Combining similarity and divergence measures for intuitionistic fuzzy information clustering

1 Introduction

Cluster analysis or clustering is to divide a family of objects into several different clusters, in which the objects in the same cluster are of some similar properties [1]. Clustering analysis is a core tool which plays the important role in various fields, such as decision making [2, 3], data minning [4, 5], knowledge discovery [6, 7], clustering ensembles [8], medical diagnosis [9], etc. In classical clustering analysis, we know that an object belongs exactly to one cluster, i.e., the so-called “hard" cluster [10]. In practice, however, it is usually very fuzzy for that whether or not an object belongs exactly to one cluster, i.e., the so-called “soft" cluster or fuzzy cluster, due to the incompleteness of information. In order to improve that, Bezdek [11] introduced the notion of fuzzy clustering that means a beginning of fuzzy clustering theory. Fuzzy sets, as important objects of study, are applied to explore the problems of fuzzy clustering. However, it is not very precise to reveal the fuzziness of objects under the complex and volatile environment since that fuzzy sets characterize the fuzziness just by the membership degree. So later, Atanassov [12] introduced again the notion of intuitionistic fuzzy set (IFS) which adds a hesitancy degree to fuzzy sets. Since then, IFS theory began to be focused on and applied to directions such as three-way decisions (3WD) [13–16], rough sets [17, 18], information systems [19–23], pattern recognition [24, 25] and image processing [26].

In the field of intuitionistic fuzzy set analysis, intuitionistic fuzzy clustering is a significant topic, has been studied for decades and plentiful results have been obtained [27–32]. For example, an early survey is due to Zhang et al. [27] who constructed an intuitionistic fuzzy equivalence matrix by using transitive closure of an intuitionistic fuzzy similarity matrix (IFSM) and finally gave an effective method of clustering samples. For instance again, Thong and Son [32] who, recently, first calculated the most proper number of clusters by using techniques of particle swarm optimization, and then gave a method of automatic picture intuitionistic fuzzy clustering. Although the previous results obtained show respectively their own advantages in clustering intuitionistic fuzzy information, they usually lead to time-consuming in computation or loss of information for that all of them are just based on an intuitionistic fuzzy equivalence matrix, the transitive closure technique or optimization methods [33]. Thus in recent years, many authors [33–37] proposed a novel technique by constructing an IFSM to cluster intuitionistic fuzzy information directly. It is pointed [33] out that this technique reduces time-consuming in computation since the clustering efforts and time are mainly determined by the IFSM calculation.

How to reduce clustering efforts and thus save the time in computation? Many authors [33–37] did lots of work on this aspect, e.g., Wang et al. [33] reduced clustering efforts by constructing an IFSM that is obtained from intuitionistic fuzzy similarity degree. It’s certain that these approaches, to some extent, save the time by reducing clustering efforts, however, it leads to a “counterintuitive result” in some cases. So how to improve the methods so that it not only saves the time but also avoids “counterintuitive results”. Thus in this paper we present a new intuitionistic fuzzy clustering approach which is based on IFSM construction.

This paper is organized as follows: In Section 2, we recall the knowledge and classical methods of IFSM construction, and analyze briefly some of their disadvantages. Section 3 gives a similarity measure between intuitionistic fuzzy numbers, the similarity and divergence measures between IFSs. Then based on that, we construct an IFSM. In Section 4, an intuitionistic fuzzy clustering method is presented by IFSM construction. In Section 5, some examples are given to illustrate the advantages of our method. Finally we end with this paper in Section 6.

2 Preliminary

Let us recall briefly some basic concepts concerning intuitionistic fuzzy theory and some classical approaches to IFSM construction.

In this section, we propose an approach to constructing an intuitionistic fuzzy similarity matrix. A similarity measure between intuitionistic fuzzy numbers (IFNs) is first given as:

Definition 4. Let a₁ = (μ₁, ν₁) and a₂ = (μ₂, ν₂) be two IFNs. A similarity measure s (a₁, a₂) between a₁ and a₂ is defined as: s ( a 1 , a 2 ) = 1 - 1 2 ( | μ 1 - μ 2 | + | ν 1 - ν 2 | + | π 1 - π 2 | ) ,(11) where π _i = 1 - μ _i - ν _i denotes the hesitation degree of the IFN a _i for any i = 1, 2.

