilation. The knowledge of process parameter setting is encoded as a string with a fuzzy rule set and the associated membership functions. The historical process data forming a combined string is used as the initial knowledge population, which is then ready for knowledge assimilation. A genetic algorithm is used to generate an optimal or nearly optimal fuzzy set and membership functions for the process parameters. The originality of this research is that the proposed system is equipped with the ability to take advantage of assessing the loss which is caused by discrepancy with a process target, thereby enabling the identification of the best set of process parameters. The approach is demonstrated by the use of an experimental example drawn from a semiconductor manufacturer and the results show us that the suggested approach is able to achieve an optimal solution for a process parameter setting problem.     

Measures of general fuzzy rough sets on a probabilistic space

Fuzzy rough set is a generalization of crisp rough set, which deals with both fuzziness and vagueness in data. The measures of fuzzy rough sets aim to dig its numeral characters in order to analyze data effectively. In this paper we first develop a method to compute the cardinality of fuzzy set on a probabilistic space, and then propose a real number valued function for each approximation operator of the general fuzzy rough sets on a probabilistic space to measure its approximate accuracy. The functions of lower and upper approximation operators are natural generalizations of the belief function and plausibility function in Dempster-Shafer theory of evidence, respectively. By using these functions, accuracy measure, roughness degree, dependency function, entropy and conditional entropy of general fuzzy rough set are proposed, and the relative reduction of fuzzy decision system is also developed by using the dependency function and characterized by the conditional entropy. At last, these measure functions for approximation operators are characterized by axiomatic approaches.     

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.     

可调节模糊粗糙集:模型与属性约简

模糊粗糙集是经典粗糙集为适应实际应用需求所进行的拓展,然而目前很多的模糊粗糙集模型都仅仅使用多个二元关系的简单融合方式,不具备调节功能。为解决这一问题,使用参数化的二元算子,提出了一种可调节的模糊粗糙集模型。在此基础上,将近似质量作为度量标准,使用启发式算法来求解可调节模糊粗糙集的约简。最后对可调节模糊粗糙集的近似质量和约简与强模糊粗糙集、弱模糊粗糙集的结果进行了比较分析。实验结果表明,可调节模糊粗糙集通过使用不同的参数,具有很好的调节作用,是强模糊粗糙集和弱模糊粗糙集的一种泛化形式。     

基于距离比值尺度的模糊粗糙集属性约简

属性约简能有效地去除不必要属性,提高分类器的性能。模糊粗糙集是处理不确定信息的重要范式,能有效地应用于属性约简。在模糊粗糙集中,样本分布的不确定性会影响对象的近似集,进而影响有效属性约简的获取。为有效地定义近似集,文中提出了基于距离比值尺度的模糊粗糙集,该模型引入了基于距离比值尺度的样本集的定义,通过对距离比值尺度的控制,避免了样本分布不确定性对近似集的影响;给出了该模型的基本性质,定义了新的依赖度函数,进而设计了属性约简算法;以SVM,NaiveBayes和J48作为测试分类器,在UCI数据集上评测所提算法的性能。实验结果表明,所提出的属性约简算法能够有效获取约简并提高分类的精度。     

A fuzzy neural network with fuzzy impact grades

Fuzzy rule derivation is often difficult and time-consuming, and requires expert knowledge. This creates a common bottleneck in fuzzy system design. In order to solve this problem, many fuzzy systems that automatically generate fuzzy rules from numerical data have been proposed. In this paper, we propose a fuzzy neural network based on mutual subsethood (MSBFNN) and its fuzzy rule identification algorithms. In our approach, fuzzy rules are described by different fuzzy sets. For each fuzzy set representing a fuzzy rule, the universe of discourse is defined as the summation of weighted membership grades of input linguistic terms that associate with the given fuzzy rule. In this manner, MSBFNN fully considers the contribution of input variables to the joint firing strength of fuzzy rules. Afterwards, the proposed fuzzy neural network quantifies the impacts of fuzzy rules on the consequent parts by fuzzy connections based on mutual subsethood. Furthermore, to enhance the knowledge representation and interpretation of the rules, a linear transformation from consequent parts to output is incorporated into MSBFNN so that higher accuracy can be achieved. In the parameter identification phase, the backpropagation algorithm is employed, and proper linear transformation is also determined dynamically. To demonstrate the capability of the MSBFNN, simulations in different areas including classification, regression and time series prediction are conducted. The proposed MSBFNN shows encouraging performance when benchmarked against other models.     

