Inclusion measures of probabilistic linguistic term sets and their application in classifying cities in the Economic Zone of Chengdu Plain

The probabilistic linguistic term set is a powerful tool to express and characterize people’s cognitive complex information and thus has obtained a great development in the last several years. To better use the probabilistic linguistic term sets in decision making, information measures such as the distance measure, similarity measure, entropy measure and correlation measure should be defined. However, as an important kind of information measure, the inclusion measure has not been defined by scholars. This study aims to propose the inclusion measure for probabilistic linguistic term sets. Formulas to calculate the inclusion degrees are put forward Then, we introduce the normalized axiomatic definitions of the distance, similarity and entropy measures of probabilistic linguistic term sets to construct a unified framework of information measures for probabilistic linguistic term sets. Based on these definitions, we present the relationships and transformation functions among the distance, similarity, entropy and inclusion measures. We believe that more formulas to calculate the distance, similarity, inclusion degree and entropy can be induced based on these transformation functions. Finally, we put forward an orthogonal clustering algorithm based on the inclusion measure and use it in classifying cities in the Economic Zone of Chengdu Plain, China.     

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。     

Some results on information measures for complex intuitionistic fuzzy sets

Complex intuitionistic fuzzy sets (CIFSs), modeled by complex-valued membership and nonmembership functions with codomain the unit disc in a complex plane, handle two-dimensional information in a single set. Under this environment, the primary objective of the present study is to introduce some novel formulae of information measures (similarity measures, distance measures, entropies, and inclusion measures) and discuss the transformation relationships among them. To demonstrate the efficiency of the proposed similarity measures, we apply it to pattern recognition problem and a detailed comparative analysis is conducted with some of the existing measures. Further, algorithms based on proposed measures are developed for handing multicriteria decision-making problems and their working is illustrated with the help of an example. Besides this, the practicality of the proposed similarity measure is demonstrated by developing a clustering algorithm under CIFS environment.     

基于犹豫模糊语言术语集的正交模糊聚类算法

犹豫模糊语言术语集（Hesitance Fuzzy Linguistic Term Sets,HFLTSs）允许决策者们用几个可能的语言术语来评估一个属性.近来,采用HFLTSs来进行模糊聚类分析的问题越来越受关注.考虑到目前基于HFLTSs的模糊聚类算法还存在计算复杂度高的问题,提出了一种新的正交模糊聚类算法：首先计算样本之间的距离测度得到距离测度矩阵,接着计算其等价矩阵;然后确定置信水平值,通过置信水平值对等价矩阵进行切割;最后根据切割矩阵的列向量之间的正交关系来确定对应样本是否可以放在同一个类别,以此得到聚类结果.该算法步骤简单,计算复杂度低,并且适合于数据量大的模糊聚类问题.本文末尾将通过一个实例结合k-means聚类算法证明该算法的可行性和高效性.     

Pythagorean Fuzzy Information Measures and Their Applications

Pythagorean fuzzy set (PFS), originally proposed by Yager, is more capable than intuitionistic fuzzy set (IFS) to handle vagueness in the real world. The main purpose of this paper is to investigate the relationship between the distance measure, the similarity measure, the entropy, and the inclusion measure for PFSs. The primary goal of the study is to suggest the systematic transformation of information measures (distance measure, similarity measure, entropy, inclusion measure) for PFSs. For achieving this goal, some new formulae for information measures of PFSs are introduced. To show the efficiency of the proposed similarity measure, we apply it to pattern recognition, clustering analysis, and medical diagnosis. Some illustrative examples are given to support the findings and also demonstrate their practicality and effectiveness of similarity measure between PFSs.     

Novel hesitant fuzzy linguistic entropy and cross-entropy measures in multiple criteria decision making

In this contribution, we mainly investigate how new entropy and cross entropy measures of hesitant fuzzy linguistic term sets (HFLTSs) can be designed by using the counterparts proposed for linguistic term sets (LTSs). In this circumstance, we intend to point out some drawbacks of the existing entropies, and then extend the theory of entropy and cross entropy measures of HFLTSs by constructing a number of new entropies. Furthermore, we compare the results of the approach being proposed based on the new entropy and cross entropy measures with that of the weight-determining method and the hesitant fuzzy linguistic alternative queuing method (HFL-AQM).     

Mining of mixed data with application to catalog marketing

Clustering is one of the most popular techniques in data mining. The goal of clustering is to identify distinct groups in a dataset. Many clustering algorithms have been published so far, but often limited to numeric or categorical data. However, most real world data are mixed, numeric and categorical. In this paper, we propose a clustering algorithm CAVE which is based on variance and entropy, and is capable of mining mixed data. The variance is used to measure the similarity of the numeric part of the data. To express the similarity between categorical values, distance hierarchy has been proposed. Accordingly, the similarity of the categorical part is measured based on entropy weighted by the distances in the hierarchies. A new validity index for evaluating the clustering results has also been proposed. The effectiveness of CAVE is demonstrated by a series of experiments on synthetic and real datasets in comparison with that of several traditional clustering algorithms. An application of mining a mixed dataset for customer segmentation and catalog marketing is also presented.     

