A novel similarity degree of intuitionistic fuzzy sets induced by triangular norm and its application in pattern recognition

1. Introduction

Intuitionistic fuzzy sets originally proposed by Atanassov [1], which is a generalization of fuzzy sets [31]. It makes descriptions of the objective world become more realistic and accurate. It is a suitable tool for modelling the hesitancy arising from the imprecise information. This hesitation is due to the lack of knowledge or personal error in defining the membership function. Intuitionistic fuzzy sets defined by two characteristic functions, namely the membership function and the non-membership function, describing the belongingness or non-belongingness of an element respectively. Intuitionistic fuzzy sets have gained much attention from past and latter researchers for applications in various fields such as decision-making problems [2, 27], medical diagnostics [4, 30] and pattern recognition [3, 23]. It is indispensable to measure the similarity between intuitionistic fuzzy sets in the process of pattern recognition. A similarity measure is used for estimating the degree of similarity between two sets. Since Zadeh [31] proposed fuzzy sets, many scholars have conducted research on similarity measures between fuzzy sets from all kinds of viewpoints. Meanwhile, some scholars have proposed many varieties of similarity measure of intuitionistic fuzzy sets and given some computation formula [13, 34]. Li and Cheng [13] discussed some similarity measures on intuitionistic fuzzy sets and then proposed a suitable similarity measure between intuitionistic fuzzy sets which is the first to be applied to pattern recognition problems. Later, Liang and Shi [14] proposed several similarity measures and discussed the relationships among these measures. Furthermore, Mitchell [17] interpreted intuitionistic fuzzy sets as ensembles of ordered fuzzy sets from a statistical viewpoint to modify Li and Cheng’s measures. On the other hand, Hung and Yang [9] proposed another method to calculate the distance between intuitionistic fuzzy sets based on the Hausdorff distance. Then they used this distance to generate several similarity measures between intuitionistic fuzzy sets that are suited to be used in linguistic variables. Xu et al. [28] defined the concepts of association matrix and equivalent association matrix, using the association coefficients of intuitionistic fuzzy sets, and proposed a clustering algorithm for intuitionistic fuzzy sets. In most of the medical diagnosis and pattern recognition problems, there exist some base patterns, and experts usually make decisions based on the similarity between unknown sample and the base patterns [12, 30]. In [12], Khatibi et al. took an intelligent approach towards the bacteria classification problem by using five similarity measures of fuzzy sets and intuitionistic fuzzy sets in the medical pattern recognition. In [30], a cosine similarity measure and a weighted cosine similarity measure between intuitionistic fuzzy sets are proposed and the efficiency of the cosine similarity measures for pattern recognition and medical diagnosis is demonstrated. Iancu [10] extended some crisp cardinality measures to measures for intuitionistic fuzzy sets and proposed several similarity measures based on Frank t-norms family. Deng et al. [7] discussed the monotonicity of similarity measures between intuitionistic fuzzy sets and presented three types of monotonic similarity measures between intuitionistic fuzzy sets and investigated their properties and relationship with entropy and inclusion measure. Chen et al. [5] proposed a new similarity measure between intuitionistic fuzzy values based on the centroid points of transformed right-angled triangular fuzzy numbers, and have applied the proposed similarity measure between intuitionistic fuzzy sets to deal with pattern recognition problems. At the same time, grounded on native concepts from activation detection in medical image analysis, a model for determining the degree of similarity between intuitionistic fuzzy sets is proposed [18]. As it is very difficult to find a certain measure that can be successfully applied to all kinds of images comparisons-related problems in the same time, it is appropriate to look for new approaches for measuring the similarity. Hassaballah and Ghareeb [8] introduced a framework for using the similarity measures on intuitionistic fuzzy sets in image processing field, specifically for image comparison. Recently, some new similarity measures between intuitionistic fuzzy sets are also proposed [16, 35]. Luo and Zhao [16] gave a new distance measure between intuitionistic fuzzy sets, which is based on a matrix norm and a strictly increasing (or decreasing) binary function. Ngan et al. [19] analyzed the disadvantages of existing distance measures of intuitionistic fuzzy sets and proposed a new distance measure called H-max of intuitionistic fuzzy sets. Jiang et al. [11] proposed a novel similarity/distance measure between intuitionistic fuzzy sets according to the intersections of the transformed isosceles triangles from intuitionistic fuzzy sets.

In this paper, we attempt to present a novel similarity measure from a new point of view. We show that in a intuitionistic fuzzy approximation space, a triangular norm can induce an inclusion degree, and the similarity measure between intuitionistic fuzzy sets can be given according to this triangular norm and inclusion degree.

Rest of this paper is organized as follows: Section 2 introduces some basic concepts related to intuitionistic fuzzy sets. Section 3 presents the similarity measure of intuitionistic fuzzy sets based on triangular norm. Section 4 gives an intuitionistic fuzzy clustering algorithm based on similarity degree of intuitionistic fuzzy sets. In Section 5, the similarity degree of intuitionistic fuzzy sets is applied to pattern recognition. Finally, conclusions are made in Section 6.

