Evaluation method based on fuzzy relations between Dempster–Shafer belief structure

Many relations in the real world can be described by mathematical language. Fuzzy set theory can transform human language into mathematical language and use membership degree function to describe relations between events. Dempster–Shafer evidence theory provides basic probability assignment (BPA), which can describe the occurrence rate of attributes in basic events. Based on the known membership degree function and BPA distribution, a new evaluation method is proposed in this paper to analyze decision making. Given the relations among relevant events, which are expressed by BPA distribution and membership degree function, the relations among basic events and top event can be obtained. The Dempster's combination rule and pignistic probability transformation are used to transform BPA distribution into probability distribution. The belief measure is applied to deal with these fuzzy relations. Some numerical examples are given in this paper to illustrate the proposed evaluation methodology.     

非单点证据体数据融合的Dempster组合规则适用性评价方法

针对待融合的证据体包含基本概率分配值相差较大的非单点证据时,基于pignistic距离的Dempster组合规则适用性评价方法判定结果存在模糊性甚至不准确的问题,提出了一种表示证据体之间关联性的改进pignistic距离,并将改进的pignistic距离与经典冲突系数相结合,提出了对Dempster组合规则适用性评价的新方法。在新方法中,定义了一种新的证据体冲突衡量系数用于判定Dempster规则的适用性。当经典冲突系数为0时,新系数与改进pignistic距离一致;当经典冲突系数不为0时,新系数与改进pignistic距离和经典冲突系数之和的平均值一致。算例分析的结果表明,与基于pignistic距离的Dempster规则适用性评价方法相比,新的基于改进pignistic距离的Dempster组合规则适用性评价方法有较好的适用性和合理性。     

A new distance between two bodies of evidence

We present a measure of performance (MOP) for identification algorithms based on the evidential theory of Dempster–Shafer. As an MOP, we introduce a principled distance between two basic probability assignments (BPAs) (or two bodies of evidence) based on a quantification of the similarity between sets. We give a geometrical interpretation of BPA and show that the proposed distance satisfies all the requirements for a metric. We also show the link with the quantification of Dempster's weight of conflict proposed by George and Pal. We compare this MOP to that described by Fixsen and Mahler and illustrate the behaviors of the two MOPs with numerical examples.     

A new probability transformation method based on a correlation coefficient of belief functions

The Dempster-Shafer evidence theory is widely used in many fields of information fusion because of its advantage in handling uncertain information. One of the key issues in this theory is how to make decision based on a basic probability assignment (BPA). Currently, a feasible scheme is transforming a BPA to a distribution of probabilities. However, little attention was paid to the correlation between BPA and probability distribution. In this paper, a novel method about the probability transformation based on a correlation coefficient of belief functions is proposed. The correlation coefficient is a new measurement, which can effectively measure the correlation between BPAs. The proposed method aims at maximizing the correlation coefficient between the given BPA and the transformed probability distribution. On the basis of this idea, the corresponding probability distribution can be obtained and could reflect the original information of the given BPA to the maximum extent. It is valid to consider that the proposed probability transformation method is reasonable and effective. Numerical examples are given to show the effectiveness of the proposed method.     

Intuitionistic fuzzy evidential power aggregation operator and its application in multiple criteria decision-making

The theory of intuitionistic fuzzy sets (IFS) is widely used for dealing with vagueness and the Dempster--Shafer (D-S) evidence theory has a widespread use in multiple criteria decision-making problems under uncertain situation. However, there are many methods to aggregate intuitionistic fuzzy numbers (IFNs), but the aggregation operator to fuse basic probability assignment (BPA) is rare. Power average (P-A) operator, as a powerful operator, is useful and important in information fusion. Motivated by the idea of P-A power, in this paper, a new operator based on the IFS and D-S evidence theory is proposed, which is named as intuitionistic fuzzy evidential power average (IFEPA) aggregation operator. First, an IFN is converted into a BPA, and the uncertainty is measured in D-S evidence theory. Second, the difference between BPAs is measured by Jousselme distance and a satisfying support function is proposed to get the support degree between each other effectively. Then the IFEPA operator is used for aggregating the original IFN and make a more reasonable decision. The proposed method is objective and reasonable because it is completely driven by data once some parameters are required. At the same time, it is novel and interesting. Finally, an application of developed models to the ‘One Belt, One road’ investment decision-making problems is presented to illustrate the effectiveness and feasibility of the proposed operator.     

