A Cosine Similarity Measure Based on the Choquet Integral for Intuitionistic Fuzzy Sets and Its Applications to Pattern Recognition

Abstract

There exist various types of similarity measures for intuitionistic fuzzy sets in the literature. However, in many studies the interactions among the elements are ignored in the construction of the similarity measure. This paper presents a cosine similarity measure for intuitionistic fuzzy sets by using a Choquet integral model in which the interactions between elements are considered. The proposed similarity measure is applied to some pattern recognition problems and the results are compared with some existing results to demonstrate the effectiveness of this new similarity measure.

1 Introduction

Zadeh (1965) introduced the concept of fuzzy set by using a membership function and Atanassov (1986) expanded this concept to the concept of intuitionistic fuzzy set (IFS) by using both a membership function and a non-membership function. The theory of intuitionistic fuzzy sets has been extensively studied by many authors (see e.g. Szmidt and Kacprzyk, 2000, 2003; Xu and Yager, 2006; Ye, 2009, 2010, 2018; Xu, 2010; Melo-Pinto et al. 2013; Balasubramaniam and Ananthi, 2014; Mani and Jerome, 2014; Lu and Ye, 2016; Das et al., 2016; Luo and Zhao, 2018). The concept of similarity measure is one of these study areas. A similarity measure is an important tool for measuring the degree of similarity between two IFSs and various versions of the concept of the similarity measure have been applied to various fields such as pattern recognition, medical diagnosis, decision making, face recognition systems, clustering (see e.g. De et al., 2001; Hung and Yang, 2004; Liu, 2005; Vlachos and Sergiadis, 2007; Li et al., 2007; Xu and Chen, 2008; Ye, 2012; Luo and Ren, 2016; Ngan, 2016; Hwang et al., 2018; Song et al., 2019). Many similarity measures of IFSs have been investigated in the literature and the concept of cosine similarity measure is one of them. The concept of cosine similarity measure for fuzzy sets is defined as the inner product of two vectors divided by the product of their lengths, that is, the cosine of the angle between the vector representations of fuzzy sets and it has been investigated by many researchers (Gerstenkorn and Manko, 1991; Ye, 2011, 2016; Garg, 2018; Wei, 2018; Wei and Wei, 2018; Wei et al., 2019; Wang et al., 2019).

The concepts of fuzzy measure (or capacity or non-additive measure) and the Choquet integral were introduced by Choquet (1953). Since that time, fuzzy measures and integrals have been studied on a rather mathematical point of view, especially in multi criteria decision making field. The purpose of the multi criteria decision making field is to order alternatives based on multiple contradictory criteria and choose the best alternative (see e.g. Grabisch, 1996; Ünver et al., 2018). Actually a fuzzy integral is a generalization of weighted mean. Yang and Ha (2008) introduced a similarity measure for IFSs by using a Choquet integral model where a generalized fuzzy measure was used to characterize interactions among elements of IFSs. Also, the Choquet integral model instead of a weighted average model was used to compute this similarity measure.

The purpose of a pattern recognition problem is to decide whether an object belongs to a set or a class. Many studies have been made to solve such a problem and many similarity measures have been proposed by several researchers (see e.g. Dengfeng and Chuntian, 2002; Liang and Shi, 2003; Mitchell, 2003; Liu, 2005; Wei and Ye, 2010; Hwang and Yang, 2013; Zhu and Ye, 2013; Song et al., 2014; Boran and Akay, 2014; Chen and Chang, 2015; Jiang et al., 2019).

In this study, we provide a new cosine similarity measure for IFSs by considering the Choquet integral, inspired by a weighted cosine similarity measure for IFSs which has been given by Ye (2011). First of all, we give the basic definitions that are used throughout the study, and then we recall the concepts of fuzzy measure and Choquet integral. Then we propose a new cosine similarity measure based on the Choquet integral for IFSs. To demonstrate the effectiveness of the new similarity measure, we apply it to a pattern recognition and a medical diagnosis problem. We also compare the results of the proposed cosine similarity measure with some previous results in the literature.

2 Preliminaries

In this section we recall some definitions of fuzzy set theory, correlation coefficient, some cosine similarity measures and some weighted cosine similarity measures for IFSs.

Definition 1 (Zadeh, 1965).

A fuzzy set A of a given set X is defined with a function which is called the membership function of A and is denoted by $$\begin{array}{c}\mathit{A}=\{⟨\mathit{x}\mathrm{,}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}⟩|\mathit{x}\in \mathit{X}\}.\end{array}$$ The value ${\mathit{\mu }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}$ is said to be the grade of membership of the element x to the set.

In the following definition we recall the concept of IFS that was introduced by Atanassov (1986).

Definition 2 (Atanassov, 1986).

An IFS A in X is given by $$\begin{array}{c}\mathit{A}=\{⟨\mathit{x}\mathrm{,}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}{\mathit{\nu }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}⟩|\mathit{x}\in \mathit{X}\}\mathrm{,}\end{array}$$ where ${\mathit{\mu }}_{\mathit{A}}:\mathit{X}\to [0\mathrm{,}1]$ is a function which is called the membership function and ${\mathit{\nu }}_{\mathit{A}}:\mathit{X}\to [0\mathrm{,}1]$ is a function which is called the non-membership function with ${\mathit{\mu }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}⩽1$ , for any $\mathit{x}\in \mathit{X}$ . The values ${\mathit{\mu }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}$ and ${\mathit{\nu }}_{\mathit{A}}\mathrm{(}\mathit{x}\mathrm{)}$ are the grade of membership and the grade of non-membership of the element x to the set A, respectively.

It is clear that each fuzzy set is an intuitionistic fuzzy set, but the converse is not true in general. Now, we recall a correlation coefficient between IFSs which is actually the motivation of the cosine similarity measure between two IFSs.

Definition 3 (Gerstenkorn and Manko, 1991).