2. Basic concepts

Let U be a classical set of objects, called the universe of discourse, where an element of U is denoted by x.

Definition 2.1. [31] A fuzzy set F = {(x, μ _F  (x)) | x ∈ X} on a universe of discourse U is characterized by a membership function, μ _F  (x), as follows: μ F : U → [ 0 , 1 ] .

Definition 2.2. [1] An intuitionistic fuzzy set A on U is an object having the form A = { ( x , μ A ( x ) , ν A ( x ) ) | x ∈ U } where μ _A  (x) , ν _A  (x) : x → [0, 1] denotes membership function and non-membership function, respectively, of A and satisfy 0 ≤ μ A ( x ) + ν A ( x ) ≤ 1 for every x ∈ U.

μ _A  (x) is the lowest bound of membership degree derived from proofs of supporting x, ν _A  (x) is the lowest bound of non-membership degree derived from proofs of rejecting x. π A ( x ) = 1 - μ A ( x ) - ν A ( x ) is called the hesitation degree of x to A. For example, let A be an intuitionistic fuzzy set with membership function μ _A  (x) and non-membership function ν _A  (x), respectively. If μ _A  (x) =0.5 and ν _A  (x) =0.3, then we have π _A  (x) =1 - 0.5 - 0.3 = 0.2. It could be interpreted as the degree that the object x belongs to A is 0.5, the degree that the object x does not belong to A is 0.3 and the degree of hesitation is 0.2. If the π _A  (x) is small, knowledge about x is more certain. If π _A  (x) is great, knowledge about x is more uncertain.

Obviously, when μ _A  (x) =1 - ν _A  (x) for all elements of the universe, the intuitionistic fuzzy set would degenerated into fuzzy set.

For simplicity, we call α = (μ _α , ν _α ) is the intuitionistic fuzzy number, where μ _α  ∈ [0, 1] is called membership degree, ν _α  ∈ [0, 1] is called non-membership degree and μ _α  + ν _α  ≤ 1. The hesitation degree of intuitionistic fuzzy number α is denoted by π _α  = 1 - μ _α  - ν _α  .

Definition 2.3. [12] Let α₁ = (μ _{α 1} , ν _{α 1} ) and α₂ = (μ _{α 2} , ν _{α 2} ) be two intuitionistic fuzzy numbers, then

(1) α₁ = α₂ if and only if μ _{α 1}  = μ _{α 2} and ν _{α 1}  = ν _{α 2} ,

(2) α₁ ≤ α₂ if and only if μ _{α 1}  ≤ μ _{α 2} and ν _{α 1}  ≥ ν _{α 2} .

Definition 2.4. [1] Suppose that A, B are two intuitionistic fuzzy sets on U, and denote A = ∑ i ( μ A ( x i ) , ν A ( x i ) ) x i , x i ∈ U , B = ∑ i ( μ B ( x i ) , ν B ( x i ) ) x i , x i ∈ U , then

(1) A = B if and only if μ _A  (x _i ) = μ _B  (x _i ), ν _A  (x _i ) = ν _B  (x _i ),

(2) A ⊆ B if and only if μ _A  (x _i ) ≤ μ _B  (x _i ), ν _A  (x _i ) ≥ ν _B  (x _i ).

Definition 2.5. [32] A binary function T : [0, 1] ² → [0, 1] is called a triangular norm, if T satisfies the following properties:

(1) T (a, 1) = a  (a ∈ [0, 1]) ;

(2) T (a, b) = T (b, a)  (a, b ∈ [0, 1]) ;

(3) T (T (a, b) , c) = T (a, T (b, c))  (a, b, c ∈ [0, 1]) ;

(4) T (a, b) ≤ T (c, d)   (a ≤ c, b ≤ d, a, b, c, d ∈ [0, 1]) .

Originally, the notion of triangular norm introduced within the framework of probabilistic metric spaces. Its main idea is to extend the classical triangle inequality in metric spaces towards probabilistic metric spaces. However, nowadays triangular norm has played an important role in the proceeding of the basic logic and fuzzy logics. The most popular continuous triangular norms are as follows:

•Standard min operator T M ( a , b ) = min { a , b } , ∀ a , b ∈ [ 0 , 1 ] ;

•Algebraic product T P ( a , b ) = a ∗ b , ∀ a , b ∈ [ 0 , 1 ] ;

•Lukasiewicz triangular norm T L ( a , b ) = max { 0 , a + b - 1 } , ∀ a , b ∈ [ 0 , 1 ] .