Some notes on betting commitment distance in evidence theory

The distance of evidence, which represents the degree of dissimilarity between bodies of evidence, has attracted more and more interest and has found extensive uses in many realms. In this paper some notes on a widely used distance of evidence, i.e., betting commitment distance, are provided, including the arguments on the rationality of its definition, some misuses and some counter-intuitive behaviors of betting commitment distance. Several numerical examples are also provided to support and verify our arguments.     

An Evidential Spam-Filtering Framework

Spam, also known as unsolicited bulk e-mail (UBE), has recently become a serious threat that negatively impacts the usability of legitimate mails. In this article, an evidential spam-filtering framework is proposed. As a useful tool to handle uncertainty, the Dempster–Shafer theory of evidence (D–S theory) is integrated into the proposed approach. Five representative features from an e-mail header are analyzed. With a machine-learning algorithm, e-mail headers with known classifications are used to train the framework. When using the framework for a given e-mail header, its representative features are quantified. Although in classical probability theory, possibilities are forcedly assigned even when information is not adequate, in our approach, for every word in an e-mail subject, basic probability assignments (BPA) are assigned in a more flexible way, thus providing a more reasonable result. Finally, BPAs are combined and transformed into pignistic probabilities for decision-making. Empirical trials on real-world datasets show the efficiency of the proposed framework.     

Combination of sources of evidence with different discounting factors based on a new dissimilarity measure

The sources of evidence may have different reliability and importance in real applications for decision making. The estimation of the discounting (weighting) factors when the prior knowledge is unknown have been regularly studied until recently. In the past, the determination of the weighting factors focused only on reliability discounting rule and it was mainly dependent on the dissimilarity measure between basic belief assignments (bba's) represented by an evidential distance. Nevertheless, it is very difficult to characterize efficiently the dissimilarity only through an evidential distance. Thus, both a distance and a conflict coefficient based on probabilistic transformations BetP are proposed to characterize the dissimilarity. The distance represents the difference between bba's, whereas the conflict coefficient reveals the divergence degree of the hypotheses that two belief functions strongly support. These two aspects of dissimilarity are complementary in a certain sense, and their fusion is used as the dissimilarity measure. Then, a new estimation method of weighting factors is presented by using the proposed dissimilarity measure. In the evaluation of weight of a source, both its dissimilarity with other sources and their weighting factors are considered. The weighting factors can be applied in the both importance and reliability discounting rules, but the selection of the adapted discounting rule should depend on the actual application. Simple numerical examples are given to illustrate the interest of the proposed approach.     

A dissimilarity measure for an arbitrary number of probability distributions

In this article we introduce a new function whose value is measure of the dissimilarity between several probability distributions. After a brief review of some previous measures for two probability distributions, a new function is introduced and its useful characteristics are presented. The most interesting feature of this new function is that it can be used for an arbitrary number of probability distributions.     

基于新的距离度量的K-Modes聚类算法

传统的K-Modes聚类算法采用简单的0-1匹配差异方法来计算同一分类属性下两个属性值之间的距离, 没有充分考虑其相似性. 对此, 基于粗糙集理论, 提出了一种新的距离度量. 该距离度量在度量同一分类属性下两个属性值之间的差异时, 克服了简单0-1匹配差异法的不足, 既考虑了它们本身的异同, 又考虑了其他相关分类属性对它们的区分性. 并将提出的距离度量应用于传统K-Modes聚类算法中. 通过与基于其他距离度量的K-Modes聚类算法进行实验比较, 结果表明新的距离度量是更加有效的.     