From (109), we know that s (a₁, a₂) is a real number between 0 and 1 (see Property 1). The bigger (smaller) the value of s (a₁, a₂) is, the higher (lower) similarity is between a₁ and a₂; if s (a₁, a₂) =1, the similarity reaches the maximum between them, namely, a₁ = a₂, which implies their identity.

As is analyzed above, some properties of the similarity of IFNs are further obtained.

Property 1. Let a₁ = (μ₁, ν₁) and a₂ = (μ₂, ν₂) be two IFNs. Then

( 1 ) 0 ≤ s ( a 1 , a 2 ) ≤ 1 ; ( 2 ) s ( a 1 , a 2 ) = s ( a 2 , a 1 ) ; ( 3 ) s ( a 1 , a 2 ) = s ( a 1 c , a 2 c ) ; ( 4 ) s ( a 1 , a 2 ) = 1 ⇔ a 1 = a 2 .

Proof. (1) Let us first prove that 0 ≤ s (a₁, a₂) ≤1.(i) The conclusion is clear by (109) if π₁ = π₂. (ii) When π₁ ≠ π₂, (a) if μ₁ = μ₂ and ν₁ = ν₂, then 0 ≤ s (a₁, a₂) ≤1 holds; (b) if μ₁ = μ₂ and ν₁ ≠ ν₂, we obtain 0 ≤ s (a₁, a₂) ≤1; (c) if μ₁ ≠ μ₂ and ν₁ = ν₂, it yields 0 ≤ s (a₁, a₂) ≤1; (d) if μ₁ ≠ μ₂ and ν₁ ≠ ν₂, it deduces respectively 0 ≤ s (a₁, a₂) ≤1 from four cases as follows: textcircled1 μ₁ > μ₂ and ν₁ > ν₂; textcircled2 μ₁ > μ₂ and ν₁ < ν₂; textcircled3 μ₁ < μ₂ and ν₁ > ν₂; textcircled4 μ₁ < μ₂ and ν₁ < ν₂, which is corresponding to the case (i), (ii), (iii) and (iv) in Fig. 1, respectively.

Without loss of generality, we only prove 0 ≤ s (a₁, a₂) ≤1 under the first case, and the same for the others. It is well known that an IFN can be explained by two-dimensional coordinates [47], shown in the case (i) of Fig. 1. The membership degree (μ), non-membership degree (ν) and hesitation degree (π) are denoted by a point inside the triangle FOE, thus the point D and C are represented by D = (μ₁, ν₁, π₁) and C = (μ₂, ν₂, π₂), respectively. For convenience, assume that h = |μ₁ - μ₂|, l₁ = |ν₁ - ν₂| and l₂ = |π₁ - π₂|, let S be the area of the right trapezoid CHDG, then |DH| = l₁, |GC| = l₂, |CH| = h. It follows S = 1 2 ( l 1 + l 2 ) h by the formula of trapezoid area and l₂ = h + l₁ with l₁, l₂, h ∈ [0, 1], which deduces 2 S h = l 1 + l 2 . Analogously, S = S PCH + S PHDG = 1 2 h 2 + l 1 h , namely, 2 S h = h + 2 l 1 , which yields l₁ + l₂ = h + 2l₁. So we can obtain s ( a 1 , a 2 ) = 1 - 1 2 ( h + l 1 + l 2 ) = 1 - ( h + l 1 ) = 1 - l 2 ≥ 0 that leads to 0 ≤ s (a₁, a₂) ≤1. In addition, (2), (3) and (4) are immediate by (109).

OPEN IN VIEWER

Fig.1

Geometrical representation of IFNs.

From Property 1, we can see that the definition of the similarity measure of IFNs coincides with human’s cognition. Using Definition 4 and Definition 2, we extend the similarity and dissimilarity measures of IFNs to the IFSs and give some new definitions.