3. The similarity degree of intuitionistic fuzzy sets based on triangular norm

Similarity measure is an important tool for determining the degree of similarity between two objects. For the similarity measures between intuitionistic fuzzy sets, we note that there exist a lot of different calculating methods [5]. Let U be the universe of discourse, where U = {x₁, x₂, ⋯ , x _n }, and let A = {(x _i , μ _A  (x _i ) , ν _A  (x _i )) |1 ≤ i ≤ n} and B = {(x _i , μ _B  (x _i ) , ν _B  (x _i )) |1 ≤ i ≤ n} be two intuitionistic fuzzy sets on U. The existing representative similarity measures between intuitionistic fuzzy sets A and B are as follows:

(1) Boran and Akay’s similarity measure S _BA [3]:

S B A ( A , B ) = 1 − ∑ i = 1 n | t ( μ A ( x i ) − μ B ( x i ) ) − ( ν A ( x i ) − ν B ( x i ) ) | p + | t ( ν A ( x i ) − ν B ( x i ) ) − ( μ A ( x i ) − μ B ( x i ) ) | p 2 n ( t + 1 ) p p . where p is the L _p norm and t is a parameter for adjusting the effect of hesitation margin, 1≤ p < ∞ and 2≤ t < ∞.

(2) Li and Cheng’s similarity measure S _LC [13]: S LC ( A , B ) = 1 - ∑ i = 1 n | ψ A ( x i ) - ψ B ( x i ) | n p . where ψ A ( x i ) = μ A ( x i ) + 1 - ν A ( x i ) 2 , ψ B ( x i ) = μ B ( x i ) + 1 - ν B ( x i ) 2 , 1≤ p < ∞ and 1 ≤ i < n.

(3) Hung and Yang’s similarity measures S_HY1, S_HY2 and S_HY3 [9]: S HY 1 ( A , B ) = 1 - ∑ i = 1 n max ( | μ A ( x i - μ B ( x i | , | ν A ( x i - ν B ( x i | ) n , S HY 2 ( A , B ) = e S HY 1 ( A , B ) - 1 - e - 1 1 - e - 1 , S HY 3 ( A , B ) = S HY 1 ( A , B ) 2 - S HY 1 ( A , B ) .

(4) Liang and Shi’s similarity measures S _LS [14]: S LS ( A , B ) = 1 - ∑ i = 1 n | φ μ ( x i ) + φ ν ( x i ) | p n p . where φ μ ( x i ) = | μ A ( x i ) - μ B ( x i ) | 2 , φ ν ( x i ) = | 1 - ν A ( x i ) - ( 1 - ν B ( x i ) ) | 2 .

(5) Mitchell’s similarity measure S _M [17]: S M ( A , B ) = ρ μ ( A , B ) + ρ ν ( A , B ) . where ρ μ ( A , B ) = 1 - ∑ i = 1 n | μ A ( x i ) - μ B ( x i ) | p n p , ρ ν ( A , B ) = 1 - ∑ i = 1 n | ν A ( x i ) - ν B ( x i ) | p n p , 1≤ p < ∞.

(6) Ye’s similarity measure S _Y [30]: S Y ( A , B ) = 1 n ∑ i = 1 n μ A ( x i ) μ B ( x i ) + ν A ( x i ) ν B ( x i ) μ A 2 ( x i ) + ν A 2 ( x i ) μ B 2 ( x i ) + ν B 2 ( x i ) .

(7) Chen et al. similarity measure S _CCL [5]: S CCL ( A , B ) = 1 - | 2 ( μ A ( x i ) - μ B ( x i ) ) - ( ν A ( x i ) - ν B ( x i ) ) | 3 × ( 1 - π A ( x i ) + π B ( x i ) 2 ) - | 2 ( ν A ( x i ) - ν B ( x i ) ) - ( μ A ( x i ) - μ B ( x i ) ) | 3 × ( π A ( x i ) + π B ( x i ) 2 ) .

Comparing the existing similarity measures listed above, we can find that these similarity measures are all given in the sense of distance. And these similarity measures are not perfect in every respect. As mentioned in [5, 19], some existing similarity measures even have drawbacks such as above S _BA , S_HY1, S_HY2, S_HY3, S _LS , S _Y maybe get unreasonable results in some situations.

In fact, it is very difficult to find a certain measure that can be successfully applied to all kinds of pattern recognition problems. Therefore, in order to deal with some specific problems reasonably, some scholars constantly put forward new similarity measures. However, it is undeniable that different similarity measure formulas can be obtained by using different distances.

Based on the existing similarity measures, this section we present a novel way to obtain the similarity measure between intuitionistic fuzzy sets from a new perspective. Our main purpose is to show that a similarity measure can be obtained from a corresponding triangular norm. A triangular norm can induce an inclusion degree, and using this triangular norm and the induced inclusion degree, a similarity degree can be obtained.

Definition 3.1. [33] (L, ≤) is called a partially ordered set if the relation ≤ on L satisfies the following properties:

(1) α ≤ α (α ∈ L);

(2) α ≤ β, β≤ α ⇒α = β (α, β ∈ L);

(3) α ≤ β ≤ γ ⇒ α ≤ γ (α, β, γ ∈ L).

If F ( U ) stands for the set of all intuitionistic fuzzy sets on U, then ( F ( U ) , ⊆ ) is a partially ordered set. Especially, ([0, 1] , ≤) is also a partially ordered set.

Definition 3.2. [33] Let (L, ≤) be a partially ordered set, L₀ ⊆ L. A real function D : L₀ × L₀ → [0, 1] is called an inclusion degree on L₀ if for any α, β, γ ∈ L₀, D satisfies the following properties:

(1) 0≤ D (β/α) ≤1 ;

(2) α≤ β ⇒ D (β/α) =1 ;

(3) α ≤ β ≤ γ ⇒ D (α/γ) ≤ D (β/γ) .

In Definition 3.2, (1) is normalization for inclusion degree; (2) states the property of consistency between inclusion degree and standard inclusion; (3) states the property of monotonicity of inclusion degree.

Example 3.1. Let U be a finite set, G = {X|X ⊆ U}, and ⊆ is a partially ordered relation on G, for ∀X, Y ∈ G, we define D ( Y / X ) = { | Y ∩ X | | X | , X ≠ ∅ 1 , X = ∅ where |X| denotes the cardinality of X.

It is easy to see that D is an inclusion degree on G.

Definition 3.3. Let U be a nonempty and finite set call the universe of discourse, F ( U ) stands for the set of all intuitionistic fuzzy sets on U. A real function D : F ( U ) × F ( U ) → [ 0 , 1 ] is called an inclusion degree on F ( U ) if for any A , B , C ∈ F ( U ) , D satisfies the following properties:

(1) 0≤ D (B/A) ≤1 ;

(2) A⊆ B ⇒ D (B/A) =1 ;

(3) A ⊆ B ⊆ C ⇒ D (A/C) ≤ D (B/C) .

Theorem 3.1. Let T be a triangular norm on [0, 1], for any a, b ∈ [0, 1], we define θ T ′ ( b / a ) = sup { c ∈ [ 0 , 1 ] | T ( a , c ) ≤ b } , then θ ′ T is an inclusion degree on [0, 1].

Example 3.2. (1) If T (a, b) = min {a, b}, then θ ′ T ( b / a ) = { 1 , a ≤ b b , a > b

(2) If T (a, b) = a ∗ b, then θ T ′ ( b / a ) = { 1 , a ≤ b b / a , a > b

(3) If T (a, b) = max {0, a + b - 1}, then θ T ' ( b / a ) = 1 ∧ ( 1 − a + b ) .

Theorem 3.2. Let U = {x₁, x₂, ⋯ , x _n } be the universe of discourse. A and B are two intuitionistic fuzzy sets on U, and A = {(x, μ _A  (x) , ν _A  (x)) | x ∈ U} , B = {(x, μ _B  (x) , ν _B  (x)) | x ∈ U} .

We define θ T ( B / A ) = 1 2 n ∑ i = 1 n [ θ T ' ( μ B ( x i ) / μ A ( x i ) ) ​ + ​ θ T ' ( ν A ( x i ) / ν B ( x i ) ) ] , then θ _T is an inclusion degree on F ( U ) .

Proof.

(1) Known by the definition of θ T ' , 0 ≤ θ T ' ( μ B ( x i ) / μ A ( x i ) ) ≤ 1 and 0 ≤ θ T ' ( ν A ( x i ) / ν B ( x i ) ) ≤ 1 are obvious. Therefore, 0 ≤ θ T ( B / A ) ≤ 1 .

(2) If A ⊆ B, then for x ∈ U, μ _A  (x) ≤ μ _B  (x) and ν _A  (x) ≥ ν _B  (x). Thus, we have θ T ' ( μ B ( x i ) / μ A ( x i ) ) = 1 , θ T ' ( ν A ( x i ) / ν B ( x i ) ) = 1. Consequently, θ _T  (B/A) =1 .

(3) Suppose that C is also an intuitionistic fuzzy set on U and C = {(x, μ _C  (x) , ν _C  (x)) | x ∈ U} . If A ⊆ B ⊆ C, then for every x ∈ U, we have μ A ( x ) ≤ μ B ( x ) ≤ μ C ( x ) , ν A ( x ) ≥ ν B ( x ) ≥ ν C ( x ) . Therefore, θ T ′ ( μ A ( x i ) / μ C ( x i ) ) ≤ θ T ′ ( μ B ( x i ) / μ C ( x i ) ) , θ T ′ ( ν C ( x i ) / ν A ( x i ) ) ≤ θ T ′ ( ν C ( x i ) / ν B ( x i ) ) . It follows that θ T ( A / C ) ≤ θ T ( B / C ) .

Definition 3.4. Let U be the universe of discourse, ℱ ( U ) stands for the set of all intuitionistic fuzzy sets on U. A real function S : F ( U ) × F ( U ) → [ 0 , 1 ] is called a similarity degree on ℱ ( U ) if for any A , B , C ∈ ℱ ( U ) , S satisfies the following properties:

(1) 0≤ S (A, B) ≤1, S (A, A) =1 ;

(2) S (A, B) = S (B, A) ;

(3) A ⊆ B ⊆ C ⇒ S (A, C) ≤ S (B, C) .

Theorem 3.3. Let U = {x₁, x₂, ⋯ , x _n } be the universe of discourse, T be a triangular norm. A and B are two intuitionistic fuzzy sets on U, and A = {(x, μ _A  (x) , ν _A  (x)) | x ∈ U} , B = {(x, μ _B  (x) , ν _B  (x)) | x ∈ U} . Then S ( A , B ) = T ( θ T ( B / A ) , θ T ( A / B ) ) is a similarity degree between A and B, where θ _T is an inclusion degree on ℱ ( U ) .

Proof.

(1) By the definition of triangular norm, we know that 0 ≤ S (A, B) ≤1 holds. By θ _T  (A/A) =1, we have S (A, A) =1.

(2) By the symmetric of triangular norm T, we can obtain

S (A, B)

= T (θ _T  (B/A) , θ _T  (A/B))

= T (θ _T  (A/B) , θ _T  (B/A))

= S (B, A) .

(3) Suppose that C is also an intuitionistic fuzzy set on U and C = {(x, μ _C  (x) , ν _C  (x)) | x ∈ U} . If A ⊆ B ⊆ C, then for every x ∈ U, we have μ _A  (x) ≤ μ _B  (x) ≤ μ _C  (x) , ν _A  (x) ≥ ν _B  (x) ≥ ν _C  (x) . Since θ _T is an inclusion degree on ℱ ( U ) , we infer that θ _T  (A/C) ≤ θ _T  (B/C) .

On the other hand,

S (A, C)

= T (θ _T  (C/A) , θ _T  (A/C))

= T (1, θ _T  (A/C))

= θ _T  (A/C) ,

S (B, C)

= T (θ _T  (C/B) , θ _T  (B/C))

= T (1, θ _T  (B/C))

= θ _T  (B/C) .

Hence, S (A, C) ≤ S (B, C) .

Now, we work out a numerical example to illustrate the similarity degree between intuitionistic fuzzy sets based on triangular norm.

Example 3.3. Suppose that U = {x₁, x₂, ⋯ , x₆} is the universe of discourse. A, B and C are three intuitionistic fuzzy sets on U, where

A = ( 0.3 , 0.6 ) x 1 + ( 0.5 , 0.3 ) x 2 + ( 0.6 , 0.3 ) x 3 + ( 0.8 , 0.1 ) x 4 + ( 0.9 , 0.1 ) x 5 + ( 0.7 , 0.2 ) x 6 , B = ( 0.4 , 0.6 ) x 1 + ( 0.2 , 0.8 ) x 2 + ( 0.6 , 0.3 ) x 3 + ( 0.7 , 0.3 ) x 4 + ( 0.8 , 0.2 ) x 5 + ( 0.7 , 0.3 ) x 6 , C = ( 0.8 , 0.1 ) x 1 + ( 0.7 , 0.2 ) x 2 + ( 0.2 , 0.7 ) x 3 + ( 0.3 , 0.4 ) x 4 + ( 0.4 , 0.5 ) x 5 + ( 0.5 , 0.4 ) x 6 .

At the same time, let T (a, b) = min {a, b}, then θ T ' ( b / a ) = { 1 , a ≤ b , b , a > b .

Thus, the similarity degree between A and B can be calculated as follows:

S ( A , B ) = T ( θ T ( B / A ) , θ T ( A / B ) ) = 1 12 ( 1 + 1 + 0.2 + 0.3 + 1 + 1 + 0.7 + 0.1 + 0.8 + 0.1 + 1 + 0.2 ) ∧ 1 12 ( 0.3 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ) =0 .617 .

Similarly, the similarity degree between A and C can be calculated as follows:

S ( A , C ) = T ( θ T ( C / A ) , θ T ( A / C ) ) = 1 12 ( 1 + 1 + 1 + 1 + 0.2 + 0.3 + 0.3 + 0.1 + 0.4 + 0.1 + 0.5 + 0.2 ) ∧ 1 12 ( 0.3 + 0.1 + 0.5 + 0.2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ) =0 .508 .

The similarity degree between B and C can be calculated as follows:

S ( B , C ) = T ( θ T ( C / B ) , θ T ( B / C ) ) = 1 12 ( 1 + 1 + 1 + 1 + 0.2 + 0.3 + 0.3 + 0.4 + 0.4 + 0.2 + 0.5 + 0.3 ) ∧ 1 12 ( 0.4 + 0.1 + 0.2 + 0.2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ) =0 .550 .

4. A intuitionistic fuzzy clustering algorithm based on similarity degree induced by triangular norm

Clustering analysis is a fundamental but important tool in statistical data analysis. Clustering is an essential data mining tool that aims to discover inherent cluster structure in data. The fuzzy approach to the clustering problem involves the concept of similarity degree between fuzzy sets. In this section, based on similarity degree induced by triangular norm, we present an intuitionistic fuzzy clustering algorithm different from [28].

Definition 4.1. [29] Let Z = (z _ij ) _m×n be a matrix. If all elements of z _ij are intuitionistic fuzzy numbers, then Z is called the intuitionistic fuzzy matrix.

Especially, if 0 ≤ z _ij  ≤ 1 (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n), then Z is called the fuzzy matrix.

Definition 4.2. [25] A matrix R = (r _ij ) _n×n is called a fuzzy similarity matrix if it satisfies the following properties:

(1)  0 ≤ r _ij  ≤ 1 for all i, j = 1, 2, ⋯ , n;

(2)  r _ij  = 1 if and only if i = j; (Reflexivity)

(3)  r _ij  = r _ji for all i, j = 1, 2, ⋯ , n. (Symmetry)

Definition 4.3. [25] Let R = (r _ij ) _n×n be a fuzzy similarity matrix, if R = R ∘ R, then R has transitivity and calls a fuzzy equivalent matrix, where R ° R = ( r ¯ i j ) n × n and r ¯ i j = max k { min { r i k , r k j } } , i , j = 1 , 2 , ⋯ , n .

Theorem 4.1. If R = (r _ij ) _n×n is a fuzzy similarity matrix, then R ° R = ( r ¯ i j ) n × n is also a fuzzy similarity matrix.

Proof. (1) R is a fuzzy equivalent matrix, so 0 ≤ r _ij  ≤ 1 for all i, j = 1, 2, ⋯ , n. Therefore 0 ≤ r ¯ i j = max k { min { r i k , r k j } } ≤ 1 , i , j = 1 , 2 , ⋯ , n .

(2) r _ij  = 1 if and only if i = j, it implies that r ¯ i j = max k { min { r i k , r k j } } = 1 if and only if i = k = j for some k.

(3) Since r _ij  = r _ji for all i, j = 1, 2, ⋯ , n, then r ¯ i j = max k { min { r i k , r k j } } = 1 r ¯ i j = max k { min { r i k , r k j } } = max k { min { r k i , r j k } } = max k { min { r j k , r k i } } = r ¯ j i .

By Theorem 4.1, we can obtain that if R is a fuzzy similarity matrix, then for any positive integer k, R ^k is also a fuzzy similarity matrix. At the same time, the following conclusion can be obtained easily:

Theorem 4.2. If R = (r _ij ) _n×n is a fuzzy similarity matrix, then for any nonnegative integer i, R^{2 i}  ∘ R^{2 i}  = R²ⁱ⁺¹ is also a fuzzy similarity matrix.

Proof. The Proof is similar to the Proof of Theorem 1.

Definition 4.4. [25] Let R = (r _ij ) _n×n be a fuzzy equivalent matrix, then R_λ = (r^λ _ij )_n×n is called the λ-cutting matrix of R, where r i j λ = { 0 , r i j < λ , 1 , r i j ≥ λ . and λ is the confidence level with λ ∈ [0, 1] .

Let U = {x₁, x₂, ⋯ , x _n } be a discrete universe of discourse which is composed of n objects. And every objects is evaluated by m attributes and the evaluation values characterized by intuitionistic fuzzy numbers. In practical problem such as pattern recognition, the objects in U need to be classified in terms of m attributes. We call this problem intuitionistic fuzzy clustering.

According to the similarity degree induced by triangular norm, we present the intuitionistic fuzzy clustering algorithm as follows:

Step 1. Construct original intuitionistic fuzzy matrix: Z = ( z 11 z 12 ⋯ z 1 m z 21 z 22 ⋯ z 2 m ⋮ ⋮ ⋮ z n 1 z n 2 ⋯ z n m ) , where z _ij  = (μ _ij , ν _ij )   (i = 1, 2, ⋯ , n ; j = 1, 2, ⋯ , m) is intuitionistic fuzzy number, expresses the evaluation value of x _i under jth attribute.

Step 2. Build fuzzy similarity matrix.

According to the approach proposed in Section 3, calculate similarity degree between objects and obtain the fuzzy similarity matrix: M = ( x 11 x 12 ⋯ x 1 n x 21 x 22 ⋯ x 2 n ⋮ ⋮ ⋮ x n 1 x n 2 ⋯ x n n ) ​. Denote x i = ( z i 1 , z i 2 , ⋯ , z im ) = ( ( μ i 1 , ν i 1 ) , ( μ i 2 , ν i 2 ) , ⋯ , ( μ im , ν im ) ) , x j = ( z j 1 , z j 2 , ⋯ , z jm ) = ( ( μ j 1 , ν j 1 ) , ( μ j 2 , ν j 2 ) , ⋯ , ( μ jm , ν jm ) ) , then here x i j = S ( x i , x j ) = T ( 1 2 m ∑ k = 1 m ( θ T ' ( μ j k / μ i k ) + θ T ' ( ν i k / ν j k ) ) , 1 2 m ∑ k = 1 m ( θ T ' ( μ i k / μ j k ) + θ T ' ( ν j k / ν i k ) ) ) where T is a triangular norm and θ T ' is an inclusion degree induced by T. x _ij expresses the similarity degree between x _i and x _j .

Step 3. Construct fuzzy equivalent matrix.

For fuzzy similarity matrix M obtained in Step 2, using square method as follows: M → M 2 → M 4 → ⋯ → M 2 i → ⋯ , after finite times of composition, if M^{2 i}  ∘ M^{2 i}  = M^{2 i} take place first time, then it implies that M^{2 i} has transitivity and is a fuzzy equivalent matrix, denoted as R.

In fact, if M is a fuzzy similarity matrix, we can prove that M^{2 i} is a fuzzy equivalent matrix as follows:

On the one hand, by Theorem 4.2, we can obtain that M^{2 i} is a fuzzy similarity matrix, that is to say M^{2 i} has reflexivity and symmetry.

On the other hand, by M^{2 i}  ∘ M^{2 i}  = M^{2 i} , we know that M^{2 i} has transitivity. Consequently, M^{2 i} is a fuzzy equivalent matrix.

Step 4. For a confidence level λ, construct R _λ .

Step 5. According to the λ-cutting matrix of R, classify the objects in U, if all elements of the ith line (column) are the same as the jth line (column) in R _λ , then the objets x _i and x _j are of the same type.

Example 4.1. Suppose that there are six suburbs (x₁, x₂, ⋯ , x₆) of one city, due to the modernization of the city, these suburbs are all polluted in some certain, but the pollution degree are different. The local government wants to draw up environmental protection programs, they need to classify six suburbs and carry out same programs to similar suburbs. The pollution of six suburbs is described by four attributes: (1) air pollution (y₁); (2) water pollution (y₂); (3) soil pollution (y₃); (4) crop pollution (y₄). The characteristics of six suburbs under four attributes are represented by the intuitionistic fuzzy matrix, as shown in Table 4.1.

Suppose that T (a, b) = min {a, b}, then θ T ' ( b / a ) = { 1 , a ≤ b , b , a > b .

According to Step 2, calculate the similarity degree between x _i and x _j (i, j = 1, 2, ⋯ , 6) and build the fuzzy similarity matrix as follows:

M ​ = ​ ( 1.0000 0.4875 0.5000 0.6125 0.4625 0.5625 0.4875 1.0000 0.8375 0.7375 0.7125 0.7250 0.5000 0.8375 1.0000 0.7000 0.5625 0.7125 0.6125 0.7375 0.7000 1.0000 0.7375 0.6500 0.4625 0.7125 0.5625 0.7375 1.0000 0.6000 0.5625 0.7250 0.7125 0.6500 0.6000 1.0000 ) ​ ​

Calculate

M 2 ​ = ​ ( 1.0000 0.6125 0.6125 0.6125 0.6125 0.6125 0.6125 1.0000 0.8375 0.7375 0.7375 0.7250 0.6125 0.8375 1.0000 0.7375 0.7125 0.7250 0.6125 0.7375 0.7375 1.0000 0.7375 0.7250 0.6125 0.7375 0.7125 0.7375 1.0000 0.7125 0.6125 0.7250 0.7250 0.7250 0.7125 1.0000 ) ​

M² = M does not hold, i.e., the matrix M is not an fuzzy equivalent matrix. Thus, by Step 3, we further calculate

​ M 4 ​ = ( 1 .0000 0 .6125 0 .6125 0 .6125 0 .6125 0 .6125 0 .6125 1 .0000 0 .8375 0 .7375 0 .7375 0 .7250 0 .6125 0 .8375 1 .0000 0 .7375 0 .7375 0 .7250 0 .6125 0 .7375 0 .7375 1 .0000 0 .7375 0 .7250 0 .6125 0 .7375 0 .7375 0 .7375 1 .0000 0 .7250 0 .6125 0 .7250 0 .7250 0 .7250 0 .7250 1 .0000 ) ,

M 8 ​ = ​ ( 1 .0000 0 .6125 0 .6125 0 .6125 0 .6125 0 .6125 0 .6125 1 .0000 0 .8375 0 .7375 0 .7375 0 .7250 0 .6125 0 .8375 1 .0000 0 .7375 0 .7375 0 .7250 0 .6125 0 .7375 0 .7375 1 .0000 0 .7375 0 .7250 0 .6125 0 .7375 0 .7375 0 .7375 1 .0000 0 .7250 0 .6125 0 .7250 0 .7250 0 .7250 0 .7250 1 .0000 ) , thus M⁴ ∘ M⁴ = M⁴, i.e., M⁴ is a fuzzy equivalent matrix, denoted as R (R = M⁴).

If confidence level λ = 0.8375, then R λ = ( 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ) .

That is, under this confidence level, six suburbs are classified into the following five types:

  {x₁} , {x₂, x₃} , {x₄} , {x₅} , {x₆} .

In this case, the local government only need to draw up five environmental protection programs, that is, government can implement the same program to x₂ and x₃.

Similarly,

(1) if λ = 0.7375, then six suburbs are classified into the following three types:

  {x₁} , {x₂, x₃, x₄, x₅} , {x₆} ;

(2) if λ = 0.7250, then six suburbs are classified into the following two types:

  {x₁} , {x₂, x₃, x₄, x₅, x₆} ;

(3) if λ = 0.6125, then six suburbs are of the same types:

  {x₁, x₂, x₃, x₄, x₅, x₆} .

Suppose that there are m patterns which are represented by intuitionistic fuzzy sets A _i  = {(x, μ _{A i}  (x)) , ν _{A i}  (x)) |x ∈ U} (i = 1, 2, ⋯ , m). Assume that there be a sample to be recognized which is represented by a intuitionistic fuzzy set B = {(x, μ _B  (x)) , ν _B  (x)) |x ∈ U}.

If S ( A i 0 , B ) = max 1 ≤ i ≤ m { S ( A i , B ) } , where S (A _i , B) is a novel similarity degree between A _i and B which defined in Section 3. According to the principle of the maximum membership degree of similarity between intuitionistic fuzzy sets, we can decide that the sample B belongs to the pattern A _{i 0} .

Example 5.1. Let U = {x₁, x₂, x₃, x₄}. Suppose that there are three patterns A₁, A₂ and A₃ which represented as follows:

A 1 = ( 0.4 , 0.5 ) x 1 + ( 0.3 , 0.7 ) x 2 + ( 0.6 , 0.2 ) x 3 + ( 0.8 , 0.1 ) x 4 , A 2 = ( 0.7 , 0.2 ) x 1 + ( 0.4 , 0.6 ) x 2 + ( 0.5 , 0.4 ) x 3 + ( 0.4 , 0.6 ) x 4 , A 3 = ( 0.5 , 0.3 ) x 1 + ( 0.7 , 0.2 ) x 2 + ( 0.5 , 0.4 ) x 3 + ( 0.3 , 0.6 ) x 4 .

Consider a sample B which will be recognized, here

B = ( 0.8 , 0.2 ) x 1 + ( 0.4 , 0.5 ) x 2 + ( 0.5 , 0.4 ) x 3 + ( 0.3 , 0.6 ) x 4 .

Assume that T (a, b) = min {a, b}, then θ T ' ( b / a ) = { 1 , a ≤ b , b , a > b .

Thus, the similarity degree between A₁ and B can be calculated as follows:

S ( A 1 , B ) = T ( θ T ( B / A 1 ) , θ T ( A 1 / B ) ) = 1 8 ( 1 + 1 + 1 + 1 + 0.5 + 0.2 + 0.3 + 0.1 ) ∧ 1 8 ( 0.4 + 0.2 + 0.3 + 0.5 + 1 + 1 + 1 + 1 ) =0 .6375 .

Similarly, the similarity degree between A₂ and B is

S ( A 2 , B ) = T ( θ T ( B / A 2 ) , θ T ( A 2 / B ) ) = 1 8 ( 1 + 1 + 1 + 1 + 1 + 1 + 0.3 + 1 ) ∧ 1 8 ( 0.7 + 1 + 1 + 0.5 + 1 + 1 + 1 + 1 ) =0 .9000 .

The similarity degree between A₃ and B is

S ( A 3 , B ) = T ( θ T ( B / A 3 ) , θ T ( A 3 / B ) ) = 1 8 ( 1 + 1 + 0.4 + 0.2 + 1 + 1 + 1 + 1 ) ∧ 1 8 ( 0.5 + 0.2 + 1 + 1 + 1 + 1 + 1 + 1 ) =0 .825 .

That is S (A 2 , B) > S(A 3 , B) > S(A 1 , B) .

According to the principle of the maximum membership degree, we can obtain the sample B belongs to the pattern A₂.

6. Conclusions

Although there are many formulas for calculating similarity measure of intuitionistic fuzzy sets. Comparing the existing similarity measures, we can find that these similarity measures are all given in the sense of distance. And these similarity measures even have drawbacks. In this paper, a novel measure method for intuitionistic fuzzy sets is proposed to calculate their degree of similarity. Our main purpose is to show that a similarity measure can be obtained from a corresponding triangular norm. We have shown that according to the membership and non-membership functions of intuitionistic fuzzy sets, an inclusion degree can be induced by a triangular norm. And using this triangular norm and the induced inclusion degree, a similarity degree of intuitionistic fuzzy sets can be obtained. That is, the similarity measure can be obtained from a concrete triangular norm. Thus, we proposed a new method to measure similarity between intuitionistic fuzzy sets. At the same time, we have applied the similarity degree which proposed in this paper to intuitionistic fuzzy clustering and pattern recognition. The proposed similarity measure provides us with a useful way for dealing with pattern recognition problems in intuitionistic fuzzy environments. Our next research work is to combining similarity measure in the sense of distance with similarity measure in the sense of triangular norm. And try to apply the proposed methods to other fields such as approximation reasoning and multiple attribute decision making.