Distance measure for Fermatean fuzzy linguistic term sets based on linguistic scale function: An illustration of the TODIM and TOPSIS methods

In this paper, we propose the Fermatean fuzzy linguistic term set (FFLTS) based on the linguistic scale function. A new similarity measure between FFLTSs is constructed, which not only includes the linguistic scale function but also combines the cosine similarity measure and Euclidean distance measure, and then the related properties of the similarity measure are proven. A corresponding distance measure is obtained according to the relationship between the distance measure and similarity measure. Furthermore, we extend the Tomada de Decisão Interativa Multicritério (TODIM) method and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method to the corresponding distance measure under the Fermatean fuzzy linguistic environment. The main advantages of the proposed methods are that they can not only transform linguistic information effectively in different decision environments but also improve the adaptability of FFLTS in decision-making problems. Finally, a numerical example is provided to illustrate the effectiveness and feasibility of the proposed methods, which are also compared with other existing methods. The sensitivity analysis of the parameters and the influence of linguistic scale function on the ranking results are also discussed.     

犹豫模糊信息相似性测度与群一致性测度及群决策应用

针对目前关于犹豫模糊运算与测度的研究中存在的不足,首先给出犹豫模糊熵函数的定义,并将其作为犹豫模糊信息不确定性测度,进而提出犹豫模糊信息特征向量概念,以信息特征向量为出发点对犹豫模糊距离测度和相似性测度展开研究;为优化群决策过程,提出基于完全优先关系的群一致性测度概念并研究其性质;最后,提出基于相似性测度和群一致性测度的群决策方法并结合算例验证所提出方法的有效性.     

A group decision-making model based on incomplete comparative expressions with hesitant linguistic terms

As a result of uncertainty and complexity for environments of decision-making, it is more suitable for decision makers to use hesitant fuzzy linguistic information. In this paper, a novel group decision making (GDM) model based on fuzzy linear programming is proposed for incomplete comparative expressions with hesitant fuzzy linguistic term set (HFLTSs). We establish an equivalence theorem of additive consistency between 2-tuple fuzzy linguistic preference relation (FLPR) and corresponding fuzzy preference relation. Based on this framework, a fuzzy linear programming is established to address incomplete comparative expressions with HFLTSs. It is more important that the proposed fuzzy linear programming has a double action, finding the highest consistent incomplete 2-tuple FLPR and increasing inconsistent 2-tuple FLPR to the additive consistent 2-tuple FLPR based on given incomplete comparative expressions with HFLTSs. By this means, a novel GDM model is constructed based on importance induced ordered weighted averaging operator. Finally, an investment decision-making in real-world is solved by the proposed model, which shows the result of GDM is effectiveness.     

基于残差分析的混合属性数据聚类算法

针对混合属性数据聚类结果精度不高、聚类结果对参数敏感等问题, 提出了基于残差分析的混合属性数据聚类算法(Clustering algorithm for mixed data based on residual analysis) RA-Clust.算法以改进的熵权重混合属性相似性度量对象间的相似性, 以提出的基于KNN和Parzen窗的局部密度计算方法计算每个对象的密度, 通过线性回归和残差分析进行聚类中心预选取, 然后以提出的聚类中心目标优化模型确定真正的聚类中心, 最后将其他数据对象按照距离高密度对象的最小距离划分到相应的簇中, 形成最终聚类.在合成数据集和UCI数据集上的实验结果验证了算法的有效性.与同类算法相比, RA-Clust具有较高的聚类精度.     

基于改进距离和相似度的区间值概率不确定语言决策方法

针对属性值为区间值概率不确定语言术语集(interval-valued probabilistic uncertain linguistic term set, IVPULTS)、专家权重未知的多属性群决策问题,提出一种融合距离和相似度的决策方法。首先,由于现有的IVPULTS中元素的无序性导致距离测度及决策结果不唯一,利用区间优势度方法对区间值概率进行排序,从而形成有序的IVPULTS;同时考虑到现有距离测度区分能力不高,利用不确定语言距离度量方法扩充现有距离公式。其次,基于距离与相似测度存在的对偶关系,为IVPULTS定义了距离相似度公式,并利用改进的相似—信任网络分析法确定不同专家的权重。再次,设计了基于改进距离和相似—信任网络的TOPSIS决策方法(improved distance and similarity-trust network TOPSIS,IDSTN-TOPSIS),从而得到唯一且稳定的方案排序。最后,以新冠疫情下某医疗用品制造公司熔喷布弹性供应商选择为例,验证了所提方法的有效性和优越性。     

基于改进流形距离和人工蜂群的二阶段聚类算法

以改进的流形距离为相似度测度, 结合人工蜂群算法, 提出一种二阶段聚类算法. 首先根据局部密度、最大最小距离和近邻选择对数据集初步归类并得到簇代表点; 然后将聚类归属为优化问题, 通过改进的蜂群算法对簇代表点及没归类的样本点较快地搜索到最优聚类中心, 同时根据流形距离的全局一致性特征, 对样本进行精确的类别划分; 最后将两阶段算法综合归类. 实验结果表明, 所提出的算法可以获得良好的聚类效果.