Let $\mathit{X}=\{{\mathit{x}}_1\mathrm{,}\dots \mathrm{,}{\mathit{x}}_{\mathit{n}}\}$ be a finite set and A and B be two IFSs of X. A correlation coefficient between A and B is defined by $$\begin{array}{c}\mathit{k}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}:={\displaystyle \frac{\mathit{C}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}}{\sqrt{\mathit{T}\mathrm{(}\mathit{A}\mathrm{)}·\mathit{T}\mathrm{(}\mathit{B}\mathrm{)}}}}\mathrm{,}\end{array}$$ where the correlation of A and B is given by $$\begin{array}{c}\mathit{C}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}={\displaystyle \sum }_{\mathit{i}=1}^{\mathit{n}}\mathrm{(}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\nu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}\mathrm{)}\end{array}$$ and the informational intuitionistic energies of A and B are $$\begin{array}{c}\mathit{T}\mathrm{(}\mathit{A}\mathrm{)}={\displaystyle \sum }_{\mathit{i}=1}^{\mathit{n}}\mathrm{(}{\mathit{\mu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}\mathrm{)}\end{array}$$ and $$\begin{array}{c}\mathit{T}\mathrm{(}\mathit{B}\mathrm{)}={\displaystyle \sum }_{\mathit{i}=1}^{\mathit{n}}\mathrm{(}{\mathit{\mu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}\mathrm{)}\mathrm{,}\end{array}$$ respectively.

The correlation coefficient k satisfies the following properties (Gerstenkorn and Manko, 1991):

• $\mathrm{(}{P}_1\mathrm{)}$

  $0⩽\mathit{k}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}⩽1$ ;

  $\mathrm{(}{P}_2\mathrm{)}$

  $\mathit{k}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}=\mathit{k}\mathrm{(}\mathit{B}\mathrm{,}\mathit{A}\mathrm{)}$ ;

  $\mathrm{(}{P}_3\mathrm{)}$

  If $\mathit{A}=\mathit{B}$ then $\mathit{k}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}=1$ .

  Various similarity measures were defined to quantify the degree of similarity between IFSs. Chen (1995) proposed a similarity measures between vague sets. Hong and Kim (1999) and Fan and Zhangyan (2001) pointed out limitations of the similarity measure of Chen (1995) and proposed a new similarity measure for IFSs. Later, Li et al. (2007) proposed a new similarity measure and Dengfeng and Chuntian (2002) generalized their similarity measure of IFSs by using Minkowski distance and applied this measure to the pattern recognition. Then, Mitchell (2003) modified this similarity measure. Later, Hung and Yang (2004) proposed simple and well suited similarity measures for IFSs by using Hausdorff distance and applied it to the pattern recognition. Now, we recall some cosine similarity measures from the literature.

  Definition 4 (Salton and McGill, 1983).

  Let $\mathit{X}=\{{\mathit{x}}_1\mathrm{,}\dots \mathrm{,}{\mathit{x}}_{\mathit{n}}\}$ be a finite set and let $$\begin{array}{c}\mathit{A}=\mathrm{(}⟨{\mathit{x}}_1\mathrm{,}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_1\mathrm{)}⟩\mathrm{,}⟨{\mathit{x}}_2\mathrm{,}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_2\mathrm{)}⟩\mathrm{,}\dots \mathrm{,}⟨{\mathit{x}}_{\mathit{n}}\mathrm{,}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{n}}\mathrm{)}⟩\mathrm{)}\end{array}$$ and $$\begin{array}{c}\mathit{B}=\mathrm{(}⟨{\mathit{x}}_1\mathrm{,}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_1\mathrm{)}⟩\mathrm{,}⟨{\mathit{x}}_2\mathrm{,}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_2\mathrm{)}⟩\mathrm{,}\dots \mathrm{,}⟨{\mathit{x}}_{\mathit{n}}\mathrm{,}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{n}}\mathrm{)}⟩\mathrm{)}\end{array}$$ be two fuzzy sets of a set X. A cosine similarity measure based on Bhattacharya’s distance (Bhattacharya, 1946) between A and B is defined by $$\begin{array}{c}{\mathit{C}}_{\mathit{F}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}:={\displaystyle \frac{{\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}{\sqrt{{\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{\mu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}\sqrt{{\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{\mu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}}}.\end{array}$$

  It is clear that $0⩽{\mathit{C}}_{\mathit{F}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}⩽1$ .

  Definition 5 (Ye, 2011).

  Let $\mathit{X}=\{{\mathit{x}}_1\mathrm{,}\dots \mathrm{,}{\mathit{x}}_{\mathit{n}}\}$ be a finite set and let $$\begin{array}{c}\mathit{A}=\mathrm{(}⟨{\mathit{x}}_1\mathrm{,}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_1\mathrm{)}\mathrm{,}{\mathit{\nu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_1\mathrm{)}⟩\mathrm{,}\dots \mathrm{,}⟨{\mathit{x}}_{\mathit{n}}\mathrm{,}{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{n}}\mathrm{)}\mathrm{,}{\mathit{\nu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{n}}\mathrm{)}⟩\mathrm{)}\end{array}$$ and $$\begin{array}{c}\mathit{B}=\mathrm{(}⟨{\mathit{x}}_1\mathrm{,}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_1\mathrm{)}\mathrm{,}{\mathit{\nu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_1\mathrm{)}⟩\mathrm{,}\dots \mathrm{,}⟨{\mathit{x}}_{\mathit{n}}\mathrm{,}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{n}}\mathrm{)}\mathrm{,}{\mathit{\nu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{n}}\mathrm{)}⟩\mathrm{)}\end{array}$$ be two IFSs of X. A cosine similarity measure between IFSs A and B is defined by $$\begin{array}{c}{\mathit{C}}_{\mathit{\text{IFS}}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}:={\displaystyle \frac{1}{\mathit{n}}}{\displaystyle \sum }_{\mathit{i}=1}^{\mathit{n}}{\displaystyle \frac{{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\nu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}{\sqrt{{\mathit{\mu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}\sqrt{{\mathit{\mu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}}}.\end{array}$$ For $\mathit{n}=1$ the cosine similarity measure ${\mathit{C}}_{\mathit{\text{IFS}}}$ is equivalent to the correlation coefficient k, that is ${\mathit{C}}_{\mathit{\text{IFS}}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}=\mathit{k}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}$ . The cosine similarity measure also satisfies ${\mathit{P}}_{1\mathrm{,}2\mathrm{,}3}$ .