Definition 5. Let X = {x₁, x₂, …, x _n } be an universe of discourse and A = {(x _k , μ _A (x _k ), ν _A (x _k )) |x _k ∈ X}, B = {(x _k , μ _B (x _k ), ν _B (x _k )) |x _k ∈ X} be two IFSs. The similarity measure S (A, B) between A and B is given by: S ( A , B ) = 1 n ∑ k = 1 n ( 1 - 1 2 ( | μ A ( x k ) - μ B ( x k ) |(12) + | ν A ( x k ) - ν B ( x k ) | + | π A ( x k ) - π B ( x k ) | ) ) , where π _A (x _k ) and π _B (x _k ) are the hesitation degree of x _k to A and to B, respectively.

Property 2. Based on (12), we have: ( 1 ) 0 ≤ S ( A , B ) ≤ 1 ; ( 2 ) S ( A , B ) = 1 ⇔ A = B ; ( 3 ) S ( A , B ) = S ( B , A ) and S ( A , B ) = S ( A c , B c ) ; ( 4 ) If A ⊆ B ⊆ C , then S ( A , C ) ≤ min { S ( A , B ) , S ( B , C ) } .

Proof. (1), (2) and (3) are obvious. (4) Given any three IFSs A, B and C such that A ⊆ B ⊆ C. Thus for any x _k ∈ X, it follows μ _A (x _k ) ≤ μ _B (x _k ) ≤ μ _C (x _k ) and ν _A (x _k ) ≥ ν _B (x _k ) ≥ ν _C (x _k ). Suppose Q ( A , C ) ( x k ) = | μ A ( x k ) - μ C ( x k ) | + | ν A ( x k ) - ν C ( x k ) | + | π A ( x k ) - π C ( x k ) | , Q ( A , B ) ( x k ) = | μ A ( x k ) - μ B ( x k ) | + | ν A ( x k ) - ν B ( x k ) | + | π A ( x k ) - π B ( x k ) | .     In what follows, we will prove Q (A, C) (x _k ) ≥ Q (A, B) (x _k ) for all x _k ∈ X. With respect to any π _A (x _k ), π _B (x _k ), π _C (x _k ) ∈ [0, 1], where x _k ∈ X, there are six preference relations among them as follows: (a) π _A (x _k ) ≤ π _B (x _k ) ≤ π _C (x _k ); (b) π _A (x _k ) ≤ π _C (x _k ) < π _B (x _k ); (c) π _B (x _k ) < π _A (x _k ) ≤ π _C (x _k ); (d) π _B (x _k ) ≤ π _C (x _k ) < π _A (x _k ); (e) π _C (x _k ) < π _A (x _k ) ≤ π _B (x _k ) and (f) π _C (x _k ) < π _B (x _k ) < π _A (x _k ).

Case a: If π _A (x _k ) ≤ π _B (x _k ) ≤ π _C (x _k ) for all x _k ∈ X, it follows Q (A, C) (x _k ) =2 (ν _A (x _k ) - ν _C (x _k )) and Q (A, B) (x _k ) =2 (ν _A (x _k ) - ν _B (x _k )), which yields Q (A, C) (x _k ) ≥ Q (A, B) (x _k ). Also we can obtain the same results under Case (b), Case (d) and Case (f) respectively by the similar method.

Case c: If π _B (x _k ) < π _A (x _k ) ≤ π _C (x _k ) for all x _k ∈ X, Q (A, C) (x _k ) =2 (ν _A (x _k ) - ν _C (x _k )) and Q (A, B) (x _k ) =2 (μ _B (x _k ) - μ _A (x _k )) hold. Suppose Q (A, C) (x _k ) < Q (A, B) (x _k ), i.e., ν _A (x _k ) - ν _C (x _k ) < μ _B (x _k ) - μ _A (x _k ), which deduces μ _A (x _k ) + ν _A (x _k ) < μ _B (x _k ) + ν _C (x _k ) ≤ μ _C (x _k ) + ν _C (x _k ). That is 1 - π _A (x _k ) <1 - π _C (x _k ). So we can get π _A (x _k ) > π _C (x _k ), which is contradictory with π _A (x _k ) ≤ π _C (x _k ). Therefore Q (A, C) (x _k ) ≥ Q (A, B) (x _k ) holds. Also we can obtain the same results under Case (e) by the similar method.

