On Integration Linguistic Factors to Fuzzy Similarity Measures and Intuitionistic Fuzzy Similarity Measures

On Integration Linguistic Factors to Fuzzy Similarity Measures and Intuitionistic Fuzzy Similarity Measures

Pham Hong Phong (National University of Civil Engineering, Hanoi, Viet Nam) and Vu Thi Hue (Hanoi University of Science and Technology, Hanoi, Viet Nam)
Copyright: © 2019 |Pages: 37
DOI: 10.4018/IJSE.2019010101

Abstract

The article is concerned with integrating linguistic elements into fuzzy similarity measures and intuitionistic fuzzy similarity measure. Some new concepts are proposed: a fuzzy linguistic value (FLv), a fuzzy linguistic vector (FLV), an intuitionistic fuzzy linguistic vector (ILV) and similarity measures. The proposed measures are used to build classification algorithms. As predicted theoretically, experiments show that with the same type of similarity measures, the linguistic-aggregated similarity measures produce better results in classification problems.
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1. Introduction

This section covers general knowledge of fuzzy similarity measure, intuitionistic fuzzy similarity measure, linguistic similarity measure and linguistic vector similarity measure. Some of the most basic similarity measures are mentioned. In order to make it possible to compare new measurements with old measurements, based on the expression, we classify similarity measures as follows:

  • Type 1: Hamming-distance-based similarity measures

  • Type 2: Min-max-based similarity measures

  • Type 3: Exponent-hamming-based similarity measures

  • Type 4: Exponent-square-root-based similarity measures

1.1. Fuzzy Similarity Measures and Intuitionistic Fuzzy Similarity Measures

The similarity measure is an important mathematical concept that is used to estimate the degree of similarity between objects and to be applicable in pattern recognition problems such as clustering, classification and information retrieval. In the case of inadequately defined and ambiguous objects, instead of the traditional similarity measure, fuzzy similarity or intuitionistic fuzzy similarity measures should be used.

1.1.1. Definition 1

A fuzzy set (FS) A on a universe X is an object of the form IJSE.2019010101.m01, where IJSE.2019010101.m02 is membership function (Zadeh, 1965). For each IJSE.2019010101.m03, IJSE.2019010101.m04 is called the degree of membership of x in A. The set of all FS on X is denoted by IJSE.2019010101.m05.

1.1.2. Definition 2

An intuitionistic fuzzy set (IFS) A on a universe X is an object of the form IJSE.2019010101.m06, where IJSE.2019010101.m07 and IJSE.2019010101.m08. For each IJSE.2019010101.m09, IJSE.2019010101.m10, IJSE.2019010101.m11 are respectively called the degree of membership, degree of non-membership of x in A of x in A, and following condition is satisfied (Atanassov 1986):

IJSE.2019010101.m12
,
IJSE.2019010101.m13
(1)

The set of all IFS on X is denoted by IJSE.2019010101.m14.

Consider the case where X contains n elements. For IJSE.2019010101.m15, we denote by IJSE.2019010101.m16 and IJSE.2019010101.m17 the degrees of membership and non-membership of the i-th element of X into A, respectively (IJSE.2019010101.m18, …, n). Here are some of the most commonly used fuzzy and intuitionistic fuzzy similarity measures.

1.1.3. Definition 3

Consider A, IJSE.2019010101.m19 (Baccour et al., 2014):

  • Hamming-distance-based fuzzy similarity measure:

    IJSE.2019010101.m20
    (2)

  • Min-max-based fuzzy similarity measure:

    IJSE.2019010101.m21
    (3)

It is convention that if the denominator is 0, the fraction is equal to 1.

  • Exponent-hamming-based fuzzy similarity measure:

    IJSE.2019010101.m22
    (4)

  • Exponent-square-root-based fuzzy similarity measure:

    IJSE.2019010101.m23
    (5)

1.1.4. Definition 4

Consider A, IJSE.2019010101.m24 (Baccour et al., 2016):

  • Hamming-distance-based intuitionistic fuzzy similarity measure:

    IJSE.2019010101.m25
    (6)

  • Min-max-based intuitionistic fuzzy similarity measure:

    IJSE.2019010101.m26
    (7)

It is convention that if the denominator is 0, the fraction is equal to 1.

  • Exponent-hamming-based intuitionistic fuzzy similarity measure:

    IJSE.2019010101.m27
    (8)

  • Exponent-square-root-based intuitionistic fuzzy similarity measure:

    IJSE.2019010101.m28
    (9)

Some typical application of fuzzy similarities are: image processing (Bloch 1999; Weken et al., 2005; Weken et al., 2004), fuzzy reasoning (Wang et al., 2008), shape retrieval using the SQUID data set described with Fourier descriptor (Gadi et al., 1999), shape classification (Baccour et al., 2007), handwritten Arabic sentences recognition (Baccour & Alimi 2010; Baccour & Alimi 2013) and medical diagnosis (Son & Phong, 2016).

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