  Definition 6 (Ye, 2011).

  Let $\mathit{X}=\{{\mathit{x}}_1\mathrm{,}\dots \mathrm{,}{\mathit{x}}_{\mathit{n}}\}$ be a finite set and let A and B be two IFSs of X. A weighted cosine similarity measure between IFSs A and B is defined by $$\begin{array}{c}{\mathit{W}}_{\mathit{\text{IFS}}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}:={\displaystyle \sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{i}}{\displaystyle \frac{{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\nu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}{\sqrt{{\mathit{\mu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}\sqrt{{\mathit{\mu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}}}\mathrm{,}\end{array}$$ where $\mathit{w}=\mathrm{(}{\mathit{w}}_1\mathrm{,}\dots \mathrm{,}{\mathit{w}}_{\mathit{n}}\mathrm{)}$ is the weight vector with ${\mathit{w}}_{\mathit{i}}\in [0\mathrm{,}1]$ for all $\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{n}$ and ${\sum }_{\mathit{i}=1}^{\mathit{n}}{\mathit{w}}_{\mathit{i}}=1$ . In particular, if $\mathit{w}=\mathrm{(}1\mathrm{/}\mathit{n}\mathrm{,}1\mathrm{/}\mathit{n}\mathrm{,}\dots \mathrm{,}1\mathrm{/}\mathit{n}\mathrm{)}$ , then the weighted cosine similarity measure is reduced to the cosine similarity measure between IFSs A and B, i.e. ${\mathit{W}}_{\mathit{IFS}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}={\mathit{C}}_{\mathit{IFS}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}$ . Obviously, the weighted cosine similarity measure of two IFSs A and B also satisfies ${\mathit{P}}_{1\mathrm{,}2\mathrm{,}3}$ .

  The concept of Choquet integral is a generalization of the concept of weighted mean. Yang and Ha (2008) proposed a similarity measure between IFSs by considering the Choquet integral. Now, we recall the concepts of fuzzy measure and Choquet integral.

  Definition 7 (Choquet, 1953).

  Let $\mathit{X}\ne \varnothing$ be a finite set and let $\mathit{P}\mathrm{(}\mathit{X}\mathrm{)}$ be the power set of X. If

3 Choquet Cosine Similarity Measure

In this section, by considering the Choquet integral we construct a new cosine similarity measure.

Definition 10.

Let $\mathit{X}=\{{\mathit{x}}_1\mathrm{,}\dots \mathrm{,}{\mathit{x}}_{\mathit{n}}\}$ be a finite set and let A and B be two IFSs in X and let σ be a fuzzy measure on X. A Choquet cosine similarity measure between A and B is given with $$\begin{array}{c}{\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}:=\mathrm{(}\mathit{C}\mathrm{)}{{\displaystyle \int }}_{\mathit{X}}{\mathit{f}}_{\mathit{A}\mathrm{,}\mathit{B}}\mathit{d}\mathit{\sigma }\mathrm{,}\end{array}$$ where $$\begin{array}{c}{\mathit{f}}_{\mathit{A}\mathrm{,}\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}:={\displaystyle \frac{{\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}{\mathit{\nu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}{\sqrt{{\mathit{\mu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{A}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}\sqrt{{\mathit{\mu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}+{\mathit{\nu }}_{\mathit{B}}^2\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}}}}\mathrm{,}\text{for}\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{n}.\end{array}$$

Proposition 1.

Let $\mathit{X}=\{{\mathit{x}}_1\mathrm{,}\dots \mathrm{,}{\mathit{x}}_{\mathit{n}}\}$ be a finite set, let A and B be two IFSs in X. The Choquet cosine similarity measure ${\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}$ satisfies the following properties:

• $\mathrm{(}{P}_1\mathrm{)}$

$0⩽{\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}⩽1$ ;

$\mathrm{(}{P}_2\mathrm{)}$

${\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}={\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}\mathit{B}\mathrm{,}\mathit{A}\mathrm{)}$ ;

$\mathrm{(}{P}_3\mathrm{)}$

If $\mathit{A}=\mathit{B}$ then ${\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}=1$ .

Proof.

$\mathrm{(}{P}_1\mathrm{)}$ Since ${\mathit{f}}_{\mathit{A}\mathrm{,}\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}\in [0\mathrm{,}1]$ and for any $\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{n}$ and the Choquet integral is monotone, we have $0⩽{\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}⩽1$ .

$\mathrm{(}{P}_2\mathrm{)}$ It is trivial that since ${\mathit{f}}_{\mathit{A}\mathrm{,}\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}={\mathit{f}}_{\mathit{B}\mathrm{,}\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}$ for any $\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{n}$ .

$\mathrm{(}{P}_3\mathrm{)}$ If $\mathit{A}=\mathit{B}$ , then ${\mathit{\mu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}={\mathit{\mu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}$ and ${\mathit{\nu }}_{\mathit{A}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}={\mathit{\nu }}_{\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}$ for $\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{n}$ . We have ${\mathit{f}}_{\mathit{A}\mathrm{,}\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}=1$ and so ${\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}\mathit{A}\mathrm{,}\mathit{B}\mathrm{)}=1$ . Thus, the proof is completed.  □

4 Applications

The proposed cosine similarity measure can be used to evaluate the degree of similarity between two IFSs. Therefore, it can be applied to pattern recognition problems with the intuitionistic fuzzy information. It is more delicate than the cosine similarity measures in the literature since it considers interactions between criteria.

Let $\mathit{X}=\{{\mathit{x}}_1\mathrm{,}\dots \mathrm{,}{\mathit{x}}_{\mathit{n}}\}$ be a finite set of attributes and let $\mathit{C}=\{{\mathit{C}}_1\mathrm{,}\dots \mathrm{,}{\mathit{C}}_{\mathit{m}}\}$ be a finite set of alternatives. Here, Q needs to be classified in one of ${\mathit{C}}_{\mathit{i}}$ , $\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{m}$ . Now, we give the implementation steps of the promised Choquet integral model.

