Existing Distance Measures Research Articles

A novel distance-based multiple attribute decision-making with hesitant fuzzy sets

Up to now, various types of distance measures have been developed and investigated in-depth for hesitant fuzzy sets (HFSs). The analytical study of the existing distance measures for HFSs shows that they have still some limitations. In an attempt to overcome the limitations, this study develops a class of Hausdorff-based distances to measure the distance among HFSs which are not restricted to the same length of their hesitant fuzzy elements (HFEs) and of course the arranging order of values in the HFEs. Furthermore, these HFS distance measures do satisfy all well-known and essential axioms, specially, the triangle inequality property. Eventually, we present some examples to illustrate the efficiency of the new developed HFS distance measures together with a comparative analysis with other existing ones.

Measuring the Directional or Non-directional Distance Between Type-1 and Type-2 Fuzzy Sets With Complex Membership Functions

Fuzzy sets (FSs) may have complex, non-normal, or non-convex membership functions that occur, for example, in the output of a fuzzy logic system or when automatically generating FSs from data. Measuring the distance between such non-standard FSs can be challenging as there is no clear correct method of comparison and only limited research currently exists that systematically compares existing distance measures (DMs) for these FSs. It is useful to know the distance between these sets, which can tell us how much the results of a system change when the inputs differ, or the amount of disagreement between individual's perceptions or opinions on different concepts. In addition, understanding the direction of difference between such FSs further enables us to rank them, learning if one represents a higher output or higher ratings than another. This paper builds on previous functions of measuring directional distance and, for the first time, presents methods of measuring the directional distance between any type-1 and type-2 FSs with both normal/non-normal and convex/non-convex membership functions. In real-world applications, where data-driven, non-convex, non-normal FSs are the norm, the proposed approaches for measuring the distance enables us to systematically reason about the real-world objects captured by the FSs.

Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis

The Pythagorean fuzzy set (PFS) which is an extension of intuitionistic fuzzy set, is more capable of expressing and handling the uncertainty under uncertain environments, so that it was broadly applied in a variety of fields. Whereas, how to measure PFSs’ distance appropriately is still an open issue. It is well known that the square root of Jensen–Shannon divergence is a true metric in the probability distribution space which is a useful measure of distance. On account of this point, a novel divergence measure between PFSs is proposed by taking advantage of the Jensen–Shannon divergence in this paper, called as PFSJS distance. This is the first work to consider the divergence of PFSs for measuring the discrepancy of data from the perspective of the relative entropy. The new PFSJS distance measure has some desirable merits, in which it meets the distance measurement axiom and can better indicate the discrimination degree of PFSs. Then, numerical examples demonstrate that the PFSJS distance can avoid generating counter-intuitive results which is more feasible, reasonable and superior than existing distance measures. Additionally, a new algorithm based on the PFSJS distance measure is designed to solve the problems of medical diagnosis. By comparing the different methods in the medical diagnosis application, it is found that the new algorithm is as efficient as the other methods. These results prove that the proposed method is practical in dealing with the medical diagnosis problems.

Serum orexin A and oxytocin levels and their relationships with attachment and childhood trauma in patients with depression

Distance measures are part and parcel of many computer vision algorithms. The underlying assumption in all existing distance measures is that feature elements are independent and identically distributed. However, in real-world settings, data generally originate from heterogeneous sources even if they do possess a common data-generating mechanism. Since these sources are not identically distributed by necessity, the assumption of identical distribution is inappropriate. Here, we use statistical analysis to show that feature elements of local image descriptors are indeed non-identically distributed. To test the effect of omitting the unified distribution assumption, we created a new distance measure called the Poisson-Binomial radius (PBR). PBR is a bin-to-bin distance which accounts for the dispersion of bin-to-bin information. PBR's performance was evaluated on twelve benchmark data sets covering six different classification and recognition applications: texture, material, leaf, scene, ear biometrics and category-level image classification. Results from these experiments demonstrate that PBR outperforms state-of-the-art distance measures for most of the data sets and achieves comparable performance on the rest, suggesting that accounting for different distributions in distance measures can improve performance in classification and recognition tasks.