基于粗糙集的改进K—Modes聚类算法

传统的K-Modes算法采用简单匹配的方法来计算对象之间的距离,并没有充分考虑同一属性下的两个不同值之间的相似性.基于粗糙集中的上、下近似,提出了一种新的距离度量,并重新定义了类中心,对传统K-Modes算法进行了改进.与其他改进K-Modes算法进行了比较,实验结果表明,基于粗糙集的改进K-Modes算法有效地提高了聚类精度.     

A dissimilarity measure based on Jensen Shannon divergence measure

The need of suitable measures to find the distance between two probability distributions arises as they play an eminent role in problems based on discrimination and inferences. In this communication, we have introduced one such divergence measure based on well-known Shannon entropy and established its existence. In addition to this, a new dissimilarity measure for intuitionistic fuzzy sets corresponding to proposed divergence measure is also introduced and validated. Some major properties of the proposed dissimilarity measure are also discussed. Further, a new multiple attribute decision-making (MADM) method based on the proposed dissimilarity measure is introduced by using the concept of TOPSIS and is thoroughly explained with the help of an illustrated example on supplier selection problem. Finally, the application of proposed dissimilarity measure is given in pattern recognition and the performance is compared with some existing divergence measures in the literature.     

Mahalanobis distance based on fuzzy clustering algorithm for image segmentation

Conventional Fuzzy C-means (FCM) algorithm uses Euclidean distance to describe the dissimilarity between data and cluster prototypes. Since the Euclidean distance based dissimilarity measure only characterizes the mean information of a cluster, it is sensitive to noise and cluster divergence. In this paper, we propose a novel fuzzy clustering algorithm for image segmentation, in which the Mahalanobis distance is utilized to define the dissimilarity measure. We add a new regularization term to the objective function of the proposed algorithm, reflecting the covariance of the cluster. We experimentally demonstrate the effectiveness of the proposed algorithm on a generated 2D dataset and a subset of Berkeley benchmark images.     

A new fuzzy dempster MCDM method and its application in supplier selection

Supplier selection is a multi-criterion decision making problem under uncertain environments. Hence, it is reasonable to hand the problem in fuzzy sets theory (FST) and Dempster Shafer theory of evidence (DST). In this paper, a new MCDM methodology, using FST and DST, based on the main idea of the technique for order preference by similarity to an ideal solution (TOPSIS), is developed to deal with supplier selection problem. The basic probability assignments (BPA) can be determined by the distance to the ideal solution and the distance to the negative ideal solution. Dempster combination rule is used to combine all the criterion data to get the final scores of the alternatives in the systems. The final decision results can be drawn through the pignistic probability transformation. In traditional fuzzy TOPSIS method, the quantitative performance of criterion, such as crisp numbers, should be transformed into fuzzy numbers. The proposed method is more flexible due to the reason that the BPA can be determined without the transformation step in traditional fuzzy TOPSIS method. The performance of criterion can be represented as crisp number or fuzzy number according to the real situation in our proposed method. The numerical example about supplier selection is used to illustrate the efficiency of the proposed method.     

基于不确定度的基本概率赋值概率转换方法

基本概率赋值概率转换是合成证据用于辅助决策的主要方法之一。针对现有转换方法中存在的缺少合理转换依据,灵活性和客观性不能兼顾的问题,提出一种基于不确定度加权的概率转换方法。该方法将概率转换视为一种决策过程,选择原BPA与最特异BPA间的Jousselme距离作为不确定性量度,通过“假设-校验”的方法联合求解最特异BPA与转换概率。实验算例表明,所提出方法的处理过程与人们的决策过程一致,概率转换结果合理有效。     

New Distance Measure of Single‐Valued Neutrosophic Sets and Its Application

A single‐valued neutrosophic set (SVNS) is an instance of a neutrosophic set, which can be used to handle uncertainty, imprecise, indeterminate, and inconsistent information in real life. In this paper, a new distance measure between two SVNSs is defined by the full consideration of truth‐membership function, indeterminacy‐membership function, and falsity‐membership function for the forward and backward differences. Then the similarity measure, the entropy measure, and the index of distance are also presented. Finally, two illustrative examples are given to demonstrate the effectiveness of the proposed clustering method and multicriteria decision‐making method based on the distance (similarity) measure between SVNSs.     