As is discussed above, it follows Q (A, C) (x _k ) ≥ Q (A, B) (x _k ) for all x _k ∈ X, which deduces S ( A , C ) = 1 n ∑ k = 1 n ( 1 - Q ( A , C ) ( x k ) 2 ) ≤ 1 n ∑ k = 1 n ( 1 - Q ( A , B ) ( x k ) 2 ) = S ( A , B ) . Similarly, it follows S (A, C) ≤ S (B, C). Therefore we obtain S (A, C) ≤ min {S (A, B), S (B, C)}.

In Property 2, we note that S (A, B) satisfies (S1)-(S4). Whence, we have the corresponding definition of divergence measure.

Definition 6. Given two IFSs A and B in X. The divergence measure D (A, B) between A and B is defined as:

D ( A , B ) = 1 n ∑ k = 1 n ( 1 2 ( | μ A ( x k ) - μ B ( x k ) | + | ν A ( x k ) - ν B ( x k ) | ) ) .(13)

Similarly, as a divergence measure of IFSs, this measure should meet (D1)-(D4). Before verifying them, we first obtain the following Lemma.

Lemma 1. For any a, b, c ∈ [0, 1], then

| min { a , c } - min { b , c } | ≤ | a - b | and | max { a , c } - max { b , c } | ≤ | a - b | .(14)

Proof. Since min { a , c } = a + c - | a - c | 2 and | | a - c | - | b - c | | ≤ | a - b | . We have:

| min { a , c } - min { b , c } | = | a - b + | b - c | - | a - c | | 2 ≤ | a - b | + | | b - c | - | a - c | | 2 = | a - b | . We can also get |max {a, c} - max {b, c} | ≤ |a - b| via the similar method. Therefore the concludes hold.

On the basis of Lemma 1, the following property is further obtained:

Property 3. Based on (111), we have ( 1 ) 0 ≤ D ( A , B ) ≤ 1 ; ( 2 ) D ( A , B ) = 0 ⇔ A = B ; ( 3 ) D ( A , B ) = D ( B , A ) , D ( A , A ) = 0 and D ( A , B ) = D ( A c , B c ) ; ( 4 ) D ( A ∩ C , B ∩ C ) ≤ D ( A , B ) and D ( A ∪ C , B ∪ C ) ≤ D ( A , B ) ; ( 5 ) If A ⊆ B ⊆ C , then D ( A , C ) ≥ max { D ( A , B ) , D ( B , C ) } ; ( 6 ) If A ⊆ B ⊆ C ⊆ D , then D ( A , D ) ≥ D ( B , C ) ; ( 7 ) D ( A ∩ B , B ) = D ( A , A ∪ B ) and D ( A ∩ B , B ) ≤ D ( A , B ) ; ( 8 ) D ( A , A ∪ B ) ≤ D ( A ∩ B , A ∪ B ) . Proof. (1), (2) and (3) hold clearly.

