A bidirectional subsethood based similarity measure for fuzzy sets

Similarity measures are useful for reasoning about fuzzy sets. Hence, many classical set-theoretic similarity measures have been extended for comparing fuzzy sets. In previous work, a set-theoretic similarity measure considering the bidirectional subsethood for intervals was introduced. The measure addressed specific concerns of many common similarity measures, and it was shown to be bounded above and below by Jaccard and Dice measures respectively. Herein, we extend our prior measure from similarity on intervals to fuzzy sets. Specifically, we propose a vertical-slice extension where two fuzzy sets are compared based on their membership values.We show that the proposed extension maintains all common properties (i.e., reflexivity, symmetry, transitivity, and overlapping) of the original fuzzy similarity measure. We demonstrate and contrast its behaviour along with common fuzzy set-theoretic measures using different types of fuzzy sets (i.e., normal, non-normal, convex, and non-convex) in respect to different discretization levels

A Similarity Measure Based on Bidirectional Subsethood for Intervals

With a growing number of areas leveraging interval-valued data—including in the context of modelling human uncertainty (e.g., in Cyber Security), the capacity to accurately and systematically compare intervals for reasoning and computation is increasingly important. In practice, well established set-theoretic similarity measures such as the Jaccard and Sørensen-Dice measures are commonly used, while axiomatically a wide breadth of possible measures have been theoretically explored. This paper identifies, articulates, and addresses an inherent and so far not discussed limitation of popular measures—their tendency to be subject to aliasing—where they return the same similarity value for very different sets of intervals. The latter risks counter-intuitive results and poor automated reasoning in real-world applications dependent on systematically comparing interval-valued system variables or states. Given this, we introduce new axioms establishing desirable properties for robust similarity measures, followed by putting forward a novel set-theoretic similarity measure based on the concept of bidirectional subsethood which satisfies both the traditional and new axioms. The proposed measure is designed to be sensitive to the variation in the size of intervals, thus avoiding aliasing. The paper provides a detailed theoretical exploration of the new proposed measure, and systematically demonstrates its behaviour using an extensive set of synthetic and real-world data. Specifically, the measure is shown to return robust outputs that follow intuition—essential for real world applications. For example, we show that it is bounded above and below by the Jaccard and Sørensen-Dice similarity measures (when the minimum t-norm is used). Finally, we show that a dissimilarity or distance measure, which satisfies the properties of a metric, can easily be derived from the proposed similarity measure

How to compare uncertain data types: towards robust similarity measures

In view of the importance of effective comparison of uncertain data in real-world applications, this thesis focuses on developing new similarity measures with high accuracy. As it identifies and articulates an inherent limitation of the popular set-theoretic similarity measures for continuous intervals where they return the same similarity value for very different sets of intervals (termed as aliasing), this thesis first underpins a new axiomatic definition of a robust similarity measure and then proposes a new similarity measure for continuous intervals based on their bidirectional subsethood. Beyond establishing theoretical foundation of the new measure, the thesis also demonstrates its robust results vis-a-vis existing measures and suitability for real world applications. In the next stage, it develops a generalized framework to assess similarity between discontinuous intervals as current approaches involve loss of discontinuity information and are also affected by aliasing of the popular measures— these weaknesses impact the accuracy of similarity results. This thesis further integrates Allen’s theory with the new generalized framework to make the latter more efficient. Moving beyond intervals, this thesis extends the new similarity measure both vertically and horizontally (α-cut based) for comparing type-1 (T1) fuzzy sets as the shortcoming of popular similarity measures persists with their extension to T1 fuzzy sets. The empirical evaluation of the extended new measures with respect to key existing fuzzy set-theoretic similarity measures shows that the vertically extended new measure behaves intuitively for various types of fuzzy sets, except for non-normal fuzzy sets; however, the α-cut based extended new measure meets expectation in all cases. At the final stage, the utility of the new similarity measure is explored to improve the robustness of fuzzy integral (FI) based uncertain-data aggregation. As existing approaches to generate fuzzy measures (FMs) rely on popular similarity measures to capture the degree of similarity among individual sources (and their combinations), they are also impacted by their aforesaid limitation. Therefore, this thesis develops a new FM based on the new similarity measure which can generate intuitive aggregation outcome when used in combination with an FI

IN-cross Entropy Based MAGDM Strategy under Interval Neutrosophic Set Environment

Cross entropy measure is one of the best way to calculate the divergence of any variable from the priori one variable. We define a new cross entropy measure under interval neutrosophic set environment

Neutrosophic Cubic MCGDM Method Based on Similiarity Measure

The notion of neutrosophic cubic set is originated from the hybridization of the concept of neutrosophic set and interval valued neutrosophic set. We define similarity measure for neutrosophic cubic sets and prove some of its basic properties. We present a new multi criteria group decision making method with linguistic variables in neutrosophic cubic set environment. Finally, we present a numerical example to demonstrate the usefulness and applicability of the proposed method

A Revisit to NC-VIKOR Based MAGDM Strategy in Neutrosophic Cubic Set Environment

Multi attribute group decision making with VIKOR (VlseKriterijuska Optimizacija I Komoromisno Resenje) strategy has been widely applied to solving real-world problems. Recently, Pramianik et al. [S. Pramanik, S. Dalapati, S. Alam, and T. K. Roy. NCVIKOR based MAGDM strategy under neutrosophic cubic set environment, Neutrosophic Sets and Systems, 20 (2018), 95-108] proposed VIKOR strategy for solving MAGDM, where compromise solutions are not identified in neutrosophic cubic environment. To overcome the shortcomings of the paper, we further modify the VIKOR strategy by incorporating compromise solution in neutrosophic cubic set environment. Finally, we solve an MAGDM problem using the modified NC-VIKOR strategy to show the feasibility, applicability and effectiveness of the proposed strategy. Further, we present sensitivity analysis to show the impact of different values of the decision making mechanism coefficient on ranking order of the alternatives