Improved Density Peaks Clustering Based on Shared-Neighbors of Local Cores for Manifold Data Sets

A novel clustering algorithm by fast search and find of density peaks (DP) was proposed in Science, 2014. It has attracted much attention from researchers. It can easily select clusters centers with decision graph. However, it cannot be used to cluster manifold data sets as the existing distance measurement is not suitable to evaluate the dissimilarity between objects on manifold structure. Some researchers use graph-based distance to measure the dissimilarity between objects on manifold clusters, but computing the graph-based distance on the original data set is time consuming. An improved density peaks clustering algorithm based on shared-neighbors between local cores, SLORE-DP, is proposed in this paper. First, it finds local cores to represent the data set and redefines the graph-based distance between local cores with shared-neighbors-based distance. Then natural neighbor-based density and the new defined graph-based distance are used to construct decision graph on local cores and DP algorithm is employed to cluster local cores. Finally, the remaining points are assigned to the same cluster as their local cores belong to. Since we use the new defined graph-based distance to estimate the dissimilarity between local cores, SLORE-DP can be used to cluster manifold data sets and at the same time it only calculates the shortest path between local cores, which greatly reduces the running time of the algorithm. We do experiments on several synthetic data sets containing manifold clusters and several real data sets from UCI. The results show that SLORE-DP is more effective and efficient than other algorithms when clustering manifold data sets.

Entropy Measures for Probabilistic Hesitant Fuzzy Information

The probabilistic hesitant fuzzy set (PHFS), which is remarkable in describing the practical condition, has attracted great attention and been applied to many areas. Although lots of achievements have been obtained, there are also some fields, such as the entropy measures with respect to the uncertainty of the information, have not yet been studied. This paper aims at presenting two kinds of entropy measures for probabilistic hesitant fuzzy elements (PHFEs). First, two membership degree-based entropies for PHFEs inspired by the classical fuzzy entropies are derived. Second, the distance-based entropies for PHFEs which are inversely proportional to the distance measures among the elements and the fuzziest element are proposed. However, it is a pity that the existing distance measures for PHFEs are helpless in the description of the entropies, so a new like-distance measure related to the expectation information of the membership degrees is proposed. Then, these entropies are applied to the decision-making case for “The Belt and Road”, and their effectiveness and practicability are verified. Finally, some comparisons among these entropies are made.

Probabilistic Linguistic MULTIMOORA: A Multicriteria Decision Making Method Based on the Probabilistic Linguistic Expectation Function and the Improved Borda Rule

The probabilistic linguistic term set (PLTS) is a powerful technique in representing linguistic evaluations of individuals or groups in the process of decision making. The aim of this paper is to propose a strongly robust method to solve multiexperts multicriteria decision making problems with linguistic evaluations. To enrich the computation and to improve the measures of PLTS, we first define an expectation function of it. In addition, we advance three kinds of probabilistic linguistic distance measures reflecting on the difference of linguistic terms and probabilities at the same time to make up for the defects of the existing distance measures, and then propose the similarity and correlation measures. Integrating the subjective opinions with the correlation coefficients between criteria, we put forward a combined weight determining method. The robustness of the ranking method, MULTIMOORA, is enhanced by the improved Borda rule. Based on these research findings, a probabilistic linguistic MULTIMOORA method is proposed. Finally, the developed method is applied to an empirical example concerning the selection of shared karaoke television brands. The effectiveness of the proposed method is verified by some comparative analyses.