Step 1. Since the model is a fuzzy integral model, it is necessary to establish a fuzzy measure on a given set before we can use the model. The decision maker determines which fuzzy measure will be used according to the situation and the number of elements of the set is mostly effective in making this determination. The thing to note here is that if the weights of the singletons in the set are given, these weights could be equal to the fuzzy measure of the singletons with the condition that the sum of fuzzy measure of singletons is equal to one. Otherwise, we may use decision making methods such as analytic hierarchy process (AHP), analytic network process (ANP) to determine fuzzy measure of singletons. For example, in sets with at least 3 elements, the decision maker establishes a hypothetical measure that will be effective in relieving the decision-making process as it can directly see the interaction between the criteria. Let X be a finite set with at least 3 elements, and let σ be a fuzzy measure on X. If

(i)

$\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{i}}\mathrm{,}{\mathit{x}}_{\mathit{j}}\}\mathrm{)}=\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{i}}\}\mathrm{)}+\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{j}}\}\mathrm{)}$ , then no interaction ${\mathit{x}}_{\mathit{i}}$ between ${\mathit{x}}_{\mathit{j}}$ ,

(ii)

$\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{i}}\mathrm{,}{\mathit{x}}_{\mathit{j}}\}\mathrm{)}\mathrm{>}\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{i}}\}\mathrm{)}+\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{j}}\}\mathrm{)}$ , then ${\mathit{x}}_{\mathit{i}}$ and ${\mathit{x}}_{\mathit{j}}$ are positively interactive,

(iii)

$\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{i}}\mathrm{,}{\mathit{x}}_{\mathit{j}}\}\mathrm{)}\mathrm{<}\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{i}}\}\mathrm{)}+\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_{\mathit{j}}\}\mathrm{)}$ , then ${\mathit{x}}_{\mathit{i}}$ and ${\mathit{x}}_{\mathit{j}}$ are negatively interactive, for all ${\mathit{x}}_{\mathit{i}}\mathrm{,}{\mathit{x}}_{\mathit{j}}\in \mathit{X}$ (Grabisch, 1997).

Moreover, the decision maker can also use measures such as λ-fuzzy measures (Larbani et al., 2011), k-additive fuzzy measures (Grabisch, 1997), p-symmetric fuzzy measures (Miranda et al., 2002) in the measurement determination process in order to ease the decision making process. Here, we use hypothetical fuzzy measure and a λ-fuzzy measure in the process of solving the pattern recognition and the medical diagnosis problem, respectively.

Step 2. The function ${\mathit{f}}_{\mathit{Q}\mathrm{,}\mathit{C}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}$ is calculated by using Definition 10.

Step 3. We choose the best alternative by using the recognition principle of maximum degree of similarity between IFSs.

4.1 Pattern Recognition Problem

We consider a pattern recognition problem which is adapted from Ye (2011).

Example 1.

Let ${\mathit{C}}_1$ , ${\mathit{C}}_2$ and ${\mathit{C}}_3$ be three patterns which are represented by following IFSs in a given finite set $\mathit{X}=\{{\mathit{x}}_1\mathrm{,}{\mathit{x}}_2\mathrm{,}{\mathit{x}}_3\}$ : $$\begin{array}{r}{\mathit{C}}_1=\{⟨{\mathit{x}}_1\mathrm{,}1.0\mathrm{,}0.0⟩\mathrm{,}⟨{\mathit{x}}_2\mathrm{,}0.8\mathrm{,}0.0⟩\mathrm{,}⟨{\mathit{x}}_3\mathrm{,}0.7\mathrm{,}0.1⟩\}\mathrm{,}\\ {\mathit{C}}_2=\{⟨{\mathit{x}}_1\mathrm{,}0.8\mathrm{,}0.1⟩\mathrm{,}⟨{\mathit{x}}_2\mathrm{,}1.0\mathrm{,}0.0⟩\mathrm{,}⟨{\mathit{x}}_3\mathrm{,}0.9\mathrm{,}0.0⟩\}\mathrm{,}\\ {\mathit{C}}_3=\{⟨{\mathit{x}}_1\mathrm{,}0.6\mathrm{,}0.2⟩\mathrm{,}⟨{\mathit{x}}_2\mathrm{,}0.8\mathrm{,}0.0⟩\mathrm{,}⟨{\mathit{x}}_3\mathrm{,}1.0\mathrm{,}0.0⟩\}.\end{array}$$ Let $\mathit{Q}=\{⟨{\mathit{x}}_1\mathrm{,}0.5\mathrm{,}0.3⟩\mathrm{,}⟨{\mathit{x}}_2\mathrm{,}0.6\mathrm{,}0.2⟩\mathrm{,}⟨{\mathit{x}}_3\mathrm{,}0.8\mathrm{,}0.1⟩\}$ be a pattern that needs to be classified in one of three classes ${\mathit{C}}_1$ , ${\mathit{C}}_2$ , and ${\mathit{C}}_3$ .

Consider the following hypothetical fuzzy measure σ on X: $\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_1\}\mathrm{)}=0.5$ , $\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_2\}\mathrm{)}=0.3$ , $\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_3\}\mathrm{)}=0.2$ , $\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_1\mathrm{,}{\mathit{x}}_2\}\mathrm{)}=0.7$ , $\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_1\mathrm{,}{\mathit{x}}_3\}\mathrm{)}=0.8$ , $\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_2\mathrm{,}{\mathit{x}}_3\}\mathrm{)}=0.6$ , $\mathit{\sigma }\mathrm{(}\{{\mathit{x}}_1\mathrm{,}{\mathit{x}}_2\mathrm{,}{\mathit{x}}_3\}\mathrm{)}=1$ . For the sake of completeness, the weights used in Example 5.1 of Ye (2011) are considered here as the measures of singletons.

Table 1

Cosine values.

