An evidential view of similarity measure for Atanassov’s intuitionistic fuzzy sets

Abstract

In this paper, the construction of similarity measures for Atanassov’s intuitionistic fuzzy sets (AIFSs) is considered from the view of evidence theory. We define similarity measures for AIFSs in the framework of Dempster–Shafer evidence theory. The proposed similarity measures are applied to deal with pattern recognition and multiple criteria decision making problems. First, existing similarity measures for AIFSs are critically reviewed. Then we introduce the transformation from AIFSs to basic probability assignments (BPAs) in evidence theory. Based on Jousselme’s distance measure and cosine similarity measure between BPAs, two similarity measures between AIFSs are proposed. A composite similarity measure is constructed following the proof of properties related to our proposed similarity measures. Then, we use some contrastive examples to illustrate that the proposed similarity measure between AIFSs can overcome the drawbacks of existing similarity measures. Finally, we apply the proposed similarity measures between AIFSs to deal with pattern recognition and multiple criteria decision making problems. It is demonstrated that our proposed similarity measures can provide compatible results compared to those results obtained based on previous measures.

Keywords

Atanassov’s intuitionistic fuzzy sets Dempster–Shafer evidence theory distance measure similarity measure

1 Introduction

Since it was proposed by Zadeh [19], the theory of fuzzy set (FS) has achieved a great success due to its capability on handling uncertainty. In order to descript and process imperfect information better, several higher order fuzzy sets have been introduced in the literature. As a generation of Zadeh’s fuzzy sets, Atanassov’s intuitionistic fuzzy sets (AIFSs), first formulated by Atanassov [18], can express and process uncertainty much better by introducing a hesitation degree. In fuzzy sets, the membership value μ (x) of x in X (called the universe) is just a single real number in [0, 1], and the non-membership of x is taken as 1 − μ (x). But for AIFSs, the membership value μ (x) and the non-membership value v (x) are both taken into account to describe any x in X. Moreover, the sum of membership and non-membership is less than or equal to 1. Thus, an AIFS is expressed with ordered pair of real numbers (μ (x) , v (x)). The difference between 1 and μ (x) + v (x), i.e., 1 − μ (x) − v (x), is called the hesitancy degree. As a novel mathematical tool for handling uncertainty, AIFS is attracting more and more attentions from researchers. Theoretical research on AIFSs mainly concentrates on its relationship with other generalized fuzzy sets, intuitionistic fuzzy operations, and some important mathematical measures for AIFSs. Meanwhile, the application of AIFS has been extended to many areas.

The definition of similarity measure between two AIFSs is one of the most interesting topics in intuitionistic fuzzy theory. A similarity measure is defined to compare the information carried by AIFSs. Measures of similarity between AIFSs, as an important tool for decision making, pattern recognition, machine learning, and image processing, have received much attention in recent years. A recent overview of similarity measures between AIFSs is published in [20]. A similarity measure between two AIFS is a real number from the unit interval representing the alignment of the two sets. The typical definition of a similarity measure is one minus the distance between vectors representing the membership and non-membership values of the elements in the sets being compared. This approach leads to various counter-intuitive numerical examples, where clearly different sets have the same degree of similarity. Szmidt and Kacprzyk [8] have pointed out the need to use hesitancy along with the membership and non-membership values themselves to overcome some of the problems. However, treating hesitancy in the same way as membership and non-membership values also poses its own problems, with many counter-intuitive examples. Beliakov et al. [11] argued that counter-intuitive similarity values were in fact inevitable under any definition of a similarity measure between AIFS as a real value. They held that a single value cannot account for both membership in a set and hesitancy in that membership simultaneously. So they proposed vector valued similarity measures for AIFSs. The vector valued similarity measures comprise a similarity measure and uncertainty measure. It is proved that the vector valued similarity measure can prevent some counter-intuitive cases. However, it brings the problem of comparing vector values in pattern recognition applications.

We hold the perspective that the similarity degree between two AIFSs should be a real number. Although much uncertainty hides in AIFSs, the similarity degree between them should be precise. In order to define a more reasonable similarity measure for AIFSs, we will address this problem from the view of Dempster-Shafer evidence theory, which has been applied in many fields, such as information fusion [29], decision making support [30], and pattern recognition [31]. It is has been proved that the interval valued representation of AIFSs is in the same spirit as the representation of the imprecision in Dempster-Shafer evidence theory [26, 28]. Here, our objective is to provide a distance measure for intuitionistic fuzzy sets in the spirit of Dempster–Shafer evidence theory with the hope of providing more alternative measures for AIFSs. Our position is that none of the distance measures can be said to be superior to the others in the absolute and that the choice of such a measure should always be guided by practical considerations relative to a specific application.

In this paper, a similarity measure for AIFSs will be defined based on the relationship between AIFS and basic probability assignments (BPAs) in evidence theory. For one-element AIFSs, they can be transformed to dichotomous BPAs. We can construct a distance measure between AIFSs based on the distance measure between BPAs. This distance measure can be naturally extended to AIFSs with more than one element. Properties of the proposed distance measure will be also proposed. Illustrative examples and applications are also employed to demonstrate the performance of the proposed similarity measure.

We organize the remainder of this paper as following. Section 2 recalls the definitions related to the AIFSs, and lists the properties of distance and similarity measures for AIFSs. Some knowledge related to Dempster–Shafer theory is also presented in this section. In Section 3, some existing similarity measures for AIFSs are reviewed. We propose several new similarity measures for AIFSs in Section 4 based on the relation between AIFSs and BPAs. In this section, we also give some axiomatic properties and proofs related to our proposed similarity measures. Comparison between the proposed similarity measures and the existing similarity measures is also carried out. The application of the proposed similarity measures is discussed in Section 5. This paper is concluded in Section 6.

2 Preliminaries

In this section, we briefly recall the basic concepts related to AIFS, and then list the properties of the axiomatic definition for similarity and distance measures. In the current paper, we will use the Dempster–Shafer evidence theory (shorten as evidence theory) for the interpretation of AIFS and as the base of a new method for defining distance measure. So we will also present a brief description on some fundamentals of evidence theory to ease subsequent analysis.

2.1 Atanassov’s intuitionistic fuzzy sets

Definition 1. [19] Let X = {x₁, x₂, ⋯ , x_n} be auniverse of discourse, then a fuzzy set A in X is defined as: $A = {〈 x, μ_{A} (x) 〉 | x \in X}$ (1) where μ_A (x) : X → [0, 1] is the membership degree.

Definition 2. [18] An intuitionistic fuzzy set A in X defined by Atanassov can be written as: $A = {〈 x, μ_{A} (x), v_{A} (x) 〉 | x \in X}$ (2) where μ_A (x) : X → [0, 1] and v_A (x) : X → [0, 1] are membership degree and non-membership degree, respectively, with the condition: $0 \leq μ_{A} (x) + v_{A} (x) \leq 1$ (3)π_A (x) determined by $π_{A} (x) = 1 - μ_{A} (x) - v_{A} (x)$ (4) is called the hesitancy degree of the element x ∈ X with respect to the AIFS A, and π_A (x) ∈ [0, 1], ∀x ∈ X.

π_A (x) is also called the intuitionistic index of x with respect to A. Greater π_A (x) indicates more vagueness on x. Obviously, when π_A (x) =0, ∀x ∈ X, A degenerates into an ordinary fuzzy set.

In the sequel, the couple 〈μ_A (x) , v_A (x)〉 is called an AIFS or intuitionistic fuzzy value (IFV) for clarity. AIFSs (X) is used to denote the set of all AIFSs in X.

Definition 3. [18] For A ∈ AIFSs (X) and B ∈ AIFSs (X), some relations between them are defined as:

A ⊆ B iff μ_A (x) ≤ μ_B (x) , v_A (x) ≥ v_B (x), ∀x ∈ X;

A = B iff μ_A (x) = μ_B (x) , v_A (x) = v_B (x), ∀x ∈ X;

A^C ={ 〈 x, v_A (x) , μ_A (x) 〉 |x ∈ X }, where A^C is the complement of A.

Definition 4. [9] Let D denote a mapping D : AIFS × AIFS → [0, 1]. If D (A, B) satisfies the following properties, D (A, B) is called a distance between A ∈ AIFSs (X) and B ∈ AIFSs (X):

0 ≤ D (A, B) ≤1;

D (A, B) =0, if and only if A = B;

D (A, B) = D (B, A);

If A ⊆ B ⊆ C, then D (A, B) ≤ D (A, C), and D (B, C) ≤ D (A, C).