A novel distance measure for time series: Maximum shifting correlation distance

Time series distance or similarity measurement is one of the most important problems in time series data mining, including representation, clustering, classification, and outlier detection. The existing distance measures may not efficiently deal with the drifts in the series in both the phase and amplitude. In this study, a novel measurement, maximum shifting correlation distance (MSCD), is proposed to improve the accuracy and efficiency of the time series distance measure. By integrating the curve registration and correlation, the misalignments or drifts in the phase and amplitude can be eliminated, respectively. In addition, the second distance of the MSCD (MSCD-2nd), which has the “shrinkage effect”, enhances the similarity of the samples within a cluster. The second distance has the same effect on the dynamic time warping (DTW) distance. The experimental results demonstrate that MSCD-2nd is the preferred measure in terms of the accuracy and efficiency for time series clustering and classification.

A multiple attribute group decision making method based on two novel intuitionistic multiplicative distance measures

Distance measure is used to measure the deviation degree between different arguments. Since the existing distance measures between intuitionistic multiplicative sets cannot represent the angle of two intuitionistic multiplicative sets, this paper introduces some new distance measures between intuitionistic multiplicative sets, which include the projection-based distance measure and the psychological distance measures. After that, a multiple attribute group decision making method is proposed to handle the problem in which the weights of experts and attributes are unknown or partially known. A case study concerning the drug supplier selection in hospital management is provided to demonstrate the calculation process of the proposed method. Finally, some comparative analyses are given to illustrate the efficiency and applicability of the proposed method.

Learning category distance metric for data clustering

Unsupervised learning of adaptive distance metrics for categorical data is currently a challenge due to the difficulties in defining an inherently meaningful measure parameterizing the heterogeneity within matched or mismatched categorical symbols. In this paper, a new distance metric called category distance and a non-center-based algorithm are proposed for categorical data clustering. The new metric is formulated based on the category weights for each categorical attribute, no more depending on the common assumption that all categories on the same attribute are independent of each other. The problem of learning the category distance is therefore transformed into the new problem of learning a set of category weights, which can be jointly optimized with the clusters optimization. A case study on DNA sequences and experimental results on ten real-world data sets from different domains are given to demonstrate the performance of the proposed methods with comparisons to the existing distance measures for categorical data.

H-max distance measure of intuitionistic fuzzy sets in decision making

Intuitionistic fuzzy sets (IFSs) are successful to handle the uncertain situations of data. Distance measures of IFSs are important in the evaluation of IFSs relationships. In this paper, we analyzed the disadvantages of existing distance measures of IFSs and proposed a new distance measure called H-max of IFSs. We continued to point out some new results on intuitionistic t-norms and intuitionistic t-conorms and evaluated distance measure between two IFSs which are basically structured from these operations. Further, we combined the classification of t-representable intuitionistic fuzzy t-norms and t-conorms with the proposed distance measure to study some interesting properties. Moreover, we studied De Morgan triplets of IFSs based on the proposed distance measure. Finally, we applied the proposed distance measure to medical diagnosis problem examples and experimental validation on real-world datasets to check the applicability and effectiveness.

Intuitive distance for intuitionistic fuzzy sets with applications in pattern recognition

Distance is an important fundamental concept of the set theory. Since the intuitionistic fuzzy sets (IFSs) was put forward, distance between IFSs has been widely concerned by some researchers and many types of measures have been proposed. However, although intuitionistic fuzzy sets have the advantage of being able to consider waver (lack of knowledge), existing distance measures have not yet considered waver and most of them have counter-intuitive cases. To deal with this problem, this paper introduces the concept of intuitive distance for IFSs, which embodies the property requirements of classical distance measure and highlights the characteristics of intuitionistic fuzzy information. Then, a new intuitive distance measure for IFSs is proposed along with its proofs. After that, a comparative analysis between the intuitive distance measure and the existing distance measures is conducted based on an extended artificial benchmark test set, in which eleven pairs of single-element IFSs are used as an illustration of six typical counter-intuitive cases. Finally, the proposed distance measure is applied to deal with pattern recognition. Results show that the proposed distance does not provide any counter-intuitive cases and the waver that brought from hesitance index can be well reflected.