${\mathit{f}}_{{\mathit{C}}_1\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_1\mathrm{)}=0.8574$	${\mathit{f}}_{{\mathit{C}}_1\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_2\mathrm{)}=0.9486$	${\mathit{f}}_{{\mathit{C}}_1\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_3\mathrm{)}=0.9998$
${\mathit{f}}_{{\mathit{C}}_2\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_1\mathrm{)}=0.9146$	${\mathit{f}}_{{\mathit{C}}_2\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_2\mathrm{)}=0.9486$	${\mathit{f}}_{{\mathit{C}}_2\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_3\mathrm{)}=0.9922$
${\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_1\mathrm{)}=0.9761$	${\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_2\mathrm{)}=0.9486$	${\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_3\mathrm{)}=0.9922$

The cosine values are given in Table 1 and the results of similarity are given in the Table 2. For example, for ${\mathit{C}}_3$ pattern we have ${\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_2\mathrm{)}⩽{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_1\mathrm{)}⩽{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_3\mathrm{)}$ and so we get $$\begin{array}{rl}{\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}{\mathit{C}}_3\mathrm{,}\mathit{Q}\mathrm{)}& =\mathrm{(}\mathit{C}\mathrm{)}\underset{\mathit{X}}{{\displaystyle \int }}{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathit{d}\mathit{\sigma }={\displaystyle \sum }_{\mathit{k}=1}^3\mathrm{(}{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_{\mathrm{(}\mathit{k}\mathrm{)}}\mathrm{)}-{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_{\mathrm{(}\mathit{k}-1\mathrm{)}}\mathrm{)}\mathrm{)}\mathit{\sigma }\mathrm{(}{\mathit{E}}_{\mathrm{(}\mathit{k}\mathrm{)}}\mathrm{)}\\ & =\mathrm{(}{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_2\mathrm{)}-{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_0\mathrm{)}\mathrm{)}\mathit{\sigma }\mathrm{(}{\mathit{E}}_{\mathrm{(}1\mathrm{)}}\mathrm{)}\\ & +\mathrm{(}{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_1\mathrm{)}-{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_2\mathrm{)}\mathrm{)}\mathit{\sigma }\mathrm{(}{\mathit{E}}_{\mathrm{(}2\mathrm{)}}\mathrm{)}\\ & +\mathrm{(}{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_3\mathrm{)}-{\mathit{f}}_{{\mathit{C}}_3\mathrm{,}\mathit{Q}}\mathrm{(}{\mathit{x}}_1\mathrm{)}\mathrm{)}\mathit{\sigma }\mathrm{(}{\mathit{E}}_{\mathrm{(}3\mathrm{)}}\mathrm{)}\\ & =0.9486\times 1+\mathrm{(}0.9761-0.9486\mathrm{)}\times 0.8\\ & +\mathrm{(}0.9922-0.9761\mathrm{)}\times 0.2\\ & =\mathrm{0.9738.}\end{array}$$

Table 2

Comparison of classification results of Example 1.

Similarity measure	Similarity scores
	$\mathrm{(}{\mathit{C}}_1\mathrm{,}\mathit{Q}\mathrm{)}$	$\mathrm{(}{\mathit{C}}_2\mathrm{,}\mathit{Q}\mathrm{)}$	$\mathrm{(}{\mathit{C}}_3\mathrm{,}\mathit{Q}\mathrm{)}$
	W (for $\mathit{\alpha }=\mathit{\beta }=\mathit{\gamma }=1\mathrm{/}3$ and $\mathit{p}=2$ , with respect to σ), (Yang and Ha, 2008)	0.7219	$\mathbf{0.8248}$	0.6294
${\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}$	0.9223	0.9437	$\mathbf{0.9738}$
${\mathit{W}}_{\mathit{IFS}}$ (Ye2011)	0.9133	0.9404	$\mathbf{0.9712}$

According to the recognition principle of maximum degree of similarity between IFSs, the process of assigning the pattern Q to ${\mathit{C}}_{\mathit{i}}$ is described by $$\begin{array}{c}\mathit{t}=\underset{1⩽\mathit{i}⩽3}{argmax}\{{\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}\mathrm{(}{\mathit{C}}_{\mathit{i}}\mathrm{,}\mathit{Q}\mathrm{)}\}.\end{array}$$(1) Therefore, we obtain $\mathit{t}={\mathit{C}}_3$ from (1) for the proposed cosine similarity measure ${\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\sigma }\mathrm{)}}$ . Namely, Q pattern belongs to the class ${\mathit{C}}_3$ . This result is in agreement with the result of Ye (2011). Moreover, we obtain again $\mathit{t}={\mathit{C}}_3$ , when we solve this problem for W (Definition 9) by using the same fuzzy measure σ.

4.2 Medical Diagnosis Problem

Medical diagnosis is the process of determining which disease explains the symptoms of a patient. In this process, patterns of symptoms are compared with patterns of disease. We take the medical diagnosis problem which is discussed by Ye (2011).

Example 2.

Let us consider a set of diagnosis and symptoms as follow. $$\begin{array}{rl}& \mathit{Q}=\left\{\begin{array}{l}{\mathit{Q}}_1\mathrm{(}\text{Viral fever}\mathrm{)}\mathrm{,}{\mathit{Q}}_2\mathrm{(}\text{Malaria}\mathrm{)}\mathrm{,}{\mathit{Q}}_3\mathrm{(}\text{Typhoid}\mathrm{)}\mathrm{,}\\ {\mathit{Q}}_4\mathrm{(}\text{Stomach problem}\mathrm{)}\mathrm{,}{\mathit{Q}}_5\mathrm{(}\text{Chest Problem}\mathrm{)}\end{array}\right\}\mathrm{,}\\ & \mathit{S}=\left\{\begin{array}{l}{\mathit{s}}_1\mathrm{(}\text{Temperature}\mathrm{)}\mathrm{,}{\mathit{s}}_2\mathrm{(}\text{Headache}\mathrm{)}\mathrm{,}{\mathit{s}}_3\mathrm{(}\text{Stomach pain}\mathrm{)}\mathrm{,}\\ {\mathit{s}}_4\mathrm{(}\text{Cough}\mathrm{)}\mathrm{,}{\mathit{s}}_5\mathrm{(}\text{Chest pain}\mathrm{)}\end{array}\right\}.\end{array}$$

