Novel similarity measures for T-spherical fuzzy sets and their applications in pattern recognition and clustering

1 Introduction

In various parts of life, in order to deal with various problems like machine learning, pattern recognition (PR), clustering, and image processing, sometimes it is necessary to compare things from a different point of view. To deal with the above said problems, there are many ways, but when the data is in fuzzy form, the method of similarity measures (SMs) is found outstanding. The under consideration article for the most part is related to the TSFSMs as it is the direct extension of fuzzy sets (FSs), intuitionistic fuzzy (IF) sets (IFSs) and picture fuzzy (PF) sets (PFSs). Therefore, it is appropriate to mention the pioneers and recent studies in terms of development and applications of FS, IFS, and PFS.

There is a plenty of unclear, ambiguous and unstable data in real life. To handle such situation Zadeh [1] introduced FS, in which each element of uncertainty is assigned with a membership grade (MG). But in real life the information is in multiple forms, so the only MG has some limitations to handle such type of information. At that stage FS has not been able to deliver. Also, when the information is clear, the complement of the MG is equal to the non-membership grade (NMG), but when there is hesitation in data the NMG is not the complement of the MG. In this situation the MG as well as NMG is needed. To overcome the situation, Atanassov [2] introduced the concept of IFS, in which each element is assigned with MG as well as NMG. Some prominent developments in these directions are done in [5–12].

In some conditions IFS has not been able to deliver. For example, if a person is given 0.6 MG and 0.5 NMG, in such situation IFS will not be able to manage it i.e. 0.6 + 0.5 = 1.1 ∉ [0, 1]. In that condition IFS has not kept in mind. In the same way, in real life matters some problems were faced where IFS also failed. To overcome the situation where IFS fails, Yager [3, 4] initiated the system of Pythagorean FSs [PyFSs], with the restriction 0 ⩽ MG² + NMG² ⩽ 1, by extending the space of IFSs.

Various persons used the PyFSs in various areas keeping in view their forceful structure. But in different conditions PyFS was not kept in mind. For example, in a situation when there are in excess of two free circumstances like in casting a ballot (choice of inclusion, restraint, resistance, and exclusion), IFSs neglected to depict the circumstance. Understanding this, Cuong [13, 14] built up a new direction known as PFS, which described the MG, hesitancy grade (HG) and NMG of an element or object in interval  [0, 1]. For some applicable work one may refer to [15–18].

PFSs extend the model of FSs and IFSs, but there is still limitation in the structure i.e. the domain is limited and no one is able to assign grades of feature functions. For example, if a person is given 0.6 MG, 0.3 HG and 0.4 NMG, in such situation PFS will not be able to manage it i.e. 0.6 + 0.3 + 0.4 = 1.3 ∉ [0, 1]. In that condition PFS has not kept in mind. In the same way, in real life matters some problems were faced where PFS failed to handle the situation. To overcome the situation where PFS fails, Mahmood et al.[19] proposed a new framework of spherical fuzzy (SF) set (SFS) to overcome the limitations of existing fuzzy structures. Further, in generalization of FSs, IFSs, PyFSs, PFSs and SFS a novel and most helpful tool in fuzziness was proposed by Mahmood et al. [20] known as TSFS which has no limitations at all. For some applicable work in this direction, we may refer to [21–28].

As research hotspots of PFSs, SM is needed to calculate the resemblance between two objects. Distance measures (DMs) and SM are necessary and are used to show differences between two matters. Recently, the examination of DMs and SMs between PFSs has offered a lot of achievements. Singh et al. established two distance measures with parameters, which contain normalized Hamming distance, normalized Euclidean distance and normalized Hausdorff distance as exceptional states [29]. Cantered on these distances, new SMs is developed and applied it to calculate flood disaster risk. Further a generalized DM is discussed in [30] and applied for clustering, pointing out the limitations, a new DM is developed [31] and used it for medical diagnostic problems. Some picture fuzzy weighted, ordered weighted and hybrid weighted DMs and their usage in MADM problems are discussed in [32]. A generalized Dice SM with parameter and its application to building material recognition is done in [33]. Recently Luo and Zhang proposed SMs [34] based on the constituent functions of a PFS, and applied it to pattern recognition. Grey, cosine and set theoretic SMs are discussed in [35, 36].