Novel single-valued neutrosophic decision-making approaches based on prospect theory and their applications in physician selection

The selection of a proper physician in a network environment is an important event for most patients, and single-valued neutrosophic sets (SVNSs) can depict the uncertainty and fuzziness of online evaluation information for physicians. In addition, the psychological behavior of patients is a significant factor that influences physician selection. This paper proposes two multi-attribute decision-making methods based on prospect theory (PT) with single-valued neutrosophic information and utilizes them to solve physician selection problems. Firstly, a new distance measure for SVNSs is developed to overcome the drawbacks of existing distance measures. Secondly, we develop extended TODIM and extended ELECTRE III approaches based on PT and new distance. We utilize the two proposed approaches to handle a physician selection case that uses data from Vitals.com. Parameter sensitivity and comparative analyses of the proposed methods are presented. Findings indicate that the two proposed methods can yield reasonable and credible solutions. With complementary advantages, the two methods are flexible for patients with various behavior preferences to select and use.

Decision Making in Medical Diagnosis via Distance Measures on Interval Valued Fuzzy Sets

The uncertain and sometimes vague, imprecise nature of medical documentation and information make the field of medical diagnosis is the most important and interesting area for applications of fuzzy set theory (FST), intuitionistic fuzzy set (IFS) and interval valued fuzzy set (IVFS). In this present study, first resemblance between IFS and IVFS has been established along with reviewed some existing distance measures for IFSs. Later, an attempt has been made to derive distance measures for IVFSs from IFSs and establish some properties on distance measures of IVFSs. Finally, medical diagnosis has been carried out and exhibits the techniques with a case study under this setting.

A Case-Based Reasoning Decision-Making Model for Hesitant Fuzzy Linguistic Information

In some complicated decision-making problems, because of time pressure or the lack of necessary information, decision makers (DMs) infrequently select optimal alternatives, but acquire satisfactory alternatives that can be obtained by analyzing the correlation between the decision problems and past similar cases. Case-based reasoning (CBR) is an effective approach to obtain preferential information for DMs from past successful decision cases. Using the CBR approach, we aim to process hesitant fuzzy linguistic information, and classify and rank the alternatives according to past successful decision cases. We first sum the distance measures for hesitant fuzzy linguistic term sets (HFLTSs) and then propose a new axiomatic definition for HFLTSs, which are compared with existing distance measures from relationships and properties. Furthermore, based on our proposed distance measure, we propose a CBR decision model for hesitant fuzzy linguistic information to calculate the weights of criteria and classifying thresholds. We then classify and rank the alternatives according to the most satisfactory solution in past successful decision cases. Finally, we consider an example to demonstrate the effectiveness and advantages of our proposed method.

Covariance effect analysis of similarity measurement methods for early construction cost estimation using case-based reasoning

Abstract Quality source data are essential to secure the reliability of early cost estimation, accuracy, and transparency of the estimation process. To improve the reliability of cost estimation results using CBR, a recurring issue in similarity measurement has been to accurately compute the distance among attributes and cases for the retrieval of the most similar singular or plural cases. However, these existing similarity measures have limitations in taking the covariance among attributes into consideration, and reflecting the effects of covariance in the computation of distances among attributes. To deal with this challenging issue, this research examines the weighted Mahalanobis distance based similarity measure applied to CBR cost estimation, and carries out comparative research on the existing distance measurement methods of CBR. To validate the suggested CBR cost model, simulation data test and case studies using 99 multi-family housing complexes are carried out. As a consequence, this research is expected to improve academic value and mathematical and statistical logicality of similarity measurement, especially in application to CBR cost estimation. The results can be further referenced to not only the area of construction cost estimation, but also international market selection, planning and scheduling, noise control and management, and energy management where case-based reasoning can be applied.