Suppose that a patient, with respect to all the symptoms, is represented by the following IFS: $$\begin{array}{c}\mathit{P}\mathrm{(}\text{Patient}\mathrm{)}=\left\{\begin{array}{l}⟨{\mathit{s}}_1\mathrm{,}0.8\mathrm{,}0.1⟩\mathrm{,}⟨{\mathit{s}}_2\mathrm{,}0.6\mathrm{,}0.1⟩\mathrm{,}⟨{\mathit{s}}_3\mathrm{,}0.2\mathrm{,}0.8⟩\mathrm{,}\\ ⟨{\mathit{s}}_4\mathrm{,}0.6\mathrm{,}0.1⟩\mathrm{,}⟨{\mathit{s}}_5\mathrm{,}0.1\mathrm{,}0.6⟩\end{array}\right\}.\end{array}$$

Moreover, assume that each diagnosis ${\mathit{Q}}_{\mathit{i}}$ for $\mathit{i}=1\mathrm{,}2\mathrm{,}3\mathrm{,}4\mathrm{,}5$ is given as an IFSs with respect to all the symptoms as follow. $$\begin{array}{rl}& {\mathit{Q}}_1\mathrm{(}\text{Viral fever}\mathrm{)}=\left\{\begin{array}{l}⟨{\mathit{s}}_1\mathrm{,}0.4\mathrm{,}0.0⟩\mathrm{,}⟨{\mathit{s}}_2\mathrm{,}0.3\mathrm{,}0.5⟩\mathrm{,}⟨{\mathit{s}}_3\mathrm{,}0.1\mathrm{,}0.7⟩\mathrm{,}\\ ⟨{\mathit{s}}_4\mathrm{,}0.4\mathrm{,}0.3⟩\mathrm{,}⟨{\mathit{s}}_5\mathrm{,}0.1\mathrm{,}0.7⟩\end{array}\right\}\mathrm{,}\\ & {\mathit{Q}}_2\mathrm{(}\text{Malaria}\mathrm{)}=\left\{\begin{array}{l}⟨{\mathit{s}}_1\mathrm{,}0.7\mathrm{,}0.0⟩\mathrm{,}⟨{\mathit{s}}_2\mathrm{,}0.2\mathrm{,}0.6⟩\mathrm{,}⟨{\mathit{s}}_3\mathrm{,}0.0\mathrm{,}0.9⟩\mathrm{,}\\ ⟨{\mathit{s}}_4\mathrm{,}0.7\mathrm{,}0.0⟩\mathrm{,}⟨{\mathit{s}}_5\mathrm{,}0.1\mathrm{,}0.8⟩\end{array}\right\}\mathrm{,}\\ & {\mathit{Q}}_3\mathrm{(}\text{Typhoid}\mathrm{)}=\left\{\begin{array}{l}⟨{\mathit{s}}_1\mathrm{,}0.3\mathrm{,}0.3⟩\mathrm{,}⟨{\mathit{s}}_2\mathrm{,}0.6\mathrm{,}0.1⟩\mathrm{,}⟨{\mathit{s}}_3\mathrm{,}0.2\mathrm{,}0.7⟩\mathrm{,}\\ ⟨{\mathit{s}}_4\mathrm{,}0.2\mathrm{,}0.6⟩\mathrm{,}⟨{\mathit{s}}_5\mathrm{,}0.1\mathrm{,}0.9⟩\end{array}\right\}\mathrm{,}\\ & {\mathit{Q}}_4\mathrm{(}\text{Stomach problem}\mathrm{)}=\left\{\begin{array}{l}⟨{\mathit{s}}_1\mathrm{,}0.1\mathrm{,}0.7⟩\mathrm{,}⟨{\mathit{s}}_2\mathrm{,}0.2\mathrm{,}0.4⟩\mathrm{,}⟨{\mathit{s}}_3\mathrm{,}0.8\mathrm{,}0.0⟩\mathrm{,}\\ ⟨{\mathit{s}}_4\mathrm{,}0.2\mathrm{,}0.7⟩\mathrm{,}⟨{\mathit{s}}_5\mathrm{,}0.2\mathrm{,}0.7⟩\end{array}\right\}\mathrm{,}\\ & {\mathit{Q}}_5\mathrm{(}\text{Chest Problem}\mathrm{)}=\left\{\begin{array}{l}⟨{\mathit{s}}_1\mathrm{,}0.1\mathrm{,}0.8⟩\mathrm{,}⟨{\mathit{s}}_2\mathrm{,}0.0\mathrm{,}0.8⟩\mathrm{,}⟨{\mathit{s}}_3\mathrm{,}0.2\mathrm{,}0.8⟩\mathrm{,}\\ ⟨{\mathit{s}}_4\mathrm{,}0.2\mathrm{,}0.8⟩\mathrm{,}⟨{\mathit{s}}_5\mathrm{,}0.8\mathrm{,}0.1⟩\end{array}\right\}.\end{array}$$

The goal is to classify P into one of the diagnosis ${\mathit{Q}}_{\mathit{i}}$ $\mathrm{(}\mathit{i}=1\mathrm{,}2\mathrm{,}3\mathrm{,}4\mathrm{,}5\mathrm{)}$ with respect to all symptoms. For this purpose we need a fuzzy measure. We calculate the measures of singletons by Analytic Hierarchy Process (AHP) (Saaty, 1980). In AHP we use the following Standard Preference (Table 3).

Table 3

Standard preference table.

Importance scale	Definition
1	Equal importance
3	Moderate importance of one over another
5	Essential or strong importance
7	Very strong importance
9	Extreme importance
2, 4, 6 and 8	Intermediate values between two adjacent judgments

Table 4 depends on the expert view and it is giving the consistent reciprocal comparison matrix with consistency index 0.0941903 used in (AHP).

Table 4

Reciprocal comparison matrix.