Sensor dynamic reliability evaluation based on evidence theory and intuitionistic fuzzy sets

This paper presents a new method for sensor dynamic reliability evaluation based on evidence theory and intuitionistic fuzzy sets when the prior knowledge is unknown. The dynamic reliability of sensors is evaluated based on supporting degree between basic probability assignments (BPAs) provided by sensors. First, the concept of asymmetric supporting degree is proposed. By transforming BPAs to intuitionistic fuzzy sets, supporting degree between BPAs is calculated based on intuitionistic fuzzy operations and similarity measure. Then the relationship between dynamic reliability and supporting degree is analyzed. The process of dynamic reliability evaluation is proposed. Finally, the proposed dynamic reliability evaluation is applied to evidence combination. A new evidence combination rule is proposed based on evidence discounting operation and Dempster’s rule. Comparative analysis on the performance of the proposed reliability evaluation method and evidence combination rule is carried out based on numerical examples. The proposed method for data fusion is also applied in target recognition to show its feasibility and validity.     

Approximating probability distribution of circuit performance function for parametric yield estimation using transferable belief model

This paper applies the transferable belief model （TBM） interpretation of the Dempster-Shafer theory of evidence to approximate distribution of circuit performance function for parametric yield estimation. Treating input parameters of performance function as credal variables defined on a continuous frame of real numbers, the suggested approach constructs a random set-type evidence for these parameters. The corresponding random set of the function output is obtained by extension principle of random set. Within the TBM framework, the random set of the function output in the credal state can be transformed to a pignistic state where it is represented by the pignistic cumulative distribution. As an approximation to the actual cumulative distribution, it can be used to estimate yield according to circuit response specifications. The advantage of the proposed method over Monte Carlo （MC） methods lies in its ability to implement just once simulation process to obtain an available approximate value of yield which has a deterministic estimation error. Given the same error, the new method needs less number of calculations than MC methods. A track circuit of high-speed railway and a numerical eight-dimensional quadratic function examples are included to demonstrate the efficiency of this technique.     

Combining supervised and unsupervised learning for data clustering

Clustering aims to partition a data set into homogenous groups which gather similar objects. Object similarity, or more often object dissimilarity, is usually expressed in terms of some distance function. This approach, however, is not viable when dissimilarity is conceptual rather than metric. In this paper, we propose to extract the dissimilarity relation directly from the available data. To this aim, we train a feedforward neural network with some pairs of points with known dissimilarity. Then, we use the dissimilarity measure generated by the network to guide a new unsupervised fuzzy relational clustering algorithm. An artificial data set and a real data set are used to show how the clustering algorithm based on the neural dissimilarity outperforms some widely used (possibly partially supervised) clustering algorithms based on spatial dissimilarity.     

基于证据推理和VIKOR的二维语言多准则群决策方法

针对专家给出二维语言评价信息的多准则群决策问题,提出基于证据推理和VIKOR的决策方法。首先, 从专家的心理认知和二维语言评价信息的语义出发,设定函数将二维语言信息映射为信度结构;接着基于所提出的广义信度结构,及证据的Pignistic概率距离,定义广义信度结构的距离;最后将专家给出的二维语言决策矩阵转化为信度决策矩阵,用证据推理算子集结为综合信度决策矩阵,并利用VIKOR方法对其求解,获取方案排序。实例分析表明了所提出方法的有效性和实用性。