Rough set theory for the interval-valued fuzzy information systems

The notion of a rough set was originally proposed by Pawlak [Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (5) (1982) 341-356]. Later on, Dubois and Prade [D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System 17 (2-3) (1990) 191-209] introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets. This paper deals with an interval-valued fuzzy information system by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory and discusses the basic rough set theory for the interval-valued fuzzy information systems. In this paper we firstly define the rough approximation of an interval-valued fuzzy set on the universe U in the classical Pawlak approximation space and the generalized approximation space respectively, i.e., the space on which the interval-valued rough fuzzy set model is built. Secondly several interesting properties of the approximation operators are examined, and the interrelationships of the interval-valued rough fuzzy set models in the classical Pawlak approximation space and the generalized approximation space are investigated. Thirdly we discuss the attribute reduction of the interval-valued fuzzy information systems. Finally, the methods of the knowledge discovery for the interval-valued fuzzy information systems are presented with an example.     

Prediction of service life of large centrifugal compressor remanufactured impeller based on clustering rough set and fuzzy Bandelet neural network

In order to predict the service life of large centrifugal compressor impeller correctly, the rough set and fuzzy Bandelet neural network are combined to construct the novel prediction model which can give full play to theirs advantages. The attribute reduction algorithm based rough set and clustering method is firstly designed to optimize the inputting variables of fuzzy Bandelet neural network. And then the prediction model based on fuzzy Bandelet neural network is proposed, the Bandelet function is used as the excitation function of hidden layer and is combined with fuzzy theory to improve the prediction effectiveness of the prediction model. The training algorithm of fuzzy Bandelet neural network is designed based on improved genetic algorithm, the improved genetic algorithm introduces the adaptive differential evolution method into the traditional genetic algorithm, which can effectively optimize the parameters of fuzzy Bandelet neural network. Finally, the original 30 input variables of fuzzy Bandelet neural network are reduced to 9 input nodes based on rough set using 500 remanufacturing impellers as research objects. The service life of remanufacturing impeller is predicted based on three prediction models, and simulation results show that the fuzzy Bandelet neural network optimized by improved genetic algorithm has highest prediction precision and efficiency, which can correctly predict the service life of remanufacturing impeller.     

On fuzzy approximation operators in attribute reduction with fuzzy rough sets

Generally speaking, there are four fuzzy approximation operators defined on a general triangular norm (t-norm) framework in fuzzy rough sets. Different types of t-norms specify various approximation operators. One issue whether and how the different fuzzy approximation operators affect the result of attribute reduction is then arisen. This paper addresses this issue from the theoretical viewpoint by reviewing attribute reduction with fuzzy rough sets and then describing and proving some theorems which demonstrate the effects of the fuzzy approximation operators on the results of attribute reduction. First, we review some notions of attribute reduction with fuzzy rough sets, such as positive region, dependency degree and attribute reduction. We then present and prove some theorems which describe how and to what degree fuzzy approximation operators impact the performance of attribute reduction. Finally, we report some experimental simulation results which demonstrate the effectiveness and correctness of the theoretical contributions. One main contribution in this paper is that we have described and proven that each attribute reduction obtained using one type of fuzzy lower approximation operator always contains one reduction obtained using the other type of fuzzy lower approximation operator.     

A multi-objective decision-making model with fuzzy rough coefficients and its application to the inventory problem

In this paper, we concentrate on developing a fuzzy rough multi-objective decision-making model according to uncertainty theory. We present some equivalent models and a traditional algorithm based on an interactive fuzzy satisfying method, which is similar to the interactive fuzzy rough satisfying method, in order to obtain a satisfying solution for the decision maker. In addition, the technique of fuzzy rough simulation is applied to deal with general fuzzy rough objective functions and fuzzy rough constraints which are usually difficult to convert into their equivalents. Furthermore, combined with the techniques of fuzzy rough simulation, a genetic algorithm using the compromise approach is designed for solving a fuzzy rough multi-objective programming problem. Finally, a model is applied to an inventory problem to illustrate the usefulness of the proposed model and algorithm, and then a sensitivity analysis is made.     

应用近似模糊熵的不完备信息系统属性约简

属性约简是Rough集理论的重要研究内容,基于信息熵的属性约简是一种有效的属性约简方法。在实际应用中,获取的信息系统通常是不完备的。针对这种问题,在容差关系下对个体进行分类时,基于属性子集redu与CAttr(属性全集)-redu之间的内在联系,定义了一种新的知识熵,提出了一种新的应用近似模糊熵的不完备信息系统属性约简算法(newS算法),其时间复杂度是O(｜C｜²∑mi=1(k^p_i)²)。最后,在ROSE和UCI data中的6个数据集上进行了实验仿真,结果表明newS算法是可行的,并且在同等约简效果下与其他算法相比具有更高的属性约简效率。     

Soft fuzzy rough sets for robust feature evaluation and selection

The fuzzy dependency function proposed in the fuzzy rough set model is widely employed in feature evaluation and attribute reduction. It is shown that this function is not robust to noisy information in this paper. As datasets in real-world applications are usually contaminated by noise, robustness of data analysis models is very important in practice. In this work, we develop a new model of fuzzy rough sets, called soft fuzzy rough sets, which can reduce the influence of noise. We discuss the properties of the model and construct a new dependence function from the model. Then we use the function to evaluate and select features. The presented experimental results show the effectiveness of the new model.