	${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{s}}_3$	${\mathit{s}}_4$	${\mathit{s}}_5$
${\mathit{s}}_1$	1	3	2	7	8
${\mathit{s}}_2$	0.333333	1	0.333333	6	6
${\mathit{s}}_3$	0.5	3	1	4	5
${\mathit{s}}_4$	0.142857	0.166667	0.25	1	3
${\mathit{s}}_5$	0.125	0.166667	0.2	0.333333	1

The weights that are calculated by (AHP) are given in Table 5.

Table 5

Weight of singletons.

${\mathit{W}}_{{\mathit{s}}_1}=0.428156$	${\mathit{W}}_{{\mathit{s}}_2}=0.191606$
${\mathit{W}}_{{\mathit{s}}_3}=0.280161$	${\mathit{W}}_{{\mathit{s}}_4}=0.062917$
${\mathit{W}}_{{\mathit{s}}_5}=0.0371593$

Now, we are ready to construct a fuzzy measure which is a λ-fuzzy measure (see, Takahagi, 2000). By taking $\mathit{\lambda }=0.75$ we construct the fuzzy measure σ given in Table 6.

Table 6

Fuzzy measure.

$\mathit{\eta }\mathrm{(}\varnothing \mathrm{)}=0$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\}\mathrm{)}=0.685876$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_2\}\mathrm{)}=0.386562$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_3\}\mathrm{)}=0.51713$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_4\}\mathrm{)}=0.145252$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_5\}\mathrm{)}=0.0882035$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_2\}\mathrm{)}=0.836764$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_3\}\mathrm{)}=0.887728$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_4\}\mathrm{)}=0.742572$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.720304$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_2\mathrm{,}{\mathit{s}}_3\}\mathrm{)}=0.726001$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_2\mathrm{,}{\mathit{s}}_4\}\mathrm{)}=0.481904$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_2\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.444458$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_3\mathrm{,}{\mathit{s}}_4\}\mathrm{)}=0.595614$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_3\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.564789$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_4\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.222067$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_2\mathrm{,}{\mathit{s}}_3\}\mathrm{)}=0.969258$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_2\mathrm{,}{\mathit{s}}_4\}\mathrm{)}=0.873979$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_2\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.859362$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_4\}\mathrm{)}=0.918363$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.906331$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_4\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.772556$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_2\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_4\}\mathrm{)}=0.777517$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_2\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.757284$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_2\mathrm{,}{\mathit{s}}_4\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.532325$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_3\mathrm{,}{\mathit{s}}_4\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.637119$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_2\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_4\}\mathrm{)}=0.989366$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_2\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.981468$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_2\mathrm{,}{\mathit{s}}_4\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.893659$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_4\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.934564$
$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_2\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_4\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=0.80476$	$\mathit{\eta }\mathrm{(}\{{\mathit{s}}_1\mathrm{,}{\mathit{s}}_2\mathrm{,}{\mathit{s}}_3\mathrm{,}{\mathit{s}}_4\mathrm{,}{\mathit{s}}_5\}\mathrm{)}=1$

We calculate the Choquet cosine similarity measure of the IFSs. For this purpose we need the cosine values obtained in Table 7.

Table 7

Cosine values.

${\mathit{f}}_{{\mathit{Q}}_1\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_1\mathrm{)}=0.9922$	${\mathit{f}}_{{\mathit{Q}}_1\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_2\mathrm{)}=0.6484$	${\mathit{f}}_{{\mathit{Q}}_1\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_3\mathrm{)}=0.9946$	${\mathit{f}}_{{\mathit{Q}}_1\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_4\mathrm{)}=0.8877$
${\mathit{f}}_{{\mathit{Q}}_1\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_5\mathrm{)}=0.9997$	${\mathit{f}}_{{\mathit{Q}}_2\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_1\mathrm{)}=0.9922$	${\mathit{f}}_{{\mathit{Q}}_2\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_2\mathrm{)}=0.4678$	${\mathit{f}}_{{\mathit{Q}}_2\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_3\mathrm{)}=0.9701$
${\mathit{f}}_{{\mathit{Q}}_2\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_4\mathrm{)}=0.9863$	${\mathit{f}}_{{\mathit{Q}}_2\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_5\mathrm{)}=0.9991$	${\mathit{f}}_{{\mathit{Q}}_3\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_1\mathrm{)}=0.7893$	${\mathit{f}}_{{\mathit{Q}}_3\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_2\mathrm{)}=1$
${\mathit{f}}_{{\mathit{Q}}_3\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_3\mathrm{)}=0.9994$	${\mathit{f}}_{{\mathit{Q}}_3\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_4\mathrm{)}=0.4678$	${\mathit{f}}_{{\mathit{Q}}_3\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_5\mathrm{)}=0.9985$	${\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_1\mathrm{)}=0.2631$
${\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_2\mathrm{)}=0.5881$	${\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_3\mathrm{)}=0.2425$	${\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_4\mathrm{)}=0.4290$	${\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_5\mathrm{)}=0.9936$
${\mathit{f}}_{{\mathit{Q}}_5\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_1\mathrm{)}=0.2461$	${\mathit{f}}_{{\mathit{Q}}_5\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_2\mathrm{)}=0.1643$	${\mathit{f}}_{{\mathit{Q}}_5\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_3\mathrm{)}=1$	${\mathit{f}}_{{\mathit{Q}}_5\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_4\mathrm{)}=0.3987$
${\mathit{f}}_{{\mathit{Q}}_5\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_5\mathrm{)}=0.2824$