     

Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS

Pythagorean fuzzy sets (PFSs) were proposed by Yager in 2013 to treat imprecise and vague information in daily life more rigorously and efficiently with higher precision than intuitionistic fuzzy sets. In this paper, we construct new distance and similarity measures of PFSs based on the Hausdorff metric. We first develop a method to calculate a distance between PFSs based on the Hasudorff metric, along with proving several properties and theorems. We then consider a generalization of other distance measures, such as the Hamming distance, the Euclidean distance, and their normalized versions. On the basis of the proposed distances for PFSs, we give new similarity measures to compute the similarity degree of PFSs. Some examples related to pattern recognition and linguistic variables are used to validate the proposed distance and similarity measures. Finally, we apply the proposed methods to multicriteria decision-making by constructing a Pythagorean fuzzy Technique for Order Preference by Similarity to an Ideal Solution and then present a practical example to address an important issue related to social sector. Numerical results indicate that the proposed methods are reasonable and applicable and also that they are well suited in pattern recognition, linguistic variables, and multicriteria decision-making with PFSs.     

一种基于MapReduce的文本聚类方法研究

在文本聚类中,相似性度量是影响聚类效果的重要因素。常用的相似性度量测度,如欧氏距离、相关系数等,只能描述文本间的低阶相关性,而文本间的关系非常复杂,基于低阶相关测度的聚类效果不太理想。一些基于复杂测度的文本聚类方法已被提出,但随着数据规模的扩展,文本聚类的计算量不断增加,传统的聚类方法已不适用于大规模文本聚类。针对上述问题,提出一种基于MapReduce的分布式聚类方法,该方法对传统K-means算法进行了改进,采用了基于信息损失量的相似性度量。为进一步提高聚类的效率,将该方法与基于MapReduce的主成分分析方法相结合,以降低文本特征向量的维数。实例分析表明,提出的大规模文本聚类方法的 聚类性能 比已有的聚类方法更好。     

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 new hesitant fuzzy linguistic approach for multiple attribute decision making based on Dempster–Shafer evidence theory

Hesitant fuzzy linguistic term sets (HFLTSs) are useful tool to represent qualitative information in multiple attribute decision making (MADM), and Dempster–Shafer evidence theory (DSET) has some advantages in denoting and fusing uncertain information. The goal of this paper is to develop a new hesitant fuzzy linguistic (HFL) MADM approach based on the DSET. To realize this goal, we propose a method of converting the original decision matrix expressed by HFLTSs into the evidence matrix with HFLTSs, and develop a weight-determining model for MADM problems with HFL information. Further, in order to integrate the evidences with HFLTSs under all attributes, we propose a combination algorithm for MADM problems based on the combination rule of DSET. Based on these studies, we develop a HFL-DSET approach for MADM problems with unknown weights. Furthermore, an applicable example for supplier selection is used to illustrate the proposed approach. Lastly, some comparative analyses with other HFL-MADM methods are conducted to show the feasibility and superiority of the proposed approach.     

基于共享最近邻的密度自适应邻域谱聚类算法

在谱聚类算法没有先验信息的情况下,对于具有复杂形状和不同密度变化的数据集很难构建合适的相似图,且基于欧氏距离的高斯核函数的相似性度量忽略了全局一致性。针对该问题,提出一种基于共享最近邻的密度自适应邻域谱聚类算法（SC-DANSN）。通过一种无参数的密度自适应邻域构建方法构建无向图,将共享最近邻作为衡量样本之间的相似性度量进而消除参数对构建相似图的影响,体现全局和局部的一致性。实验结果表明,SC-DANSN算法相比K-means算法和基于K最近邻的谱聚类算法（SC-KNN）具有更高的聚类精度,同时相比SC-KNN算法对参数的选取敏感性更低。     

Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship

In this paper, we introduce an axiomatic definition of an interval-valued fuzzy sets’ inclusion measure which is different from Bustince’s [H. Bustince, Indicator of inclusion grade for interval-valued fuzzy sets, Applications to approximate reasoning based on interval-valued fuzzy sets, International Journal of Approximate Reasoning, 23 (2000) 137-209]. The relationship among the normalized distance, the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets is investigated in detail. Furthermore, six theorems are proposed showing how the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets can be deduced by the interval-valued fuzzy sets’ normalized distance based on their axiomatic definitions. Some formulas have also been put forward to calculate the similarity measure, the inclusion measure, and the entropy of interval-valued fuzzy sets.