(4) Since A ∩ C = { ( x k , min { μ A ( x k ) , μ C ( x k ) } , max { ν A ( x k ) , ν C ( x k ) } ) | x k ∈ X } , B ∩ C = { ( x k , min { μ B ( x k ) , μ C ( x k ) } , max { ν B ( x k ) , ν C ( x k ) } ) | x k ∈ X } . According to Lemma 1, then D ( A ∩ C , B ∩ C ) = 1 n ∑ k = 1 n ( 1 2 ( | μ A ∩ C ( x k ) - μ B ∩ C ( x k ) | + | ν A ∩ C ( x k ) - ν B ∩ C ( x k ) | ) ) = 1 n ∑ k = 1 n ( 1 2 | min { μ A ( x k ) , μ C ( x k ) } - min { μ B ( x k ) , μ C ( x k ) } | ) + 1 n ∑ k = 1 n ( 1 2 | max { ν A ( x k ) , ν C ( x k ) } - max { ν B ( x k ) , ν C ( x k ) } | ) ≤ 1 n ∑ k = 1 n ( 1 2 ( | μ A ( x k ) - μ B ( x k ) | + | ν A ( x k ) - ν B ( x k ) | ) ) = D ( A , B ) , which leads to D (A ∩ C, B ∩ C) ≤ D (A, B) and we can obtain D (A ∪ C, B ∪ C) ≤ D (A, B) by the similar method. (5) Given three IFSs A, B and C such that A ⊆ B ⊆ C, then μ _A (x _k ) ≤ μ _B (x _k ) ≤ μ _C (x _k ) and ν _A (x _k ) ≥ ν _B (x _k ) ≥ ν _C (x _k ) for all x _k ∈ X. Thus, D ( A , B ) = 1 n ∑ k = 1 n ( 1 2 ( | μ A ( x k ) - μ B ( x k ) | + | ν A ( x k ) - ν B ( x k ) | ) ) ≤ 1 n ∑ k = 1 n ( 1 2 ( | μ A ( x k ) - μ C ( x k ) | + | ν A ( x k ) - ν C ( x k ) | ) ) = D ( A , C ) . Similarly, we can also obtain D (B, C) ≤ D (A, C), which follows D (A, C) ≥ max {D (A, B), D (B, C)}. In addition, according to (4), (6), (7) and (8) hold.

In Property 3, it shows that the new definition of divergence measure of IFSs meets (D1)-(D4). Therefore based on the similarity and divergence measures, some relations of both measures are revealed as follows:

Theorem 1. Let A and B be two IFSs on X. Assume S (A, B) and D (A, B) are the similarity and divergence measures of A and B, respectively. Then we have:

( 1 ) 0 ≤ S ( A , B ) + D ( A , B ) ≤ 1 ; ( 2 ) S ( A , B ) = S ( B , A ) and D ( A , B ) = D ( B , A ) ; ( 3 ) If S ( A , B ) = 1 , then D ( A , B ) = 0 and vice versa ; ( 4 ) A = B ⇔ S ( A , B ) = 1 and D ( A , B ) = 0 . Proof. (1) Since S ( A , B ) + D ( A , B ) = 1 n ∑ k = 1 n ( 1 - 1 2 ( | μ A ( x k ) - μ B ( x k ) | + | ν A ( x k ) - ν B ( x k ) | + | π A ( x k ) - π B ( x k ) | ) + 1 n ∑ k = 1 n ( 1 2 ( | μ A ( x k ) - μ B ( x k ) | + | ν A ( x k ) - ν B ( x k ) | ) ) = 1 n ∑ k = 1 n ( 1 - 1 2 | π A ( x k ) - π B ( x k ) | ) ≤ 1 . Also S (A, B) + D (A, B) ≥0. Thus 0 ≤ S (A, B) + D (A, B) ≤1. It is clear that (2), (3) and (4) hold.

In Theorem 1, the relations between similarity and divergence measures of IFSs are characterized, which are of the practical semantics of similarity and divergence features and also coincides with human’s cognition. In what follows we construct a tuple called a closeness degree of IFSs.

Definition 7. Let A and B be two IFSs on X, and R (A, B) be a binary relation on X × X. We call R (A, B) a closeness degree between A and B, if R ( A , B ) = ( S R ( A , B ) , D R ( A , B ) ) .(15) where S _R (A, B) and D _R (A, B) are the similarity and divergence measures between A and B under the intuitionistic fuzzy relation R, respectively.

Theorem 2. The function R (A, B) in Definition 7 is an intuitionistic fuzzy similarity degree.