The results of the cosine similarity measure are given in Table 8. For example, for ${\mathit{Q}}_4$ symptom we obtain ${\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_3\mathrm{)}⩽{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_1\mathrm{)}⩽{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_4\mathrm{)}⩽{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_2\mathrm{)}⩽{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_5\mathrm{)}$ and so we have $$\begin{array}{rl}{\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\eta }\mathrm{)}}\mathrm{(}{\mathit{Q}}_4\mathrm{,}\mathit{P}\mathrm{)}& =\mathrm{(}\mathit{C}\mathrm{)}\underset{\mathit{X}}{{\displaystyle \int }}{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathit{d}\mathit{\eta }={\displaystyle \sum }_{\mathit{k}=1}^5\mathrm{(}{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_{\mathrm{(}\mathit{k}\mathrm{)}}\mathrm{)}-{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_{\mathrm{(}\mathit{k}-1\mathrm{)}}\mathrm{)}\mathrm{)}\mathit{\eta }\mathrm{(}{\mathit{E}}_{\mathrm{(}\mathit{k}\mathrm{)}}\mathrm{)}\\ & =\mathrm{(}{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_3\mathrm{)}-{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_0\mathrm{)}\mathrm{)}\mathit{\eta }\mathrm{(}{\mathit{E}}_{\mathrm{(}1\mathrm{)}}\mathrm{)}\\ & +\mathrm{(}{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_1\mathrm{)}-{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_3\mathrm{)}\mathrm{)}\mathit{\eta }\mathrm{(}{\mathit{E}}_{\mathrm{(}2\mathrm{)}}\mathrm{)}\\ & +\mathrm{(}{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_4\mathrm{)}-{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_1\mathrm{)}\mathrm{)}\mathit{\eta }\mathrm{(}{\mathit{E}}_{\mathrm{(}3\mathrm{)}}\mathrm{)}\\ & +\mathrm{(}{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_2\mathrm{)}-{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_4\mathrm{)}\mathrm{)}\mathit{\eta }\mathrm{(}{\mathit{E}}_{\mathrm{(}4\mathrm{)}}\mathrm{)}\\ & +\mathrm{(}{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_5\mathrm{)}-{\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}\mathrm{(}{\mathit{s}}_2\mathrm{)}\mathrm{)}\mathit{\eta }\mathrm{(}{\mathit{E}}_{\mathrm{(}5\mathrm{)}}\mathrm{)}\\ & =0.2425\times 1+\mathrm{(}0.2631-0.2425\mathrm{)}\times 0.893659\\ & +\mathrm{(}0.4290-0.2631\mathrm{)}\times 0.532325\\ & +\mathrm{(}0.5881-0.4290\mathrm{)}\times 0.444458\\ & +\mathrm{(}0.9936-0.5881\mathrm{)}\times 0.0882035\\ & =\mathrm{0.4557.}\end{array}$$

Therefore, we obtain $\mathit{t}={\mathit{Q}}_1$ from (1) for the proposed cosine similarity measure ${\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\eta }\mathrm{)}}$ . Namely, P patient has viral fever. This result is in agreement with the one obtained in Ye (2011). Moreover, we obtain again $\mathit{t}={\mathit{Q}}_1$ , when we solve this problem for W (Definition 9) by using the proposed fuzzy measure η.

Table 8

Comparison of classification results of Example 2.

Similarity measure	Similarity scores
	$\mathrm{(}{\mathit{Q}}_1\mathrm{,}\mathit{P}\mathrm{)}$	$\mathrm{(}{\mathit{Q}}_2\mathrm{,}\mathit{P}\mathrm{)}$	$\mathrm{(}{\mathit{Q}}_3\mathrm{,}\mathit{P}\mathrm{)}$	$\mathrm{(}{\mathit{Q}}_4\mathrm{,}\mathit{P}\mathrm{)}$	$\mathrm{(}{\mathit{Q}}_5\mathrm{,}\mathit{P}\mathrm{)}$
	W (for $\mathit{\alpha }=\mathit{\beta }=\mathit{\gamma }=1\mathrm{/}3$ and $\mathit{p}=2$ , with respect to η), (Yang and Ha, 2008)	0.6579	$\mathbf{0.7420}$	0.6855	0.4544	0.4692
${\mathit{W}}_{\mathit{IFS}}^{\mathrm{(}\mathit{C}\mathrm{,}\mathit{\eta }\mathrm{)}}$	$\mathbf{0.9685}$	0.9546	0.9426	0.4557	0.6441
${\mathit{C}}_{\mathit{IFS}}$ (Ye, 2011)	$\mathbf{0.9046}$	0.0.8602	0.8510	0.5033	0.4542

The numerical results presented in Table 2 and Table 8 show that the result of the proposed cosine similarity measure is consistent with Ye (2011). The difference of this cosine similarity measure from the existing similarity measures is that it is established by considering the interaction between the criteria. Indeed, in both similarity measures proposed by Yang and Ha (2008) and the cosine similarity measure algorithms proposed in this study, the interaction between the criteria using the fuzzy measure is considered. However, as the variables α, β, γ and p change, the result may change and the decision maker has to explain what he/she pays attention to when choosing these variables. When these variables are independent of special selection, both ordering of their $\mathit{f}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}$ values and the calculation of the Choquet integral become difficult and as the number of elements of the set increases, it becomes very hard to make comparisons between these calculations. In the Choquet cosine similarity measure proposed in this paper ${\mathit{f}}_{\mathit{A}\mathrm{,}\mathit{B}}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}$ is equal to $cos{\mathit{\theta }}_{\mathit{i}}$ for any $\mathit{i}=1\mathrm{,}\dots \mathrm{,}\mathit{n}$ . In this case, the decision maker will not have difficulty while ordering functions and calculating the proposed cosine similarity measure.

Figure 1 illustrates the calculation of the Choquet integral of ${\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}$ in Example 2. Choquet integral value is the sum of 5 rectangle areas.

Fig. 1

Choquet integral of ${\mathit{f}}_{{\mathit{Q}}_4\mathrm{,}\mathit{P}}$ .

5 Conclusion

In this paper, we propose a new cosine similarity measure based on the Choquet integral, inspired by the cosine similarity measure given for intuitionistic fuzzy sets in the literature. In addition, we apply this new similarity measure to a pattern recognition and a medical diagnosis problem and we obtain results that are consistent with the results obtained in the past. If we consider the sensitivity of Choquet integral compared with the weighted mean, we can say that the similarity measure proposed in this paper is more sensitive than the one proposed in the past. In the future, different kind of similarity measures and fuzzy sets can be considered. The applications can be extended to some other real life areas such as face recognition systems and classification.