Similarity measure for aggregated fuzzy numbers from interval-valued data

Abstract

This paper presents a method to compute the degree of similarity between two aggregated fuzzy numbers from intervals using the Interval Agreement Approach (IAA). The similarity measure proposed within this study contains several features and attributes, of which are novel to aggregated fuzzy numbers. The attributes completely redefined or modified within this study include area, perimeter, centroids, quartiles and the agreement ratio. The recommended weighting for each feature has been learned using Principal Component Analysis (PCA). Furthermore, an illustrative example is provided to detail the application and potential future use of the similarity measure.

Methods of ranking for aggregated fuzzy numbers from interval-valued data

“…The following is a synthetic example that was produced and explored by [9] to illustrate the use of its attribute similarity measure. The primary aspects of this illustrative example will be explained in brief, refer to [9] for a further description of why this dataset was utilized.…”

“…In contrast to [7], Gunn et al [9] propose a similarity measure for IAA Fuzzy Numbers that utilizes a collection of attributes as features, along with each weight of said feature calculated by Principal Component Analysis (PCA); this alternative technique was inspired by Khorshidi and Nikfalazar's similarity measure for Generalized Fuzzy Numbers in [10]. Note that the proposals of [9] are in their current state only applicable to Type-1 fuzzy numbers. Various types of fuzzy numbers have seen proposals for both similarity measures and ranking methods [10,11,12,13], which has provided opportunities for their further practical application.…”

“…Wagner et al [7] propose a similarity measure for Type-1 Fuzzy Numbers which utilizes the Jaccard similarity coefficient and is applicable to IAA Fuzzy Numbers [7,8]. In contrast to [7], Gunn et al [9] propose a similarity measure for IAA Fuzzy Numbers that utilizes a collection of attributes as features, along with each weight of said feature calculated by Principal Component Analysis (PCA); this alternative technique was inspired by Khorshidi and Nikfalazar's similarity measure for Generalized Fuzzy Numbers in [10]. Note that the proposals of [9] are in their current state only applicable to Type-1 fuzzy numbers.…”

“…Whilst it is common that similarity measures may sometimes be also applied as methods of ranking, the similarity measures of previous IAA literature have shortcomings in this regard and require modification. Building upon the findings in [7,9], we propose a new similarity measure and two ranking methods for IAA fuzzy numbers and provide both an update to the previous illustrative example in [9], as well as explore the results from the application in a real-world case study [14,15] as a means to demonstrate their potential use. This paper is organized as follows.…”

“…In this section, we provide a very brief introduction into the IAA method for Type-1 fuzzy numbers [1,2], and include the outlines of the similarity measures proposed in [7,9]. A crisp interval is defined as 𝐴 ̅ = [𝑙 𝐴 ̅ , 𝑟 𝐴 ̅ ], where 𝑙 is the 'left' start value and 𝑟 is 'right' end value, and thus a set of these intervals is defined as 𝐴 = {𝐴 ̅ 1 , … , 𝐴 ̅ 𝑛 }.…”