Definition 5. [9] A mapping S : AIFS × AIFS → [0, 1] is called a degree of similarity between A ∈ AIFSs (X) and B ∈ AIFSs (X), if S (A, B) satisfies the following properties:

0 ≤ S (A, B) ≤1;

S (A, B) =1, if and only if A = B;

S (A, B) = S (B, A);

If A ⊆ B ⊆ C, then S (A, B) ≥ S (A, C), and S (B, C) ≥ S (A, C).

We can note that distance and similarity measures are a couple of complementary concepts. So similarity measures can be used to define distance measures, and vice versa. In many research papers, they are investigated simultaneously.

2.2 Dempster–Shafer evidence theory

Dempster–Shafer evidence theory, which is also known as belief function theory, was modeled based on a finite set of mutually exclusive elements, called the frame of discernment denoted by Ω. The power set of Ω, denoted by 2^Ω, contains all possible unions of the sets in Ω including Ω itself. Singleton sets in the discernment frame Ω are called atomic sets because they do not contain nonempty subsets. It is assumed that only one atomic set can be true at any one time. If a set is assumed to be true, then all supersets are considered true as well. The following definitions are central in Dempster–Shafer evidence theory.

Definition 6. [12] Let Ω = {A₁, A₂, ⋯ , A_n} be the frame of discernment. A basic probability assignment (BPA) is a function m : 2^Ω → [0, 1], satisfying the following two conditions: $m (\emptyset) = 0$ (5) $\sum_{A \subseteq Ω} m (A) = 1$ (6) where ∅ denotes the empty set, and A is any subset of Ω. Such function is also called a basic belief assignment or a belief structure. For each subset A ⊆ Ω, the value taken by the BPA at A is called the basic probability mass of A, denoted by m (A).

Definition 7. [12] A subset A of Ω is called the focal element of a belief structure m if m (A) >0.

Definition 8. [12] A Bayesian belief structure (BBS) on Ω is a belief structure on Ω whose focal elements are atomic sets (singletons) of Ω.

Definition 9. [22] Given a belief structure m on Ω, the belief function and plausibility function which are in one-to-one correspondence with m can be defined respectively as: $Bel (A) = \sum_{B \subseteq A} m (B)$ (7) $Pl (A) = \sum_{B \cap A \neq \emptyset} m (B) = 1 - \sum_{B \cap A = \emptyset} m (B)$ (8)

Definition 10. [22] The pignistic transformation maps a belief structure m to so called pignistic probability function. The pignistic transformation of a belief structure m on Ω = {A₁, A₂, ⋯ , A_n} is given by $BetP (A) = \sum_{B \subseteq Ω} \frac{| A \cap B |}{| B |} \frac{m (B)}{1 - m (\emptyset)}, \forall A \subseteq Ω$ (9) where |A| is the cardinality of set A.

Definition 11. [12] Given two belief structures m₁ and m₂ in Ω, the belief structure that results from the application of Dempster’s combination rule, denoted as m₁ ⊕ m₂, or m₁₂ for short, is given by:

$\begin{matrix} m_{1} \oplus m_{2} (A) \\ = {\begin{matrix} \frac{\sum_{B \cap C = A} m_{1} (B) m_{2} (C)}{1 - \sum_{B \cap C = \emptyset} m_{1} (B) m_{2} (C)}, \forall A \subseteq Ω, A \neq \emptyset \\ 0, A = \emptyset \end{matrix} \end{matrix}$ (10)

3 Existing similarity measures

Since the introduction of Atanassov’s intuitionistic fuzzy sets, many similarity measures between AIFSs have been defined in the technical literature. In this section, we will give a survey of existing similarity measures. We hope this survey is exhaustive and apologize for any forgotten contribution.

Let X = {x₁, x₂, ⋯ , x_n} be the universe of discourse. A ∈ AIFSs (X) and B ∈ AIFSs (X) are two AIFSs in X, denoted by A = {〈x, μ_A (x) , v_A (x) 〉|x ∈ X} and B ={ 〈 x, μ_B (x) , v_B (x) 〉 |x ∈ X }, respectively.

Considering the analysis presented by Bustine and Burillo [14], which concluded that the intuitionistic fuzzy sets and the vague sets are similar, Chen [23] proposed a similarity measure for AIFSs defined as: $\begin{matrix} S_{C} (A, B) \\ = 1 - \frac{\sum_{i = 1}^{n} | (μ_{A} (x_{i}) - v_{A} (x_{i})) - (μ_{B} (x_{i}) - v_{B} (x_{i})) |}{2 n} \end{matrix}$ (11)

This similarity measure leads to many counter-intuitive cases identified in [7 , 27]. The fact that ∀x_i ∈ X, μ_A (x_i) − v_A (x_i) = μ_B (x_i) − v_B (x_i) ⇒ S_C (A, B) =1 can lead to a series of cases satisfying μ_A (x_i) − v_A (x_i) = μ_B (x_i) − v_B (x_i) and then S_C (A, B) =1. All of these are counter-intuitive except the case of A = B. The existence of counter-intuitive cases results from the fact that S_C (A, B) does not satisfy property condition SP2. For example, if A = {< x, 0.1, 0.1 >} and B = {< x, 0.6, 0.6 >}, we have S_C (A, B) =1, but in fact A ≠ B.

To overcome the deficiency of Chen’s similarity measure, Hong and Kim [7] proposed the following similarity measure: $\begin{matrix} S_{H} (A, B) \\ = 1 - \frac{1}{2 n} \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | v_{A} (x_{i}) - v_{B} (x_{i}) |) \end{matrix}$ (12)

S_H cannot distinguish positive difference from negative difference, so there are still counter-intuitive cases for it [27]. For example, given A = {< x, 0, 0 >} and B = {< x, 0.3, 0.5 >}, then we have S_H (A, B) =0.6, which is not reasonable. An example of another type of counter-intuitive case is given in [27]. If A = {< x, 1, 0 >}, B = {< x, 0, 0 >}, and C = {< x, 0.5, 0.5 >}, then, S_H (A, B) = S_H (B, C) =0.5.

Following the work of Hong and Kim, Li and Xu [10] proposed a new similarity measure as:

$\begin{matrix} S_{L} (A, B) \\ = - \frac{\sum_{i = 1}^{n} | (μ_{A} (x_{i}) - v_{A} (x_{i})) - (μ_{B} (x_{i}) - v_{B} (x_{i})) |}{4 n} \\ - \frac{\sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | + | v_{A} (x_{i}) - v_{B} (x_{i}) |)}{4 n} \end{matrix}$ (13)

Although S_L can overcome the counter cases of S_H and S_C, it is troubled by other limitations. The example in [27] shows the limitationsof S_L. Considering A = {< x, 0.4, 0.2 >}, B = {< x, 0.5, 0.3 >}, and C = {< x, 0.5, 0.2 >}, we have S_L (A, B) = S_L (A, C) =0.95, this result is not consistent with intuitive analysis.

Taking the membership distance and the non-membership distance between two AIFSs into consideration, Li et al. [27] proposed a similarity measure as: $\begin{matrix} S_{O} (A, B) \\ = 1 - \sqrt{\frac{\sum_{i = 1}^{n} ({(μ_{A} (x_{i}) - μ_{B} (x_{i}))}^{2} + {(v_{A} (x_{i}) - v_{B} (x_{i}))}^{2})}{2 n}} \end{matrix}$ (14)

This is essentially induced from the Euclidean distance. Since |μ_A (x_i) − μ_B (x_i) | = |μ_C (x_i) − μ_D (x_i) | and |v_A (x_i) − v_B (x_i) | = |v_C (x_i) − v_D (x_i) | will lead to S_O (A, B) = S_O (C, D), it is not sensitive to the change of AIFSs. For example, given A = {< x, 0.3, 0.4 >}, B = {< x, 0.5, 0.2 >}, and C = {< x, 0.1, 0.6 >}, we have S_O (A, B) = S_O (A, C) =0.8, which is not helpful in the application of pattern recognition.

Li and Cheng [6] pointed out that some similarity measures could not deal with the similarity between AIFSs well. So they defined a new similarity measure for pattern recognition: $S_{d}^{p} (A, B) = 1 - \sqrt[p]{\frac{\sum_{i = 1}^{n} {| φ_{A} (x_{i}) - φ_{B} (x_{i}) |}^{p}}{n}}$ (15) where $φ_{A} (x_{i}) = \frac{μ_{A} (x_{i}) + 1 - v_{A} (x_{i})}{2}$ , $φ_{B} (x_{i}) = \frac{μ_{B} (x_{i}) + 1 - v_{B} (x_{i})}{2}$ .