Proof. (1) Referring (R1)-(R4), let us first prove that R (A, B) is an IFN. It follows that 0 ≤ S _R (A, B) ≤1, 0 ≤ D _R (A, B) ≤1 and 0 ≤ S _R (A, B) + D _R (A, B) ≤1, respectively. Therefore R (A, B) is an intuitionistic fuzzy number. (2) R (A, B) = (1, 0) ⇔ A = B is immediate due to Theorem 1. (3) Obviously, S _R (A, B) = S _R (B, A) and D _R (A, B) = D _R (B, A). Thus R (A, B) = R (B, A). (4) If A ⊆ B ⊆ C, then S _R (A, C) ≤ S _R (A, B) and S _R (A, C) ≤ S _R (B, C) hold. Also it yields D _R (A, C) ≥ D _R (A, B) and D _R (A, C) ≥ D _R (B, C), thus R (A, C) ⊆ R (A, B) and R (A, C) ⊆ R (B, C).

It is found that the new definition of intuitionistic fuzzy similarity degree can deal with the shortcomings of “counterintuitive results” in some cases. For example, returning to the question in section 2: Let A₁ = {(x, 0.1, 0.9) |x ∈ X}, A₂ = {(x, 0.5, 0.3) |x ∈ X} and A₃ = {(x, 0.6, 0.3) |x ∈ X}, where X = {x₁}. We have A₁ ⊂ A₂ ⊂ A₃. Based on (112), it follows R (A₁, A₂) = (0.4, 0.5) and R (A₁, A₃) = (0.4, 0.55). From the semantics viewpoint of both degrees, the similarity degree between A₁ and A₂ is the same as the one between A₁ and A₃, while the divergence degree between A₁ and A₂ is lower than the one between A₁ and A₃. It leads to R (A₁, A₂) ⊃ R (A₁, A₃), which indicates that the closeness degree between A₁ and A₂ is bigger than the one between A₁ and A₃ under the condition of A₁ ⊂ A₂ ⊂ A₃, which is consistent with human’s intuition.

Definition 8. Let A = {A₁, A₂, …, A _m } be a set of m (m ≥ 2) IFSs and R be a binary relation. Let R (A _i , A _j ) = (S _R (A _i , A _j ), D _R (A _i , A _j )) = (S _{R ij} , D _{R ij} ) be an intuitionistic fuzzy similarity degree between A _i and A _j (1 ≤ i, j ≤ m). An intuitionistic fuzzy similarity matrix Z = (z _ij ) _m×m is defined as: Z = [ ( S R 11 , D R 11 ) ( S R 12 , D R 12 ) … ( S R 1 m , D R 1 m ) ( S R 21 , D R 21 ) ( S R 22 , D R 22 ) … ( S R 2 m , D R 2 m ) ⋮ ⋮ ⋮ ⋮ ( S R m 1 , D R m 1 ) ( S R m 2 , D R m 2 ) … ( S R mm , D R mm ) ] . From Definition 8, S _{R ij} can be understood as the degree of A _i similar to A _j and D _{R ij} is the degree of A _i dissimilar to A _j . Furthermore, we call π _{R ij} = 1 - S _{R ij} - D _{R ij} (i, j = 1, 2, …, m) the uncertainty of A _i similar to A _j . Compared with (4)-(10), the proposed approach to IFSM construction z _ij is based on the defined similarity and divergence measures of IFSs served as its membership and non-membership degrees, respectively, which is of practical implicatios and may be not provided with the features in (4)-(10). Considering its computational efforts and time, the proposed method takes a less computational time than some existing methods, which is illustrated in Example 2 in details.

Definition 9. Let A = {A₁, A₂, …, A _m } be a set of m (m ≥ 2) IFSs, R be an intuitionistic fuzzy similarity degree and Z = ((S _{R ij} , D _{R ij} )) _m×m be an intuitionistic fuzzy similarity matrix. For any α, β ∈ [0, 1] with α + β ≤ 1, the (α, β)-level cut-sets R α β of R is defined as: R α β = { ( A i , A j ) ∈ A × A | S R ij ≥ α ∧ D R ij ≤ β } ,(16) i , j = 1 , 2 , . . . , m , i . e . , R α β ( A i ) = { A j ∈ A | ( A i , A j ) ∈ R α β } . We further call R _α = {(A _i , A _j ) ∈ A × A|S _{R ij} ≥ α} and R ^β = {(A _i , A _j ) ∈ A × A|D _{R ij} ≤ β} the α-level cut-sets on similarity degree and β-level cut-sets on divergence degree between IFSs generated by R, respectively.