Self-adapted mixture distance measure for clustering uncertain data

Distance measure plays an important role in clustering uncertain data. However, existing distance measures for clustering uncertain data suffer from some issues. Geometric distance measure can not identify the difference between uncertain objects with different distributions heavily overlapping in locations. Probability distribution distance measure can not distinguish the difference between different pairs of completely separated uncertain objects. In this paper, we propose a self-adapted mixture distance measure for clustering uncertain data which considers the geometric distance and the probability distribution distance simultaneously, thus overcoming the issues in previous distance measures. The proposed distance measure consists of three parts: (1) The induced kernel distance: it can be used to measure the geometric distance between uncertain objects. (2) The Jensen–Shannon divergence: it can be used to measure the probability distribution distance between uncertain objects. (3) The self-adapted weight parameter: it can be used to adjust the importance degree of the induced kernel distance and the Jensen–Shannon divergence according to the location overlapping information of the dataset. The proposed distance measure is symmetric, finite and parameter adaptive. Furthermore, we integrate the self-adapted mixture distance measure into the partition-based and density-based algorithms for clustering uncertain data. Extensive experimental results on synthetic datasets, real benchmark datasets and real world uncertain datasets show that our proposed distance measure outperforms the existing distance measures for clustering uncertain data.

Hesitant distance set on hesitant fuzzy sets and its application in urban road traffic state identification

Since fuzziness lacks the distinction between a set and its complement, it is difficult to measure the distance between different hesitant fuzzy sets (HFSs) by a single value. In this study, a new concept “hesitant distance set (HDS)” is proposed, where the distance between different HFSs can be characterized by a series of different values. This study has three primary contributions. Firstly, most of the existing distance measures on HFSs are based on vector operation, while the novel proposed HDSs are based on set operation. Secondly, a statistical method is proposed to compare different HDSs, and some important properties of the comparison method are introduced. Thirdly, the characteristics of the novel HDSs and the classical hesitant distances are studied comparatively. Finally, the practicality and validity of the HDSs on HFSs are illustrated through an urban road traffic state identification example.

Distance measures for higher order dual hesitant fuzzy sets

In this study, we propose new distance measures for dual hesitant fuzzy sets (DHFSs) in terms of the mean, standard deviation of dual hesitant fuzzy elements (DHFEs), respectively, which overcome some drawbacks of the existing distance measures. Meanwhile, we extend DHFS to its higher order type and refer to it as the higher order dual hesitant fuzzy set (HODHFS). HODHFS is the actual extension of DHFS that enables us to define the membership and non-membership of a given element in terms of several possible generalized type of fuzzy sets (G-Type FSs). The rationale behind HODHFS can be seen in the case that the decision makers are not satisfied by providing exact values for the membership degrees and the non-membership degrees. To indicate HODHFSs have a good performance in decision making, we introduce several distance measures for HODHFSs based on our proposed new distance for dual hesitant fuzzy sets. Finally, we practice our proposed measures for HODHFSs in multi-attribute decision making illustrating their applicability and availability.

Semi-elliptic membership function: Representation, generation, operations, defuzzification, ranking and its application to the real-time task scheduling problem

A potentially important, yet under-stated membership function (MF) shape viz. semi-elliptic membership function (SEMF) has been elaborated in this paper so that researchers are able to see its true potential for realistic modelling of the decision variables in real-life applications. The concept of semi-elliptic membership function has the practical significance since it can nicely remove the drawbacks associated with the Gaussian membership function shape regarding its long tail with non-zero membership values. A semi-elliptic membership function can be mathematically expressed in terms of only two parameters. It is shown that, with the representation considered here, prominent arithmetic operations such as addition, subtraction, multiplication etc. can be applied on the semi-elliptic fuzzy numbers. As an extension, a novel membership function generation technique for interval type-2 semi-elliptic (IT2SE) MF has also been proposed. Then the applicability of the arithmetic operations on the IT2SE fuzzy numbers is shown. It is illustrated that semi-elliptic fuzzy numbers can suitably be ranked by existing distance measures. We have shown that the well-known INT (Improved Nie-Tan’s) defuzzification technique can be applied to the interval type-2 semi-elliptic fuzzy sets for producing defuzzified outputs. Finally, with application point of view, semi-elliptic fuzzy numbers has been applied to the real-time task scheduling problem. Results are compared with triangular fuzzy numbers. Suitable numerical examples are considered for demonstration purposes.