However, Li et al. [27] demonstrated by examples that there are some counter-intuitive cases for $S_{d}^{p}$ . So, if p = 1, $S_{d}^{p} = S_{C}$ and independent of the value of p, the similarity measure $S_{d}^{p}$ have the same type counter-intuitive cases as S_C. In the same way, Mitchell [15], Liang and Shi [32] presented other counter-intuitive cases for $S_{d}^{p}$ similarly in nature to those of S_C, which illustrated that $S_{d}^{p} (A, B)$ does not satisfy the property condition SP2.

As an example, we compute the similarity $S_{d}^{p} (A, B)$ with the sets A = {< x, 0.1, 0.1 >} and B = {< x, 0.6, 0.6 >} with p = 1, we find that $S_{d}^{1} (A, B) = S_{C} (A, B) = 1$ , which violates property condition SP2.

Mitchell [15] found that the similarity measure $S_{d}^{p}$ would characterize two different AIFSs as identical. To overcome this drawback, he provided a more realistic strong similarity measure of the following form: $S_{HB} (A, B) = \frac{1}{2} (ρ_{μ} (A, B) + ρ_{v} (A, B))$ (16) where $ρ_{μ} (A, B) = 1 - \sqrt[p]{\frac{\sum_{i = 1}^{n} {| μ_{A} (x_{i}) - μ_{B} (x_{i}) |}^{p}}{n}}$ ,

$ρ_{v} (A, B) = 1 - \sqrt[p]{\frac{\sum_{i = 1}^{n} {| v_{A} (x_{i}) - v_{B} (x_{i}) |}^{p}}{n}} .$

To solve the problems appeared in $S_{d}^{p}$ , Liang and Shi [32] proposed the following similarity measures: $S_{e}^{p} (A, B) = 1 - \sqrt[p]{\frac{\sum_{i = 1}^{n} {(φ_{μ} (x_{i}) + φ_{v} (x_{i}))}^{p}}{n}}$ (17) where φ_μ (x_i) = |μ_A (x_i) − μ_B (x_i) |/2, φ_v (x_i) = | (1 − v_A (x_i)) − (1 − v_B (x_i)) |/2 .

The authors mentioned that the proposed measure $S_{e}^{p}$ contains more information in AIFSs than $S_{d}^{p}$ . So, the difference between AIFSs can be discerned. Then, they gave another definition of similarity measure as: $S_{s}^{p} (A, B) = 1 - \sqrt[p]{\frac{\sum_{i = 1}^{n} {| ψ_{s 1} (x_{i}) + ψ_{s 2} (x_{i}) |}^{p}}{n}}$ (18) where $\begin{matrix} ψ_{s 1} (x_{i}) & = & | m_{A 1} (x_{i}) - m_{B 1} (x_{i}) | / 2, \\ ψ_{s 2} (x_{i}) & = & | m_{A 2} (x_{i}) - m_{B 2} (x_{i}) | / 2, \\ m_{A 1} (x_{i}) & = & | μ_{A} (x_{i}) + m_{A} (x_{i}) | / 2, \\ m_{B 1} (x_{i}) & = & | μ_{B} (x_{i}) + m_{B} (x_{i}) | / 2, \\ m_{A 2} (x_{i}) & = & | 1 - v_{A} (x_{i}) + m_{A} (x_{i}) | / 2, \\ m_{B 2} (x_{i}) & = & | 1 - v_{B} (x_{i}) + m_{B} (x_{i}) | / 2, \\ m_{A} (x_{i}) & = & | 1 - v_{A} (x_{i}) + μ_{A} (x_{i}) | / 2, \\ m_{B} (x_{i}) & = & | 1 - v_{B} (x_{i}) + μ_{B} (x_{i}) | / 2 . \end{matrix}$

The two proposed measures $S_{e}^{p}$ and $S_{s}^{p}$ cannot differentiate patterns in some cases. This can be shown by the example below.

Let two patterns be A₁ and A₂ in AIFSs(X), with X = {x₁, x₂, x₃}, defined as: $\begin{matrix} A_{1} & = & {< x_{1}, 0.2, 0.5 >, < x_{2}, 0.2, 0.5 >, \\ < x_{3}, 0.2, 0.5 >}, \\ A_{2} & = & {< x_{1}, 0.4, 0.3 >, < x_{2}, 0.4, 0.3 >, \\ < x_{3}, 0.4, 0.3 >} . \end{matrix}$

The sample to be recognized is: $\begin{matrix} B & = & {< x_{1}, 0.3, 0.4 >, < x_{2}, 0.3, 0.3 >, \\ < x_{3}, 0.3, 0.5 >} \end{matrix}$

Considering similarity measures $S_{e}^{p}$ and $S_{s}^{p}$ , we can get: $\begin{matrix} S_{e}^{1} (A_{1}, B) & = & S_{e}^{1} (A_{2}, B) = 0.68, \\ S_{s}^{1} (A_{1}, B) & = & S_{s}^{1} (A_{2}, B) = 0.68 . \end{matrix}$

So, the pattern B cannot be correctly classified using the proposed similarity measures.

Therefore, Liang and Shi [32] used other information to discern AIFSs and propose this similarity: $S_{h}^{p} (A, B) = 1 - \sqrt[p]{\frac{\sum_{i = 1}^{n} {(η_{1} (x_{i}) + η_{2} (x_{i}) + η_{3} (x_{i}))}^{p}}{3 n}}$ (19) where η₁ (x_i) = φ_μ (x_i) + φ_v (x_i) (defined in $S_{e}^{p}$ ),

η₂ (x_i) = |φ_μ (x_i) − φ_v (x_i) | (defined in $S_{d}^{p}$ ),

η₃ (x_i) = max(l_A (x_i) , l_B (x_i)) − min(l_A (x_i) , l_B (x_i)),

l_A (x_i) = (1 − μ_A (x_i) − v_A (x_i))/2,

l_B (x_i) = (1 − μ_B (x_i) − v_B (x_i))/2.

This is a relative powerful method for considering more information to measure the similarity degree of IFSs. Thus, $S_{h}^{p}$ can avoid all the counter-intuitive cases existing in above similarity measures.

Boran and Akay [9] proposed a biparametric similarity measure for AIFSs. It has the following form with two parameters: $\begin{matrix} S_{t}^{p} (A, B) \\ = 1 - \\ \sqrt[p]{\sum_{i = 1}^{n} \frac{1}{2 n (1 + p)} {{| t (Δ μ_{i}) - Δ v_{i} |}^{p} + {| t (Δ v_{i}) - Δ μ_{i} |}^{p}}} \end{matrix}$ (20)

where, Δμ_i = μ_A (x_i) − μ_B (x_i), Δv_i = v_A (x_i) − v_B (x_i), p = 1, 2, 3, ⋯ is the L_p norm and t = 2, 3, 4, ⋯ identifies the level of uncertainty.

It is demonstrated that $S_{t}^{p}$ can also avoid some counter-intuitive examples. Moreover, it can be applied to pattern recognition and classify patterns more effectively. However, it cannot discriminate some special AIFSs as shown in [24]. The lack of explicit physical meaning and the difficulty of determining additional parameters are also itslimitations.

Chen and Chang [24] have defined a new similarity measure between Atanassov’s intuitionistic fuzzy sets (AIFSs) based on transformation techniques. An AIFS A on X = {x₁, x₂, ⋯ , x_n} can be transformed to triangular fuzzy numbers A_{x
₁}, A_{x
₂}, ⋯ , A_{x
_n}, where A_{x
_i} has the following form:

$\begin{matrix} μ_{A_{x_{i}}} (z) \\ = {\begin{matrix} 1, & if z = μ_{A} (x_{i}) = 1 - v_{A} (x_{i}) \\ \frac{1 - v_{A} (x_{i}) - z}{1 - μ_{A} (x_{i}) - v_{A} (x_{i})} & if z \in [μ_{A} (x_{i}), 1 - v_{A} (x_{i})] \\ 0, & otherwise \end{matrix} \end{matrix}$ (21)

So, the similarity between AIFSs is defined as:

$\begin{matrix} S_{CC} (A, B) \\ = \sum_{i = 1}^{n} (1 - | μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ \times (1 - \frac{π_{A} (x_{i}) + π_{B} (x_{i})}{2})) \\ - \sum_{i = 1}^{n} \int_{0}^{1} | μ_{A_{x_{i}}} (z) - μ_{B_{x_{i}}} (z) | dz \\ \times (\frac{π_{A} (x_{i}) + π_{B} (x_{i})}{2}) \end{matrix}$ (22)

Ye [17] proposed a cosine similarity measure based on the concept of the cosine similarity measure for fuzzy sets: $\begin{matrix} C_{IFS} (A, B) \\ = \frac{1}{n} \sum_{i = 1}^{n} \frac{μ_{A} (x_{i}) μ_{B} (x_{i}) + v_{A} (x_{i}) v_{B} (x_{i})}{\sqrt{{(μ_{A} (x_{i}))}^{2} + {(v_{A} (x_{i}))}^{2}} \sqrt{{(μ_{B} (x_{i}))}^{2} + {(v_{B} (x_{i}))}^{2}}} \end{matrix}$ (23)

This similarity measure may lead to many counter-intuitive cases because ∀x_i ∈ X, μ_A (x_i)/v_A (x_i) = μ_B (x_i)/v_B (x_i) ⇒ C_IFS (A, B) =1. So the property condition SP2 cannot be satisfied. For example, if A = {< x, 0.1, 0.2 >} and B = {< x, 0.3, 0.6 >} then C_IFS (A, B) =1, however, A ≠ B.