Property 4. Let R and G be two intuitionistic fuzzy similarity degrees on X. α and β are a pair thresholds, satisfying 0 ≤ α + β ≤ 1. Then

( 1 ) R α β = R α ∩ R β ; ( 2 ) If 0 ≤ α 1 ≤ α 2 ≤ 1 and 0 ≤ β 2 ≤ β 1 ≤ 1 , then R α 2 β 2 ⊆ R α 1 β 1 ; ( 3 ) If R ⊆ G , then R α β ⊆ G α β ; ( 4 ) ( R ∩ G ) α = R α ∩ G α , ( R ∩ G ) β = R β ∩ G β , ( R ∩ G ) α β = R α β ∩ G α β ; ( 5 ) ( R ∪ G ) α = R α ∪ G α , ( R ∪ G ) β = R β ∪ G β , ( R ∪ G ) α β ⊇ R α β ∪ G α β .

Proof. It is not difficult to prove that Property 4 holds based on related definitions.

In this section, we observe that an intuitionistic fuzzy similarity matrix is well constructed on the basis of the proposed similarity and divergence measure of IFSs, which is of the practical semantics of the similarity and divergence. Based on this matrix, the notion of (α, β)-level cut-sets of an intuitionistic fuzzy similarity degree and its similarity class are given so as to do clustering analysis directly.

This section aims to propose a novel method for intuitionistic fuzzy information clustering directly, which is based on the similarity class under intuitionistic fuzzy (α, β)-level cut-sets. Thus the detailed steps are as follows:

Step 1: Considering an analysis issue of decision making, assume A = {A₁, A₂, …, A _m } is a discrete set of m alternatives needed to be clustered, and X = {x₁, x₂, …, x _n } is a collection of n evaluation attributes. The description of each alternative A _i with respect to all attributes x _k (k = 1, 2, …, n) is expressed by IFSs as:

A i = { ( x k , μ A i ( x k ) , ν A i ( x k ) ) | x k ∈ X } , i = 1 , 2 , . . . , m ; k = 1 , 2 , . . . , n , where μ _{A i} (x _k ) is the membership degree of x _k to the alternative A _i , and ν _{A i} (x _k ) is the non-membership degree of x _k to the alternative A _i . In addition, π _{A i} (x _k ) =1 - μ _{A i} (x _k ) - ν _{A i} (x _k ) denotes the uncertainty of x _k to A _i .

Step 2: According to (114) and (115) to calculate the intuitionistic fuzzy similarity degree z _ij = (S _{R ij} , D _{R ij} ) (i, j = 1, 2, …, m) between A _i and A _j , where S _{R ij} and D _{R ij} can be understood respectively as the similarity and divergence degrees between them. S R ij = 1 n ∑ k = 1 n ( 1 - 1 2 ( | μ A i ( x k ) - μ A j ( x k ) | + | ν A i ( x k ) - ν A j ( x k ) | + | π A i ( x k ) - π A j ( x k ) | ) )(17) and D R ij = 1 n ∑ k = 1 n ( 1 2 ( | μ A i ( x k ) - μ A j ( x k ) | + | ν A i ( x k ) - ν A j ( x k ) | ) ) .(18) And then we obtain an intuitionistic fuzzy similarity matrix Z = (z _ij ) _m×m.

Step 3: Delete all elements above the diagonal of the matrix Z = (z _ij ) _m×m and add the symbol of the alternative A _i (i = 1, 2, …, m) to the elements on the diagonal, respectively. Thus, we require the corresponding matrix Z ˜ .

Step 4: Do clustering analysis, we first determine the thresholds α and β by Z ˜ as follows: