Similarity measures of picture fuzzy sets and their applications to pattern recognition 1

1 Introduction

Zadeh [15] proposed fuzzy sets, which can represent information with fuzziness. Later, Atanassov [1] proposed intuitionistic fuzzy sets. It can represent more types of information, such as fuzziness and uncertainty than fuzzy sets. Cuong [2] proposed picture fuzzy sets (PFSs), which brought a new direction for representing different types of information, such as fuzziness, uncertainty and inconsistency. Since PFSs are more outstanding in representing information, many studies based on PFSs have been generated ([4, 13]).

Pattern recognition is a problem of processing and classifying information, therefore it has a wide range of applications in real life, for example the selection of candidates for companies, the selection of building materials, etc. The representation of information can be done with the help of PFSs, while other tools are needed to process information. Similarity measures can quantify the degree of similarity between different sets, and it has never ceased to be studied on PFSs. Gupta and Kumar ([4]) proposed two similarity metrics using fractional functions which are used to solve pattern recognition and cluster analysis. Khan et al ([6]) proposed similarity measures with parameters and applied them to pattern recognition and medical diagnosis. Li et al ([7]) proposed a similarity measure that combines Hamming distance and transformed tetrahedral centroid distance and applied it to pattern recognition. Verma and Rohtag ([13]) proposed Chi-square and Canberra picture fuzzy similarity measures and then applied them to pattern recognition and medical diagnosis.

However, when applying these similarity measures, we find that in some cases some existing similarity measures are not reasonable, which is an obstacle to solving practical problems. Secondly, similarity measures based on exponential functions have not been fully investigated. Considering the above two aspects, we propose two exponential function-based similarity measures, and they can deal with some problems that cannot be solved by some existing similarity measures in practical applications, or even better.

The rest of this paper is organized as follows: in Section 2, we review the basic definitions and properties for PFSs. In Section 3, we give two new similarity measures based on exponential function between PFSs. In Section 4, we apply the proposed similarity measures to pattern recognition. The conclusions are given in Section 5.

2 Preliminaries

In this section, we review some common concepts of picture fuzzy sets, which will be used in this paper.

Definition 2.1 ([2]) A picture fuzzy set G on universe K = { γ 1 , γ 2 , . . . , γ n } is an object of the form: G = { 〈 τ G ( γ i ) , σ G ( γ i ) , θ G ( γ i ) 〉 ∣ γ i ∈ K } , where τ _G  (γ _i ): K → [ 0 , 1 ] is called the membership degree, σ _G  (γ _i ): K → [ 0 , 1 ] is called the neutral membership degree, θ _G  (γ _i ): K → [ 0 , 1 ] is called the non-membership degree, and τ _G  (γ _i ), σ _G  (γ _i ), θ _G  (γ _i ) satisfy the condition: 0 ≤ τ _G  (γ _i ) + σ _G  (γ _i ) + θ _G  (γ _i ) ≤1, ∀ γ i ∈ K . φ _G  (γ _i ) =1 - (τ _G  (γ _i ) + σ _G  (γ _i ) + θ _G  (γ _i )) is called the degree of refusal membership. All picture fuzzy sets on universe K are denoted by PFSs ( K ) .

Definition 2.2 ([2]) Let G = { 〈 γ i , τ G ( γ i ) , σ G ( γ i ) , θ G ( γ i ) 〉 ∣ γ i ∈ K } and H = { 〈 γ i , τ H ( γ i ) , σ H ( γ i ) , θ H ( γ i ) 〉 ∣ γ i ∈ K } be two PFSs on universe K = { γ 1 , γ 2 , . . . , γ n } , the operations between G and H defined as follows:

(1) G ⊆ H iff τ _G  (γ _i ) ≤ τ _H  (γ _i ), σ _G  (γ _i ) ≤ σ _H  (γ _i ) and θ _G  (γ _i ) ≥ θ _H  (γ _i );

(2) G = H iff G ⊆ H and H ⊆ G;

Let D^* = {〈τ, σ, θ〉 ∈ [0, 1] ³, τ + σ + θ ≤ 1} is the set of picture fuzzy numbers. Let φ = 〈τ _φ , σ _φ , θ _φ 〉 and ψ = 〈τ _ψ , σ _ψ , θ _ψ 〉 ∈D^*, the ordering ≤ _{D ^*} is defined as φ ≤  _{D ^*} ψ iff τ _φ  ≤ τ _ψ , and σ _φ  ≤ σ _ψ , and θ _φ  ≥ θ _ψ ([2]).

Definition 2.3 ([11]) A function S : PFS ( K ) × PFS ( K ) → [ 0 , 1 ] with G , H ∈ PFS ( K ) is called a similarity measure between PFSs, if it satisfies:

(1) 0 ≤ S (G, H) ≤1;

(2) S (G, H) = S (H, G);

(3) S (G, H) =1 if and only if G = H;

(4) If G ⊆ H ⊆ J, then S (G, J) ≤ S (G, H) and S (G, J) ≤ S (H, J).

Let G = { 〈 γ i , τ G ( γ i ) , σ G ( γ i ) , θ G ( γ i ) 〉 ∣ γ i ∈ K } and H = { 〈 γ i , τ H ( γ i ) , σ H ( γ i ) , θ H ( γ i ) 〉 ∣ γ i ∈ K } be two PFSs on universe K = { γ 1 , γ 2 , . . . , γ n } , some similarity measures between G and H are as follows:

Dinh and Thao’s similarity measures ([2]): S d 1 ( G , H ) = 1 - 1 n ∑ i = 1 n max ( | τ G ( γ i ) - τ H ( γ i ) | , | σ G ( γ i ) - σ H ( γ i ) | , | θ G ( γ i ) - θ H ( γ i ) | )(1)

S d 2 ( G , H ) = 1 - { 1 n ∑ i = 1 n max ( | τ G ( γ i ) - τ H ( γ i ) | 2 , | σ G ( γ i ) - σ H ( γ i ) | 2 , | θ G ( γ i ) - θ H ( γ i ) | 2 ) } 1 2(2)

Liu and Zeng’s similarity measure ([8]): S z ( G , H ) = 1 - 1 3 { ∑ i = 1 n ω i ( | τ G ( γ i ) - τ H ( γ i ) | + | σ G ( γ i ) - σ H ( γ i ) | + | θ G ( γ i ) - θ H ( γ i ) | ) 2 } 1 2(3)

Thao’s similarity measure ([12]): S t ( G , H ) = 1 - 1 3 ∑ i = 1 n ω i [ | τ G ( γ i ) - τ H ( γ i ) | + | σ G ( γ i ) - σ H ( γ i ) | + | θ G ( γ i ) - θ H ( γ i ) | ](4)

Wei’s similarity measures ([14]) Equations (5), (6) and (7): S g 2 ( G , H ) = 1 n ∑ i = 1 n cos π 2 ( max ( | τ G ( γ i ) - μ H ( γ i ) | , | σ G ( γ i ) - σ H ( γ i ) | , | θ G ( γ i ) - θ H ( γ i ) | , | φ G ( γ i ) - φ H ( γ i ) | ) )(5) S g 3 ( G , H ) = 1 n ∑ i = 1 n cos π 4 ( | τ G ( γ i ) - τ H ( γ i ) | + | σ G ( γ i ) - σ H ( γ i ) | + | θ G ( γ i ) - θ H ( γ i ) | + | φ G ( γ i ) - φ H ( γ i ) | )(6)

S g 1 ( G , H ) = 1 n ∑ i = 1 n τ G ( γ i ) τ H ( γ i ) + σ G ( γ i ) σ H ( γ i ) + θ G ( γ i ) θ H ( γ i ) { τ G 2 ( γ i ) + σ G 2 ( γ i ) + θ G 2 ( γ i ) } 1 2 { τ H 2 ( γ i ) + σ H 2 ( γ i ) + θ H 2 ( γ i ) } 1 2(7)

Singh and Mishra’s similarity measures ([11]) Equations (8), (9), (10) and (11):

S p 4 ( G , H ) = 1 4 n ∑ i = 1 n min ( | τ G ( γ i ) - τ H ( γ i ) | , | σ G ( γ i ) - σ H ( γ i ) | , | θ G ( γ i ) - θ H ( γ i ) | , | Δ φ i | ) max ( | τ G ( γ i ) - μ H ( γ i ) | , | σ G ( γ i ) - σ H ( γ i ) | , | θ G ( γ i ) - θ H ( γ i ) | , | φ G ( γ i ) - φ H ( γ i ) | )(8)

S p 1 ( G , H ) = 1 - 1 4 n ∑ i = 1 n ( | τ G ( γ i ) - τ H ( γ i ) | + | σ G ( γ i ) - σ H ( γ i ) | + | θ G ( γ i ) - θ H ( γ i ) | + | φ G ( γ i ) - φ H ( γ i ) | )(9)

S p 2 ( G , H ) = 1 - 1 4 n ∑ i = 1 n max ( | τ G ( γ i ) - τ H ( γ i ) | , | σ G ( γ i ) - σ H ( γ i ) | , | θ G ( γ i ) - θ H ( γ i ) | , | φ G ( γ i ) - φ H ( γ i ) | )(10)

S p 3 ( G , H ) = 1 - { 1 4 n ∑ i = 1 n max ( τ G ( γ i ) - μ H ( γ i ) 2 , σ G ( γ i ) - σ H ( γ i ) 2 , θ G ( γ i ) - θ H ( γ i ) 2 , φ G ( γ i ) - φ H ( γ i ) 2 ) } 1 2(11)

In this section, based on exponential function, we propose two new similarity measures on PFSs.

Theorem 3.1. Let G = { ( τ G ( γ i ) , σ G ( γ i ) , θ G ( γ i ) ) ∣ γ i ∈ K } and H = { ( τ H ( γ i ) , σ H ( γ i ) , θ H ( γ i ) ) ∣ γ i ∈ K } be two PFSs on K = { γ 1 , γ 2 , . . . , γ n } , the function S 1 : PFS ( K ) × PFS ( K ) → [ 0 , 1 ] is defined as S₁ (G, H) eq(12).

S 1 ( G , H ) = ∑ i = 1 n ω i e - | τ G ( γ i ) - τ H ( γ i ) | 2 + | σ G ( γ i ) - σ H ( γ i ) | 2 - e - 2 1 - e - 2 + 2 - | σ G ( γ i ) - σ H ( γ i ) | 2 - | θ G ( γ i ) - θ H ( γ i ) | 2 3(12) Then S₁ (G, H) is a similarity measure on PFSs, where ∑ i = 1 n ω i = 1 (i = 1, 2, …, n).

Proof. Obviously, S₁ (G, H) satisfies the properties (1) , (2) of similarity measure. Now, we give the proof of the properties (3) , (4).

(3) If S₁ (G, H) =1, then e - | τ G ( γ i ) - τ H ( γ i ) | 2 + | σ G ( γ i ) - σ H ( γ i ) | 2 - e - 2 1 - e - 2 + 2 - | σ G ( γ i ) - σ H ( γ i ) | 2 - | θ G ( γ i ) - θ H ( γ i ) | 2 = 3 . Because 0 ≤ e - | τ G ( γ i ) - τ H ( γ i ) | 2 + | σ G ( γ i ) - σ H ( γ i ) | 2 - e - 2 1 - e - 2 ≤ 1 , 2 ≤ e - | τ G ( γ i ) - τ H ( γ i ) | 2 + | σ G ( γ i ) - σ H ( γ i ) | 2 - e - 2 1 - e - 2 + 2 ≤ 3 , 0 ≤ e - | τ G ( γ i ) - τ H ( γ i ) | 2 + | σ G ( γ i ) - σ H ( γ i ) | 2 - e - 2 1 - e - 2 + 2 - | σ G ( γ i ) - σ H ( γ i ) | 2 - | θ G ( γ i ) - θ H ( γ i ) | 2 = 3 , then |σ _G  (γ _i ) - σ _H  (γ _i ) |² + |θ _G  (γ _i ) - θ _H  (γ _i ) |² = 0, |σ _G  (γ _i ) - σ _H  (γ _i ) | = |θ _G  (γ _i ) - θ _H  (γ _i ) |=0, |τ _G  (γ _i ) - τ _H  (γ _i ) |² + |σ _G  (γ _i ) - σ _H  (γ _i ) |² = 0. So τ _G  (γ _i ) = τ _H  (γ _i ), σ _G  (γ _i ) = σ _H  (γ _i ), θ _G  (γ _i ) = θ _H  (γ _i ), (i = 1, 2, …, n).

If τ _G  (γ _i ) = τ _H  (γ _i ), σ _G  (γ _i ) = σ _H  (γ _i ), θ _G  (γ _i ) = θ _H  (γ _i ), (i = 1, 2, …, n), then S₁ (G, H) =1 is obvious.

(4) Let G, H, J be three PFSs. If G ⊆ H ⊆ J i.e. τ _G  ≤ τ _H  ≤ τ _J , σ _G  ≤ σ _H  ≤ σ _J , θ _G  ≥ θ _H  ≥ θ _J . Because |τ _G  (γ _i ) - τ _H  (γ _i ) | ≤ |τ _G  (γ _i ) - τ _J  (γ _i ) |, |σ _G  (γ _i ) - σ _H  (γ _i ) | ≤ |σ _G  (γ _i ) - σ _J  (γ _i ) |, |θ _G  (γ _i ) - θ _H  (γ _i ) | ≤ |θ _G  (γ _i ) - θ _J  (γ _i ) |, then - | τ G ( γ i ) - τ H ( γ i ) | 2 + | σ G ( γ i ) - σ H ( γ i ) | 2 ≥ - | τ G ( γ i ) - τ J ( γ i ) | 2 + | σ G ( γ i ) - σ J ( γ i ) | 2 , e - | τ G ( γ i ) - τ H ( γ i ) | 2 + | σ G ( γ i ) - σ H ( γ i ) | 2 - e - 2 1 - e - 2 ≥ e - | τ G ( γ i ) - τ J ( γ i ) | 2 + | σ G ( γ i ) - σ J ( γ i ) | 2 - e - 2 1 - e - 2 , - | σ G ( γ i ) - σ H ( γ i ) | 2 - | θ G ( γ i ) - θ H ( γ i ) | 2 ≥ - | σ G ( γ i ) - σ J ( γ i ) | 2 - | θ G ( γ i ) - θ J ( γ i ) | 2 , so S₁ (G, J) ≤ S₁ (G, H). Similarly, S₁ (G, J) ≤ S₁ (H, J).

Theorem 3.2. Let G = { ( τ G ( γ i ) , σ G ( γ i ) , θ G ( γ i ) ) ∣ γ i ∈ K } and H = { ( τ H ( γ i ) , σ H ( γ i ) , θ H ( γ i ) ) ∣ γ i ∈ K } be two PFSs on K = { γ 1 , γ 2 , . . . , γ n } , the function S 2 : PFS ( K ) × PFS ( K ) → [ 0 , 1 ] is defined as S 2 ( G , H ) = ∑ i = 1 n ω i e - A ( G , H ) 3 - e - 1 1 - e - 1(13) where A (G, H) = |τ _G  (γ _i ) - τ _H  (γ _i ) |² + |σ _G  (γ _i ) - σ _H  (γ _i ) |² + |θ _G  (γ _i ) - θ _H  (γ _i ) |². Then S₂ (G, H) is a similarity measure on PFSs, where ∑ i = 1 n ω i = 1 (i = 1, 2, …, n).

Proof.  (1) Because 0 ≤ | τ G ( γ i ) - τ H ( γ i ) | ≤ 1 , 0 ≤ | σ G ( γ i ) - σ H ( γ i ) | ≤ 1 , 0 ≤ | θ G ( γ i ) - θ H ( γ i ) | ≤ 1 , 0 ≤ | τ G ( γ i ) - τ H ( γ i ) | 2 + | σ G ( γ i ) - σ H ( γ i ) | 2 + | θ G ( γ i ) - θ H ( γ i ) | 2 ≤ 3 , 0 ≤ e - A ( G , H ) 3 - e - 1 1 - e - 1 ≤ 1 . Therefore, 0 ≤ S₂ (G, H) ≤1.

(2) Obviously.

(3) If S₂ (G, H) =1, i.e., e - A ( G , H ) 3 = 1 , then - A ( G , H ) 3 = 0 . Because A (G, H) ≥0, therefore, τ _G  (γ _i ) = τ _H  (γ _i ) , σ _G  (γ _i ) = σ _H  (γ _i ) , θ _G  (γ _i ) = θ _H  (γ _i ) , (i = 1, 2, …, n) .

If τ _G  (γ _i ) = τ _H  (γ _i ), σ _G  (γ _i ) = σ _H  (γ _i ), θ _G  (γ _i ) = θ _H  (γ _i ), (i = 1, 2, …, n), S₂ (G, H) =1 is obvious.

(4) Let G, H, J be three PFSs. If G ⊆ H ⊆ J i.e. τ _G  ≤ τ _H  ≤ τ _J , σ _G  ≤ σ _H  ≤ σ _J , θ _G  ≥ θ _H  ≥ θ _J . Because |τ _G  (γ _i ) - τ _H  (γ _i ) | ≤ |τ _G  (γ _i ) - τ _J  (γ _i ) |, |σ _G  (γ _i ) - σ _H  (γ _i ) | ≤ |σ _G  (γ _i ) - σ _J  (γ _i ) |, |θ _G  (γ _i ) - θ _H  (γ _i ) | ≤ |θ _G  (γ _i ) - θ _J  (γ _i ) |, then, − A ( G , H ) 3 ≥ − A ( G , J ) 3 , e - A ( G , H ) 3 - e - 1 1 - e - 1 ≥ e - A ( G , J ) 3 - e - 1 1 - e - 1 . where A (G, J) = |τ _G  (γ _i ) - τ _J  (γ _i ) |² + |σ _G  (γ _i ) - σ _J  (γ _i ) |² + |θ _G  (γ _i ) - θ _J  (γ _i ) |². Therefore, S₂ (G, J) ≤ S₂ (G, H). Similarly, S₂ (G, J) ≤ S₂ (H, J).

In this section, we first give a numerical example to illustrate the rationale for new similarity measures. Then, new similarity measures are used to solve pattern recognition problems. Finally, we compare the results of new similarity measures with some existing similarity measures to illustrate the validity of new similarity measures.

Example 4.1 G _i , H _i (i = 1, 2, 3, 4) are different PFSs. We use the proposed similarity measures S₁, S₂ to calculate and the results are shown in Table 1.

According to Table 1, we find that G₁ = G₂, H₁ ≠ H₂. The similarity measures S (G₁, H₁) and S (G₂, H₂) are not supposed to be equal. However S _t , S _z , S _{g 3} , S _{p 1} obtain that the similarity measures between G₁, H₁ and G₂, H₂ are equal, this is unreasonable; S _{d 1} , S _{d 2} , S _{g 1} , S _{g 2} , S _{p 2} , S _{p 3} have a similar case: the similarity measures between G₁, H₁ are higher than G₂, H₂. Similarly, G₃ = G₄, H₃ ≠ H₄, the similarity measures S (G₃, H₃) and S (G₄, H₄) are not supposed to be equal. However, S _{d 1} , S _{d 2} , S _{g 1} , S _{g 2} , S _{p 2} , S _{p 3} obtain that the similarity measures between G₃, H₃ and G₄, H₄ are equal, this is unreasonable; S _t , S _z , S _{g 3} , S _{p 1} have the same situation: the between G₄, H₄ are higher than G₃, H₃, S _{p 4} appears to have a denominator of 0, so the result cannot be calculated.

4.1 Algorithm and applications

4.1.1 Algorithm for pattern recognition

Let K = { γ 1 , γ 2 , . . . , γ n } be a finite universe of discourse, there are m patterns G k = { 〈 τ G k ( γ i ) , σ G k ( γ i ) , θ G k ( γ i ) 〉 ∣ γ i ∈ K } ( k = 1 , 2 , ⋯ , m ) and a unknown pattern H = { 〈 τ H ( γ i ) , σ H ( γ i ) , θ H ( γ i ) 〉 ∣ γ i ∈ K } . Which pattern does the sample H belongs to?

The recognition steps are as follows:

Step 1. Calculate the similarity measure S (G _k , H) (k = 1, 2, ⋯ , m) between G _k and H.

Step 2. Choose the maximum one S (G _{k 0} , H) from S (G _k , H) (k = 1, 2, ⋯ , m). Then the unknown pattern H is belongs to G _{k 0} according to the maximum principle of similarity measures.

Step 3. Calculate the degree of confidence Doc, Doc ( k 0 ) = ∑ k = 1 , k ≠ k 0 m | S ( G k 0 , H ) - S ( G k , H ) | ([5]). The Doc^(k₀) is larger, the result of similarity measure is more credible.

4.1.2 Applications for pattern recognition

Example 4.2 ([9]) There are three known patterns G₁, G₂, G₃ and an unknown pattern H, which are denoted by PFSs on the universe K = { γ 1 , γ 2 , . . . , γ n } as presented in Table 2. The issue is to assign an unknown pattern H to G₁, G₂, G₃.

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Table 2

Patterns and unknown pattern (Example 4.2)

	γ 1	γ 2	γ 3
G 1	(0.5, 0.1, 0.1)	(0.3, 0.1, 0.3)	(0.3, 0.1, 0.4)
G 2	(0.4, 0.3, 0.2)	(0.3, 0.2, 0.5)	(0.4, 0.1, 0.3)
G 3	(0.2, 0.4, 0.3)	(0.1, 0.4, 0.4)	(0.0, 0.0, 0.0)
H	(0.3, 0.2, 0.3)	(0.4, 0.1, 0.3)	(0.4, 0.2, 0.3)

To solve this problem, we calculate S (G₁, H), S (G₂, H), S (G₃, H) using the proposed similarity measures S₁, S₂. The results are shown in Table 3.

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Table 3

Consequences derived from different similarity measures for pattern recognition (Example 4.2)

Similarity measures	S (G1, H)	S (G2, H)	S (G3, H)	The consequences	Doc
S d 1 ([3])	0.867	0.867	0.700	\	no
S d 2 ([3])	0.859	0.859	0.689	\	no
S z ([8])	0.900	0.911	0.789	G 2	0.133
S p 1 ([11])	0.900	0.900	0.750	\	no
S p 2 ([11])	0.967	0.967	0.883	\	no
S p 3 ([11])	0.929	0.929	0.720	\	no
S p 4 ([11])	0.125	0.125	0.046	\	no
S t ([12])	0.886	0.902	0.773	G 2	0.145
S g 1 ([14])	0.919	0.952	NAN	\	no
S g 2 ([14])	0.975	0.975	0.666	\	no
S g 3 ([14])	0.943	0.943	0.639	\	no
The proposed
S 1	0.929	0.938	0.837	G 2	0.110
S 2	0.836	0.851	0.678	G 2	0.188

In Table 3, we find S _{d 1} , S _{d 2} , S _{g 1} , S _{g 2} , S _{g 3} , S _{p 1} , S _{p 2} , S _{p 3} , S _{p 4} have a similar case: S (G₁, H) = S (G₂, H); S _t , S _z classify pattern H into pattern G₂, the results of proposed similarity measures S₁, S₂ are same. The Doc results comparison about S₁, S₂ and some existing similarity measures are shown in Fig. 1. We could notice that the Doc of S₁, S₂ are higher than most other similarity measures. Therefore, the proposed similarity measures S₁, S₂ are effective.

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Fig. 1

The Doc results comparison about S₁, S₂ and some existing similarity measures (Example 4.2).

Example 4.3. There are three known patterns Q₁, Q₂, Q₃ and an unknown pattern J, which are denoted by PFSs on the universe K = { γ 1 , γ 2 , . . . , γ n } as presented in Table 4. The issue is to assign an unknown pattern J to Q₁, Q₂, Q₃.

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Table 4

Patterns and unknown pattern (Example 4.3)

	γ 1	γ 2	γ 3	γ 4	γ 5
Q 1	(0.4, 0.4, 0.2)	(0.0, 0.6, 0.3)	(0.3, 0.2, 0.1)	(0.7, 0.0, 0.1)	(0.5, 0.1, 0.1)
Q 2	(0.3, 0.0, 0.5)	(0.3, 0.7, 0.0)	(0.2, 0.4, 0.2)	(0.5, 0.1, 0.0)	(0.6, 0.1, 0.1)
Q 3	(0.6, 0.1, 0.0)	(0.7, 0.2, 0.0)	(0.3, 0.1, 0.3)	(0.4, 0.2, 0.1)	(0.8, 0.0, 0.1)
J	(0.4, 0.2, 0.1)	(0.5, 0.2, 0.3)	(0.3, 0.3, 0.2)	(0.3, 0.2, 0.1)	(0.8, 0.0, 0.1)

To solve this problem, we calculate S (Q₁, J), S (Q₂, J), S (Q₃, J) using the proposed similarity measures S₁, S₂. The results are shown in Table 5.

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Table 5

Consequences derived from different similarity measures for pattern recognition (Example 4.3)

Similarity measures	S (Q1, J)	S (Q2, J)	S (Q3, J)	The consequences	Doc
S d 1 ([3])	0.700	0.720	0.820	Q 3	0.220
S d 2 ([3])	0.688	0.694	0.810	Q 3	0.238
S z ([8])	0.840	0.827	0.913	Q 3	0.159
S p 1 ([11])	0.830	0.860	0.920	Q 3	0.150
S p 2 ([11])	0.915	0.930	0.960	Q 3	0.075
S p 3 ([11])	0.823	0.842	0.905	Q 3	0.145
S p 4 ([11])	0.000	0.012	NAN	\	no
S t ([12])	0.820	0.801	0.894	Q 3	0.167
S g 1 ([14])	0.846	0.818	0.935	Q 3	0.206
S g 2 ([14])	0.850	0.881	0.956	Q 3	0.181
S g 3 ([14])	0.850	0.881	0.956	Q 3	0.181
The proposed
S 1	0.860	0.860	0.931	Q 3	0.142
S 2	0.714	0.727	0.843	Q 3	0.245

In Table 5, we find S _{d 1} , S _{d 2} , S _t , S _z , S _{g 1} , S _{g 2} , S _{g 3} , S _{p 1} , S _{p 2} , S _{p 3} , S _{p 4} classify the sample J into pattern Q₃, the results of proposed similarity measures S₁, S₂ are same. The Doc results comparison about S₁, S₂ and some existing similarity measures are shown in Fig. 2. We could notice that the Doc of S₁, S₂ are higher than most other similarity measures. Therefore, the proposed similarity measures are effective.

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Fig. 2

The Doc results comparison about S₁, S₂ and some existing similarity measures (Example 4.3).

First, we give a numerical example, and it shows that the new similarity measures S₁ and S₂ give the accurate results in the face of small gaps between PFSs, overcoming shortcoming of some existing similarity measures. Secondly, two applications of pattern recognition illustrate that new similarity measures S₁ and S₂ can still compute reasonable results in the face of more complex situations. By comparing the degree of confidence with some existing similarity measures, the similarity measure S₂ has the higher degree of confidence than other similarity masures in pattern recognition example 4.2 and 4.3. Although the new similarity measure S₁ has lower degree of confidence than similarity measure S₂, they have the same pattern recognition results. The results of pattern recognition show that new similarity measures S₁ and S₂ can give more accurate computational results, and therefore the proposed new similarity measures are reasonable andvalid.

6 Conclusions

Recently, the study of similarity measures on PFSs has received a lot of attention, and different types of similarity measures have been proposed in large numbers. After research, we find that there is still room for research on similarity measures based on exponential functions, so we proposed two new similarity measures based on exponential functions. Then new similarity measures are used to solve the problem of pattern recognition. The results show that the proposed new similarity measures are reasonable and effective. In the next work, we will consider parameterizing new similarity measures, and construct a formula by introducing other functions to include some existing exponential similarity measures, so as to solve practical problems more effectively.