Hung and Yang [25] introduced three similarity measures based on the Hausdorff distance d_H (A, B): $S_{HY}^{1} (A, B) = 1 - d_{H} (A, B)$ (24) $S_{HY}^{2} (A, B) = \frac{e^{- d_{H} (A, B)} - e^{- 1}}{1 - e^{- 1}}$ (25) $S_{HY}^{3} (A, B) = \frac{1 - d_{H} (A, B)}{1 + d_{H} (A, B)}$ (26) where $\begin{matrix} d_{H} (A, B) & = & \frac{1}{n} \sum_{i = 1}^{n} max (| μ_{A} (x_{i}) - μ_{B} (x_{i}) |, \\ | v_{A} (x_{i}) - v_{B} (x_{i}) |) . \end{matrix}$

For $S_{HY}^{1}$ , $S_{HY}^{2}$ and $S_{HY}^{3}$ , all of them face the counter-intuitive cases of S_L and S_H. For one-element AIFSs, if max(|μ_A (x) − μ_B (x) |, |v_A (x) − v_B (x) |) = max(|μ_C (x) − μ_D (x) |, |v_C (x) − v_D (x) |), then S_HY (A, B) = S_HY (C, D). It is obvious that a lot of cases satisfy these conditions, so the S_HY series measure similarity too roughly, also leading to counter-intuitive cases.

Since similarity measure and distance measure are dual converse concepts, other widely used distance measures are applied to define similarity measure for AIFSs. The following distance measures have been proposed:

Hamming Distance: $H (A, B) = \frac{1}{2 n} \sum_{i = 1}^{n} ((| Δ μ_{i} | + | Δ v_{i} | + | Δ π_{i} |))$ (27)

Euclidean Distance: $E (A, B) = \sqrt{\frac{1}{2 n} \sum_{i = 1}^{n} ((Δ μ_{i})^{2} + (Δ v_{i})^{2})}$ (28)

We can find that similarity measures S_H and S_o are induced by Hamming Distance and Euclidean Distance, respectively. So these distance measures cannot depict the relationship between AIFSs perfectly.

By taking the hesitancy degree into account, Szdmit and Kacprzyk [8] proposed the following distance measures between AIFSs A and B based on a geometric interpretation of AIFSs.

The normalized Hamming distance $L_{IFS} (A, B) = \frac{1}{2 n} \sum_{i = 1}^{n} (| Δ μ_{i} | + | Δ v_{i} | + | Δ π_{i} |)$ (29) where Δπ_i = π_A (x_i) − π_B (x_i).

The normalized Euclidean distance:

$\begin{matrix} q_{IFS} (A, B) \\ = \sqrt{\frac{1}{2 n} \sum_{i = 1}^{n} ((Δ μ_{i})^{2} + (Δ v_{i})^{2} + (Δ π_{i})^{2})} \end{matrix}$ (30)

For L_IFS and q_IFS, more information is taken into consideration. So they are much applicable in pattern recognition. However, since membership degree, non-membership degree and hesitancy degree are treated equally in these distances, they cannot classify patterns in some cases. For example, if A = {< x, 1, 0 >}, B = {< x, 0, 1 >} and C = {< x, 0, 0 >}, we have q_IFS (A, B) = q_IFS (A, C) =0.7 and L_IFS (A, B) = L_IFS (A, C) =0.5.

Recently, Beliakov et al. [11] argued that counter-intuitive similarity values were in fact inevitable under any definition of a similarity measure between AIFSs as a real value. Their point is that a single value cannot account for both membership in a set and hesitancy in that membership simultaneously. Based on such analysis, they define a vector valued similarity consisting of a similarity measure and uncertainty measure for AIFSs. Although counter-intuitive examples can be avoided by this vector valued similarity measure, the relation between its two components is not clear.

We also hold the perspective that counter-intuitive similarity values are inevitable under any definition of a similarity measure between AIFSs as a real value. Nevertheless, the vector valued similarity measures will intensify the complexity of similarity measure, which is not helpful in applications. So our point is that defining similarity degree between AISFs as a real number is much more referential than vector values. What we need to do is try to reduce the counter-intuitive values caused by real numbered similarity measures even they are inevitable. Thus a more reasonable similarity measure between AIFSs is desirable.

4 New similarity measures for AIFSs

4.1 Interpretation of AIFSs in the framework of evidence theory

Hong and Kim [7] have proposed to use an interval representation [μ_A (x) , 1 − v_A (x)] of intuitionistic fuzzy set A in X instead of the couple 〈μ_A (x) , v_A (x)〉. This approach is equivalent to the interval valued fuzzy sets interpretation of AIFS, where μ_A (x) and 1 − v_A (x) represent the lower bound and upper bound of membership degree, respectively. Obviously, [μ_A (x) , 1 − v_A (x)] is a valid interval, since μ_A (x) ≤1 − v_A (x) always holds for μ_A (x) + v_A (x) ≤1.

Interestingly, in evidence theory, Bel (x) and Pl (x) are the lower limit and upper limits of the belief level of hypothesis x, respectively. The belief value of hypothesis x is regarded as the minimal uncertainty about x, and its plausibility value is regarded as the maximal uncertainty about x. The relation between them is: $Pl (x) = 1 - Bel (\bar{x})$ (31) $Pl (x) \geq Bel (x)$ (32) where $\bar{x}$ is the classical complement of hypothesis x.

[Bel (x) , Pl (x)] is the confidence interval which describes the uncertainty about x, and Pl (x) − Bel (x) represents the level of ignorance about x, which is identical to the hesitancy degree. If the difference between Bel (x) and Pl (x) increases, then the information quality decreases or becomes unreliable. This difference provides a measurement of uncertainty about the level of belief in decision making problem.

Obviously, the membership interval [μ_A (x) , 1 − v_A (x)] is equivalent to the confidence interval [Bel (x) , Pl (x)]. This correspondence leads to the possibility to redefine AIFSs in terms of BPAs in evidence theory.

For an one-element AIFS A ={ 〈 x, μ_A (x) , v_A (x) 〉 } on X = {x}, μ_A (x) and v_A (x) are the membership degree and non-membership degree of x belong to A. Following this original interpretation of AIFS, μ_A (x) and μ_A (x) can be regarded as the answer of “Is x belong to A?”. The hypotheses frame of the answer is {{Yes} , {No} , {Yes, No}}. In the framework of evidence theory, we have m ({Yes}) = μ_A and m ({No}) = v_A. Since π_A is the hesitation degree, we can get m ({Yes, No}) = π_A. Therefore, AIFS A can elicit a dichotomous BPA as: ${\begin{matrix} m ({Yes}) = μ_{A} \\ m ({No}) = v_{A} \\ m ({Yes, No}) = π_{A} \end{matrix}$ (33)

Therefore, the following definition can be introduced.

Definition 12. Let X = {x₁, x₂, ⋯ , x_n} be the universe of discourse, A = {〈x_i, μ_A (x_i) , v_A (x_i) 〉|x_i ∈ X} be AIFS over X. Then A can be expressed by the following form: $A = {m_{x_{i}}^{A} | x_{i} \in X}$ , where $m_{x_{i}}^{A}$ is a dichotomous BPA defined in the discernment frame Θ = {Y, N} (shorten form of {Yes, No}) as: $\begin{matrix} \begin{array}{l} m_{x_{i}}^{A} ({Y}) = μ_{A} (x_{i}), \\ m_{x_{i}}^{A} ({N}) = v_{A} (x_{i}), \end{array} \\ \begin{array}{l} m_{x_{i}}^{A} ({Y, N}) = π_{A} (x_{i}) . \end{array} \end{matrix}$

Definition 12 shows that the use of semantics in evidence theory is possible to enhance the performance of AIFS. Particularly, this approach allows us to directly use the distance measures between BPAs to construct a distance measure for AIFSs.

4.2 Construct distance measure for AIFSs based on evidence theory

Jousselme and Maupin [2] have presented a survey on the main distance measures defined so far in the mathematical framework of evidence theory. It has been highlighted that Dempster’s conflict cannot be considered as a genuine dissimilarity measure between two belief functions. An alternative distance measure based on a cosine function is proposed in [5]. Other original results include the justification of the use of two-dimensional indexes as (cosine, distance) couples and a general formulation for this class of new indexes. Jousselme and Maupin have shown that most of the dissimilarity measures so far published are based on inner products, in some cases degenerated. Considering the wide application of Jousselme’s full metric distance measure d_J [1], we will first construct a distance measure for AIFSs based on d_J. Then a similarity measure will be defined on the basis of the cosine measure of the angle between two mass vectors.

Definition 13. [1] Let m₁ and m₂ be two BPAs on the same frame of discernment X = {x₁, x₂, ⋯ , x_n}, containing n mutually exclusive and exhaustive hypotheses. The distance between m₁ and m₂ is: $d_{J} (m_{1}, m_{2}) = \sqrt{\frac{1}{2} {(m_{1} - m_{2})}^{T} Jac (m_{1} - m_{2})}$ (34) where m is the column vector with coordinates m (x_i) in the space generated by all the non-empty subsets of X. The factor 1/2 is needed in Equation (34) to normalize d_J and to guarantee that 0 ≤ d_J ≤ 1. Jac is the Jaccard index matrix whose elements are Jaccard indexes: $J (A, B) = \frac{| A \cap B |}{| A \cup B |}, for A, B \in 2^{X} ∖ \emptyset$ (35)

For the properties of d_J, we have the following theorem.

Theorem 1.Letm₁andm₂be two BPAs on the same frame of discernmentX = {x₁, x₂, ⋯ , x_n}. T he distanced_Jis a full metric distance, satisfying the following conditions:

Nonnegativity: d_J (m₁, m₂) ≥0;

Symmetry: d_J (m₁, m₂) = d_J (m₂, m₁);

Definiteness: d_J (m₁, m₂) =0 ⇔ m₁ = m₂;

Triangle inequality: d_J (m₁, m₂) ≤ d_J (m₁, m₃) + d_J (m₂, m₃), ∀m₃defined inX.

Proof. Since d_J is essentially an Euclidean distance and the matrix Jac is positive definite (see [21]), the proof is trivial. □

Given two AIFSs A ={ 〈 x, μ_A (x) , v_A (x) 〉 } and B ={ 〈 x, μ_B (x) , v_B (x) 〉 } in X = {x}, we can express them in the form of BPA in the discernment frame Θ = {Y, N} as: $\begin{matrix} \begin{array}{l} m_{x}^{A} : m_{x}^{A} ({Y}) = μ_{A}, \\ m_{x}^{A} ({N}) = v_{A}, \end{array} \\ \begin{array}{l} m_{x}^{A} ({Y, N}) = π_{A}; \\ m_{x}^{B} : m_{x}^{B} ({Y}) = μ_{B}, \end{array} \\ \begin{array}{l} m_{x}^{B} ({N}) = v_{B}, \\ m_{x}^{B} ({Y, N}) = π_{B} . \end{array} \end{matrix}$

The space generated by Θ = {Y, N} is {{Y} , {N} , {Y, N}}. So we have BPA vectors as: $m^{A} = (μ_{A}, v_{A}, π_{A})^{T}, m^{B} = (μ_{B}, v_{B}, π_{B})^{T} .$

By Equation (35), we can also get the Jaccard matrix as: $Jac = (\begin{matrix} 1 & 0 & 0.5 \\ 0 & 1 & 0.5 \\ 0.5 & 0.5 & 1 \end{matrix}) .$

For convenience, we denote this special Jaccard matrix as Q.

By Definition 13, we can get the distance between m^A and m^B: $d_{J} (m_{x}^{A}, m_{x}^{B}) = \sqrt{\frac{1}{2} {(m_{x}^{A} - m_{x}^{B})}^{T} Q (m_{x}^{A} - m_{x}^{B})} .$

This distance can reflect the difference between two AIFSs, so we can apply it to define a distance measure between two AIFSs. Such analysis is applicable for two AIFSs in X = {x₁, x₂, ⋯ , x_n}. So we can define a distance measure for AIFSs in the framework of evidence theory.

Definition 14. Let X = {x₁, x₂, ⋯ , x_n} be the universe of discourse, A = {〈x_i, μ_A (x_i) , v_A (x_i) 〉|x_i ∈ X} and B ={ 〈 x_i, μ_B (x_i) , v_B (x_i) 〉 |x_i ∈ X } be two AIFSs over X. A and B can be expressed by dichotomous BPAs defined in the discernment frame Θ = {Y, N} as: $A = {m_{x_{i}}^{A} | x_{i} \in X}$ and $B = {m_{x_{i}}^{B} | x_{i} \in X}$ . Then the distance measure between them can be defined as:

$\begin{matrix} D_{q} (A, B) \\ = \frac{1}{n} \sum_{i = 1}^{n} d_{J} (m_{x_{i}}^{A}, m_{x_{i}}^{B}) \\ = \frac{1}{n} \sum_{i = 1}^{n} \sqrt{\frac{1}{2} {(m_{x_{i}}^{A} - m_{x_{i}}^{B})}^{T} Q (m_{x_{i}}^{A} - m_{x_{i}}^{B})} \end{matrix}$ (36)

Taking a close examine on the expression of D_q (A, B), we can get:

$\begin{matrix} {(m_{x_{i}}^{A} - m_{x_{i}}^{B})}^{T} Q (m_{x_{i}}^{A} - m_{x_{i}}^{B}) \\ = {(\begin{matrix} μ_{A} (x_{i}) - μ_{B} (x_{i}) \\ v_{A} (x_{i}) - v_{B} (x_{i}) \\ π_{A} (x_{i}) - π_{B} (x_{i}) \end{matrix})}^{T} (\begin{matrix} 1 & 0 & 0.5 \\ 0 & 1 & 0.5 \\ 0.5 & 0.5 & 1 \end{matrix}) \\ (\begin{matrix} μ_{A} (x_{i}) - μ_{B} (x_{i}) \\ v_{A} (x_{i}) - v_{B} (x_{i}) \\ π_{A} (x_{i}) - π_{B} (x_{i}) \end{matrix}) \\ = {(μ_{A} (x_{i}) - μ_{B} (x_{i}))}^{2} + {(v_{A} (x_{i}) - v_{B} (x_{i}))}^{2} \end{matrix}$ (37)

So we have: $\begin{matrix} D_{q} (A, B) \\ = \frac{1}{n} \sum_{i = 1}^{n} \sqrt{\frac{1}{2} {(μ_{A} (x_{i}) - μ_{B} (x_{i}))}^{2} + {(v_{A} (x_{i}) - v_{B} (x_{i}))}^{2}}, \end{matrix}$

which is a kind of Euclidean distance.

Theorem 2.D_q (A, B) is a distance measure between AIFSsAandBinX.

Proof. Such proof is trivial because D_q (A, B) is essentially a kind of Euclidean distance. □

Theorem 3.S_Dq = 1 − D_q (A, B) is a distance measure between AIFSsAandBinX.

Proof. It is trivial. □

Besides all properties in Definition 4, D_q (A, B) also satisfies the triangle inequality: D_q (A, B) ≤ D_q (A, C) + D_q (B, C), ∀A, B, C ∈ AIFS (X). However, it cannot discriminate some special cases in the application of pattern recognition, which is similar to the problem troubling similarity measure S_O (A, B) and Euclidean distance E (A, B). Although the inevitable problem that two different AIFSs B and C have the same similarity with A is not a big issue by itself, we should try our best to reduce such cases. So we need to construct another kind of similarity to assist the construction of a composite similarity measure for AIFSs. A novel similarity measure based on the cosine similarity between mass vectors may make sense. We start with the definition of cosine similarity between belief functions.

Definition 15. [2] Let m₁ and m₂ be two BPAs on the same frame of discernment X = {x₁, x₂, ⋯ , x_n}, containing n mutually exclusive and exhaustive hypotheses. The general formulation of cosine similarity between m₁ and m₂ is: ${Cos}_{W} (m_{1}, m_{2}) = \frac{m_{1}^{T} W m_{2}}{{∥ m_{1} ∥}_{W} \cdot {∥ m_{2} ∥}_{W}}$ (38) where W is the weighting matrix constructed by the similarity between all non-empty subsets of X, ∥ m ∥ _W is the module of m defined as: ${∥ m ∥}_{W} = \sqrt{m^{T} Wm}$ .

Since the Jaccard index matrix is positive definite, we can take it as the weighting matrix. In the discernment frame Θ = {Y, N}, Jar = Q , so a modified cosine similarity for AIFSs can be defined as:

Definition 16. Let X = {x₁, x₂, ⋯ , x_n} be the universe of discourse, A = {〈x_i, μ_A (x_i) , v_A (x_i) 〉|x_i ∈ X} and B ={ 〈 x_i, μ_B (x_i) , v_B (x_i) 〉 |x_i ∈ X } be two AIFSs over X. A and B can be expressed by dichotomous BPAs defined in the discernment frame Θ = {Y, N} as: $A = {m_{x_{i}}^{A} | x_{i} \in X}$ and $B = {m_{x_{i}}^{B} | x_{i} \in X}$ . Then the similarity measure between them can be defined as: $C_{q} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \frac{(m_{x_{i}}^{A})^{T} {Qm}_{x_{i}}^{B}}{{∥ m_{x_{i}}^{A} ∥}_{Q} \cdot {∥ m_{x_{i}}^{B} ∥}_{Q}}$ (39)

Theorem 4.C_q (A, B) is the similarity measure between AIFSsAandBinX.

Proof. Since bfitQ is positive definite, bfitQ can be uniquely factorized as bfitQ = bfitU^bfitTbfitU where bfitU is upper triangular with positive diagonal entries. So we have: $\begin{matrix} (m_{x_{i}}^{A})^{T} Q m_{x_{i}}^{B} = (m_{x_{i}}^{A})^{T} U^{T} {Um}_{x_{i}}^{B} = ({Um}_{x_{i}}^{A})^{T} ({Um}_{x_{i}}^{B}), \\ {∥ m_{x_{i}}^{A} ∥}_{Q} = \sqrt{(m_{x_{i}}^{A})^{T} U^{T} {Um}_{x_{i}}^{A}} = \sqrt{({Um}_{x_{i}}^{A})^{T} ({Um}_{x_{i}}^{A})}, \\ {∥ m_{x_{i}}^{B} ∥}_{Q} = \sqrt{(m_{x_{i}}^{B})^{T} U^{T} {Um}_{x_{i}}^{B}} = \sqrt{({Um}_{x_{i}}^{B})^{T} ({Um}_{x_{i}}^{B})} . \end{matrix}$

If we designate ${Um}_{x_{i}}^{A}$ and ${Um}_{x_{i}}^{B}$ as $y_{x_{i}}^{A}$ and $y_{x_{i}}^{B}$ , respectively, C_q (A, B) can be rewritten as:

$C_{q} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \frac{(y_{x_{i}}^{A})^{T} y_{x_{i}}^{B}}{∥ y_{x_{i}}^{A} ∥ \cdot ∥ y_{x_{i}}^{B} ∥}$ , with $∥ y_{x_{i}}^{A} ∥ = \sqrt{(y_{x_{i}}^{A})^{T} y_{x_{i}}^{A}}$ and $∥ y_{x_{i}}^{B} ∥ = \sqrt{(y_{x_{i}}^{B})^{T} y_{x_{i}}^{B}}$ .

Therefore, C_q (A, B) satisfies all conditions in Definition 5, it is a similarity measure between AIFSs A and B in X. □

If we consider the weights of x_i, the weighted similarity measures between AIFSs A and B can be proposed as follows: $\begin{matrix} S_{wDq} (A, B) \\ = 1 - \sum_{i = 1}^{n} w_{i} \sqrt{\frac{1}{2} {(m_{x_{i}}^{A} - m_{x_{i}}^{B})}^{T} Q (m_{x_{i}}^{A} - m_{x_{i}}^{B})} \end{matrix}$ (40) $C_{wq} (A, B) = \sum_{i = 1}^{n} \frac{w_{i} (m_{x_{i}}^{A})^{T} Q m_{x_{i}}^{B}}{{∥ m_{x_{i}}^{A} ∥}_{Q} \cdot {∥ m_{x_{i}}^{B} ∥}_{Q}}$ (41) where w_i ∈ [0, 1], i = 1, 2, ⋯ , n, and $\sum_{i = 1}^{n} w_{i} = 1$ .

It is effortlessly to prove that S_wDq (A, B) and C_wq (A, B) satisfy all properties in Definition 5.

From above analysis, we can note that D_q (A, B) measures how far a vector is from another, while C_q (A, B) measures the angle between two vectors. It is natural that neither of them can depict the relationship between vectors perfectly because of the existence of counter-intuitive cases. Since the combination of them can capture both aspects which may be helpful in reducing the counter-intuitive cases, we can define a composite similarity for AIFSs by combining D_q (A, B) and C_q (A, B) through an algebraic formula.

Definition 17. Let A and B be two AIFSs in X = {x₁, x₂, ⋯ , x_n}. A composite similarity measure between them can be defined as: $S_{q} (A, B) = \frac{1}{2} (C_{q} (A, B) + S_{Dq} (A, B))$ (42) where $S_{Dq} (A, B) = 1 - \frac{1}{n} \sum_{i = 1}^{n} \sqrt{\frac{1}{2} {(m_{x_{i}}^{A} - m_{x_{i}}^{B})}^{T} Q (m_{x_{i}}^{A} - m_{x_{i}}^{B})}$ , $C_{q} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \frac{(m_{x_{i}}^{A})^{T} Q m_{x_{i}}^{B}}{{∥ m_{x_{i}}^{A} ∥}_{Q} \cdot {∥ m_{x_{i}}^{B} ∥}_{Q}}$ .

Theorem 5.The measureS_q (A, B) is a similarity measure between AIFSsAandB.

Proof. Since C_q (A, B) and S_Dq (A, B) are similarity measures for between AIFSs A and B, it is easy to get the result. □

Analogously, when the weights of x_i are taken into consideration, a composite weighted similarity measure can be obtained as following: $S_{wq} (A, B) = \frac{1}{2} (C_{wq} (A, B) + S_{wDq} (A, B))$ (43)

Obviously, S_wq is an axiomatic similarity measure between two AIFSs.

In order to illustrate the performance of our proposed similarity measures, a comparison between our proposed similarity measures and some existing similarity measures is conducted based on the numerical cases in [9]. Table 1 presents a comprehensive comparison of the similarity measures for AIFS with counter-intuitive examples (p = 1 for S_HB, $S_{e}^{p}$ , $S_{s}^{p}$ , $S_{h}^{p}$ and p = 1 t = 2 for $S_{t}^{p}$ ).

We can see that S_C (A, B) = S_DC (A, B) = C_IFS (A, B) =1 for two different AIFSs A =<0.3, 0.3 > and B =<0.4, 0.4 >. This indicates that the second axiom of similarity measure (SP2) is not satisfied by S_C (A, B), S_DC (A, B) and C_IFS (A, B). This also can be illustrated by S_C (A, B) = S_DC (A, B) =1when A =<0.5, 0.5 >, B =<0, 0 > and A =<0.4, 0.2 >, B =<0.5, 0.3 >. We can also find that for S_H, S_O, S_HB, $S_{e}^{p}$ , $S_{s}^{p}$ , C_q and $S_{h}^{p}$ , different pairs of A, B may provide the identical results, which is not helpful for the application of pattern recognition. In fact, as analyzed earlier, such cases are inevitable. It is common that the similarity degrees between different pairs of objects are equal. Table 1 shows that $S_{h}^{p}$ , S_CC, C_q, and S_q can describe the similarity degree between AIFSs reasonably without providing cases with equal similarity degrees. However, when other cases are employed, some results may be identical. In Table 1, we can note that the cosine similarity C_IFS cannot make sense when B = {< x, 0, 0 >} is involved.

For Case 5 and Case 6 in Table 1, if we let A = {< x, 0.4, 0.2 >}, B = {< x, 0.5, 0.3 >}, and C = {< x, 0.5, 0.2 >}, we notice that the ranking order of S (A, B) and S (A, C) varies with respect to different similarity measures. From Definition 3 we have A ⊆ C and B ⊆ C, so it is not necessary that S (A, B) ≥ S (A, C) or S (A, B) ≤ S (A, C). Hence similarity measures cannot be evaluated by the ranking result of S (A, B) and S (A, C). The similarity degrees for two different pairs of AIFSs are also not relevant. We can note that the similarity measures C_q and S_q can avoid these so-called counter-intuitive cases, just as S_CC performs. However, we cannot say they are the best similarity measures, since they are not discriminative in some special cases. There is no doubt that the similarity measure for AIFSs should satisfy the axiomatic properties in Definition 5. Moreover, our perspective is that an ideal similarity measure should have specific physical meaning with relative simple expressions and less counter-intuitive cases. Based on such sense, we can claim that our proposed similarity measures S_Dq, C_q, and S_q are reasonable.

5 Applications of similarity measures

AIFSs have been widely applied to model and process imperfect information. In this section we will present some applications to study the performance of our proposed similarity measures.

Suppose there are m patterns, which can be represented by AIFSs A_j = {〈x_i, μ_{A
_j} (x_i) , v_{A
_j} (x_i) 〉|x_i ∈ X}, A_j ∈ AIFSs (X), j = 1, 2, ⋯ , m. The sample to be recognized is B ={ 〈 x_i, μ_B (x_i) , v_B (x_i) 〉 |x_i ∈ X }. According to the recognition principle of maximum similarity degree between AIFSs, the process of assigning B to A_k is described by [17]: $k = arg max_{j = 1, 2, \dots, m} {S (A_{j}, B)}$ (44)

Example 1. There are three patterns in X = {x₁, x₂, x₃}. They are denoted as follows: $\begin{matrix} A_{1} & = & {〈 x_{1}, 0.3, 0.3 〉, 〈 x_{2}, 0.2, 0.2 〉, 〈 x_{3}, 0.1, 0.1 〉}, \\ A_{2} & = & {〈 x_{1}, 0.2, 0.2 〉, 〈 x_{2}, 0.2, 0.2 〉, 〈 x_{3}, 0.2, 0.2 〉}, \\ A_{3} & = & {〈 x_{1}, 0.4, 0.4 〉, 〈 x_{2}, 0.4, 0.4 〉, 〈 x_{3}, 0.4, 0.4 〉} . \end{matrix}$

Assume that an unknown sample B = {〈x₁, 0.3, 0.3〉, 〈x₂, 0.2, 0.2〉, 〈x₃, 0.1, 0.1〉} is to be classified.

The similarity degrees of S (A₁, B), S (A₂, B) and S (A₃, B) calculated by all existing similarity measures are shown in Table 2.

It’s obvious that B is identical to A₁. So B should be classified to A₁ explicitly. However, the similarity degrees of S (A₁, B), S (A₂, B) and S (A₃, B) are all equal to 1 when S_C, S_H, S_DC and C_IFS are employed. These four similarity measures are not capable of discriminating the difference between three patterns. We can notice that S_Dq (A_i, B), C_q (A_i, B), and S_q (A_i, B) (i = 1, 2, 3) are competent to depict the similarity between different patterns, which is helpful for pattern classification. Moreover, this means that each of our proposed similarity measures performs well in the application of pattern recognition.

Example 2. Consider three patterns represented by three AIFSs A₁, A₂ and A₃ in X = {x₁, x₂, x₃}. They are shown as following: $\begin{matrix} A_{1} & = & {〈 x_{1}, 0.1, 0.1 〉, 〈 x_{2}, 0.5, 0.1 〉, 〈 x_{3}, 0.1, 0.9 〉}, \\ A_{2} & = & {〈 x_{1}, 0.5, 0.5 〉, 〈 x_{2}, 0.7, 0.3 〉, 〈 x_{3}, 0, 0.8 〉}, \\ A_{3} & = & {〈 x_{1}, 0.7, 0.2 〉, 〈 x_{2}, 0.1, 0.8 〉, 〈 x_{3}, 0.4, 0.4 〉} . \end{matrix}$

An unknown pattern B needs to be classified to one of three patterns A₁, A₂ and A₃. B is defined by an AIFS in X = {x₁, x₂, x₃} as: $B = {〈 x_{1}, 0.4, 0.4 〉, 〈 x_{2}, 0.6, 0.2 〉, 〈 x_{3}, 0, 0.8 〉} .$

Assume that w_i = 1/3 is the weight of element x_i, i = 1, 2, 3. The similarity degrees of S (A₁, B), S (A₂, B) and S (A₃, B) calculated by all existing similarity measures and our proposed measures are shown in Table 3. We can see that the similarity measure S_C and S_DC cannot provide reasonable classification result. The ranking order of S (A₁, B), S (A₂, B) and S (A₃, B) derived from our proposed similarity measures is identical to that from other reasonable measures. So the unknown pattern B can be classified to pattern A₂ based on our proposed measures, which is consistent with the result shown in [32].

Along with the previous investigation of classification capabilities of the proposed measure, two additional practical applications will be also analyzed.

Example 3. Suppose that there are four patients Al, Bob, Joe, Ted. The set of patients are P = {Al, Bob, Joe, Ted}. Their symptoms are Temperature, Headache, Stomach pain, Cough, Chest pain, represented as S = {Temperature, Headache, Stomach pain, Cough, Chest pain}. The set of diagnoses is defined as D = {Viral fever, Malaria, Typhoid, Stomach problem, Chest problem}. The intuitionistic fuzzy relation P → S is presented in Table 7. Table 5 gives the intuitionistic fuzzy relation S → D. Each element of the tables is given as intuitionistic fuzzy numbers.

In order to make a proper diagnosis for each patient, we calculate the similarity degree between each patient and each diagnose. According to the principle of maximum similarity degree, the higher similarity degree indicates a proper diagnosis. In Tables 6 –8, similarity grades between patients and diagnoses are presented based on our proposed similarity measures. According to the similarity degrees in Tables 6 –8, we can conclude that Al suffers from Malaria, Bob suffers from Stomach problem, Joe suffers from Typhoid, and Ted suffers from Viral Fever. The diagnosis results for this case obtained in previous study have been discussed in [9]. It is clear that our diagnosis results are completely identical to those obtained by Own [3] and Wei et al. [4]. Compared with the results derives by Vlachos in [16] and Boran in [9], the diagnoses for Bob, Joe and Ted are the same, but the diagnosis for Al is different. This situation illustrates that it is very hard to diagnose whether Al suffers from Viral, Fever, or Malaria, because these two symptoms are involved with each other. Moreover, our proposed similarity measures are calculated based on the belief functions, without determining any other parameters. So it can reduce the computation complexity effectively.

Example 4. Suppose that there is an investment company to invest a sum of money in the best option. Assume that there is a panel with five alternatives to invest the money: a car company M₁, a food company M₂, a computer company M₃, an arms company M₄ and a TV company M₅. The investment company needs to take a decision according to the following four attributes: (1) G₁ is the risk analysis; (2) G₂ is the growth analysis; (3) G₃ is the social-political impact analysis; (4) G₄ is the environmental impact analysis. Let w = (0.20, 0.18, 0.32, 0.30) be the attribute weight vector calculated by the method proposed in [13]. The five possible alternatives M_i (i = 1, 2, ⋯ , 5) are to be evaluated by decision maker under above four attributes in the following form: $\begin{matrix} M_{1} & = & {〈 G_{1}, 0.5, 0.4 〉, 〈 G_{2}, 0.6, 0.3 〉, \\ 〈 G_{3}, 0.3, 0.6 〉, 〈 G_{4}, 0.2, 0.7 〉}, \\ M_{2} & = & {〈 G_{1}, 0.7, 0.3 〉, 〈 G_{2}, 0.7, 0.2 〉, \\ 〈 G_{3}, 0.7, 0.2 〉, 〈 G_{4}, 0.4, 0.5 〉}, \\ M_{3} & = & {〈 G_{1}, 0.6, 0.4 〉, 〈 G_{2}, 0.5, 0.4 〉, \\ 〈 G_{3}, 0.5, 0.3 〉, 〈 G_{4}, 0.6, 0.3 〉}, \\ M_{4} & = & {〈 G_{1}, 0.8, 0.1 〉, 〈 G_{2}, 0.6, 0.3 〉, \\ 〈 G_{3}, 0.3, 0.4 〉, 〈 G_{4}, 0.2, 0.6 〉}, \\ M_{5} & = & {〈 G_{1}, 0.6, 0.2 〉, 〈 G_{2}, 0.4, 0.3 〉, \\ 〈 G_{3}, 0.7, 0.1 〉, 〈 G_{4}, 0.5, 0.3 〉} . \end{matrix}$

In [13], the positive-ideal solution M⁺ and negative-ideal solution M^- are defined as: $M^{+} = {〈 G_{i}, μ_{i}^{+}, v_{i}^{+} 〉 | i = 1, 2, 3, 4}$ (45) $M^{-} = {〈 G_{i}, μ_{i}^{-}, v_{i}^{-} 〉 | i = 1, 2, 3, 4}$ (46) where $μ_{i}^{+} = max_{j = 1, \dots, 5} {μ_{M_{j}} (G_{i})}$ , $v_{i}^{+} = min_{j = 1, \dots, 5} {μ_{M_{j}} (G_{i})}$ , $\begin{matrix} μ_{i}^{-} & = & min_{j = 1, \dots, 5} {μ_{M_{j}} (G_{i})}, \\ v_{i}^{-} & = & max_{j = 1, \dots, 5} {μ_{M_{j}} (G_{i})}, i = 1, 2, 3, 4 . \end{matrix}$

So we can get: $\begin{matrix} M^{+} & = & {〈 G_{1}, 0.8, 0.1 〉, 〈 G_{2}, 0.7, 0.2 〉, \\ 〈 G_{3}, 0.7, 0.1 〉, 〈 G_{3}, 0.6, 0.3 〉}, \\ M^{-} & = & {〈 G_{1}, 0.5, 0.4 〉, 〈 G_{2}, 0.4, 0.4 〉, \\ 〈 G_{3}, 0.3, 0.6 〉, 〈 G_{3}, 0.2, 0.7 〉} . \end{matrix}$

The similarity grades of S (M_i, M⁺) and S (M_i, M⁻) are calculated with respect to our proposed similarity measures. The results are listed in Table 9.

The relative similarity measure S (M_i) of M_i with respect to M⁺ and M⁻ can be obtained as: $S (M_{i}) = \frac{S (M_{i}, M^{+})}{S (M_{i}, M^{+}) + S (M_{i}, M^{-})}$ (47)

So we can get the relative similarity grades of M_i calculated based on our proposed similarity measures. They are presented in Table 10. We can see that the results derived from three similarity measures all show that M₅ ≻ M₂ ≻ M₃ ≻ M₄ ≻ M₁, which is identical to the ranking order shown in [4, 13]. Thus M₅ is the most desirable alternative.

6 Conclusions

At present, many similarity measures for AIFSs have been applied to the problems based on intuitionistic fuzzy information, but many of them are troubled by counter-intuitive results. For similarity measures satisfying all axiomatic properties, their counter-intuitive results come from the lack of ability to discriminate some special AIFSs. Although counter-intuitive cases are inevitable for single-valued similarity measures, we should try to avoid it when defining similarity measures. Since the choice of similarity measure should always be guided by practical considerations relative to a specific application, more reasonable similarity measures with more explicit physical meaning and less computational burden are desirable.

In this paper, we have defined three similarity measures for AIFSs based on the connection between AIFSs and evidence theory. By transforming AIFSs to BPAs, we first proposed two similarity measure based on Jousselme’s distance measure and cosine similarity measure, respectively. After analyzing their properties, a composite similarity measure is defined. The properties of these three similarity measures are mathematically analyzed. Numerical examples are employed to compare our proposed similarity measures with other measures. It has been demonstrated that the proposed similarity measures are capable of differentiating some special AIFSs. For pattern recognition problems in Examples 1–4, the results are identical to some of those of other studies. For Example 5, compatible results have been achieved compared to the results of previous studies. In the application of multiple criteria decision making in Example 6, our proposed similarity measures can provide the most desirable alternative based on the ranking order of all alternatives. In the light of this study, we will extend the application of our proposed similarity measures to more real applications in the future research.

Footnotes

Acknowledgments

This work is supported by the Natural Science Foundation of China under Grants No. 61273275, No. 60975026, No. 61573375 and No. 61503407.

References

Jousselme

A.-L.

, Grenier

, and Bossé

, A new distance between two bodies of evidence, Information Fusion2 (2001), 91–101.

Jousselme

A.-L.

, and Maupin

, Distances in evidence theory: Comprehensive survey and generalizations, International Journal of Approximate Reasoning53 (2012), 118–145.

Own

C.M.

, Switching between type-2 fuzzy sets and intuitionistic fuzzy sets: An application in medical diagnosis, Applied Intelligence31 (2009), 283–291.

Wei

C.P.

, Wang

, and Zhang

Y.Z.

, Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications, Information Sciences181 (2011), 4273–4286.

Wen

, Wang

, and Xu

, Fuzzy information fusion algorithm of fault diagnosis based on similarity measure of evidence, in: Advances in Neural Networks, Lecture Notes in Computer Sciencevol. 5264, Springer, Berlin/ Heidelberg, 2008, pp. 506–515.

D.F.

, and Cheng

C.T.

, New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions, Pattern Recognition Letters23 (2002), 221–225.

Hong

D.H.

, and Kim

, A note on similarity measures between vague sets and between elements, Information Sciences115 (1999), 83–96.

Szmidt

, and Kacprzyk

, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems114 (2000), 505–518.

Boran

F.E.

, and Akay

, A biparametric similarity measure on intuitionistic fuzzy sets with applications to pattern recognition, Information Sciences255 (2014), 45–57.

10.

, and Xu

, Similarity measures between vague sets, Journal of Software12 (2001), 922–927.

11.

Beliakov

, Pagola

, and Wilkin

, Vector valued similarity measures for Atanassov’s intuitionistic fuzzy sets, Information Sciences280 (2014), 352–367.

12.

Shafer

, A Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ, 1976.

13.

Wei

G.W.

, Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting, Knowledge-Based Systems21 (2008), 833–836.

14.

Bustince

, and Burillo

, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems79 (1996), 403–405.

15.

Mitchell

H.B.

, On the Dengfeng–Chuntian similarity measure and its application to pattern recognition, Pattern Recognition Letters24 (2003), 3101–3104.

16.

Vlachos

I.K.

, and Sergiadis

G.D.

, Intuitionistic fuzzy information-application to pattern recognition, Pattern Recognition Letters28 (2007), 197–206.

17.

, Cosine similarity measures for intuitionistic fuzzy sets and their applications, Mathematical and Computer Modelling53 (2011), 91–97.

18.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20 (1986), 87–96.

19.

Zadeh

L.A.

, Fuzzy sets, Information Control8 (1965), 338–353.

20.

Baccour

, Alimi

, and John

R.I.

, Similarity measures for intuitionistic fuzzy sets: State of the art, Journal of Intelligent & Fuzzy Systems24 (2013), 37–49.

21.

Bouchard

, Jousselme

A.-L.

, and Doré

P.-E.

, A proof for the positive definiteness of the Jaccard index matrix, International Journal of Approximate Reasoning54 (2013), 615–626.

22.

Smets

, Analyzing the combination of conflicting belief functions, Information Fusion8 (2007), 387–412.

23.

Chen

S.M.

, Measures of similarity between vague sets, Fuzzy Sets and Systems74 (1995), 217–223.

24.

Chen

S.M.

, and Chang

C.H.

, A novel similarity measure between Atanassov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition, Information Sciences291 (2015), 96–114.

25.

Hung

W.L.

, and Yang

M.S.

, Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance, Pattern Recognition Letters25 (2004), 1603–1611.

26.

Wang

, Zhu

, Song

, and Lei

, Combination of unreliable evidence sources in intuitionistic fuzzy MCDM framework, Knowledge-Based Systems97 (2016), 24–39.

27.

, Chi

, and Yan

, Similarity measures between vague sets and vague entropy, Journal of Computer Science29 (2002), 129–132.

28.

Song

, Wang

, Lei

, and Xue

, Combination of interval-valued belief structures based on intuitionistic fuzzy set, Knowledge-Based Systems67 (2014), 61–70.

29.

Liu

Z.-g.

, Dezert

, Pan

, and Mercier

, Combination of sources of evidence with different discounting factors based on a new dissimilarity measure, Decision Support Systems52 (2011), 133–141.

30.

Liu

Z.-g.

, Pan

, Dezert

, and Martin

, Adaptive imputation of missing values for incomplete pattern recognition, Pattern Recognition52 (2016), 85–95.

31.

Liu

Z.-g.

, Pan

, Dezert

, and Mercier

, Credal c-means clustering method based on belief functions, Knowledge-Based Systems74 (2015), 119–132.

32.

Liang

Z.Z.

, and Shi

P.F.

, Similarity measures on intuitionistic fuzzy sets, Pattern Recognition Letters24 (2003), 2687–2693.