However, present SMs do not take into account the relationship between four functions of the PFS, which will lead to unreasonable results in some cases. At the end we concluded that, several studies explore the concepts of similarity, entropy, differential differences, and correlation in PFSs model. But, by above discussion some limitations have been seen in picture fuzzy model. Following this we aim to develop SMs defined in [28] in T-spherical fuzzy (TSF) environment. The current paper concerns with the investigation of TSF SMs (TSFSMs).

The main features of initiated investigation are:

Four fresh SMs are initiated along with their desired features.

Further, this article is separated into various sections. In section 2, a few essentials of the proposed work have been examined, in 3, new SMs are proposed along with numerical example, in 4, application of proposed MSs have been studied along with numerical examples in PR and clustering, in 5, a comparative study have been done with the advantage of proposed work.

2 Preliminaries

In this part, the notions discussed set up a ground for our work.

Definition 1. [2] An IFS is of the type I ={ x, t (x) , f (x) |x ∈ X }, where t (x) : X → [0, 1] and f (x) : X → [0, 1] denotes the MG and NMG of x respectively with a restriction 0 ⩽ t (x) + f (x) ⩽ 1 and r (x) = 1 - (t (x) + f (x)) is the refusal grade (RG) of x in I. Further, (t (x) , f (x)) is an IF number (IFN).

Because of the some limitations of the IFS, when we allocate the MbG and NMG, Yager [3, 4] initiated the system of PyFSs, extending the space of IFSs.

Definition 2. [3] A PyFS is of the type I ={ x, t (x) , f (x) |x ∈ X } where t (x) : X → [0, 1] and f (x) : X → [0, 1] denotes the MG and NMG of x respectively with a restriction 0 ⩽ t² (x) + f² (x) ⩽ 1 and r (x) = 1 - (t² (x) + f² (x)) is the RG of x in I. Further, (t (x) , f (x)) is a Pythagorean fuzzy number (PyFN).

PyFS, IFSs effectively improved FSs, however in such a situation when IFSs neglected to depict the circumstance. the new idea was proposed in [13, 14] known as PFS, which described the MbG, HG and NMG.

Definition 3. [13] A PFS is of the type I ={ x, t (x) , i (x) , f (x) |x ∈ X } where t (x) : X → [0, 1], i (x) : X → [0, 1] and f (x) : X → [0, 1] denotes the MG, HG and NMG of x respectively with a restriction 0 ⩽ t (x) + i (x) + f (x) ⩽ 1 and r (x) = 1 - (t (x) + i (x) + f (x)) is the RG of x in I. Further, (t (x) , i (x) , f (x)) is a PF number (PFN).

From now to onward we use t, i and f act as MG, HG and NMG respectively, and X act as whole set.

Definition 4. [28] For PFNs P = (t _P , i _P , f _P ) and Q = (t _Q , i _Q , f _Q ), the picture fuzzy similarity measures (PFSMs) can be defined as: S 1 ( P , Q ) = 1 m ∑ l = 1 m { 2 [ 1 - ( | t P ( x i ) - t Q ( x i ) | ∨ | i P ( x i ) - i Q ( x i ) | ∨ | f P ( x i ) - f Q ( x i ) | ) ] - 1 }(1) S 2 ( P , Q ) = 1 m ∑ l = 1 m { 2 [ 1 - 1 2 ( | t P ( x i ) - t Q ( x i ) | + | i P ( x i ) - i Q ( x i ) | + | f P ( x i ) - f Q ( x i ) | ) ] - 1 }(2) S 3 ( P , Q ) = 1 m ∑ l = 1 m { 2 [ 1 - ( | t P ( x i ) - t Q ( x i ) | ∨ | i P ( x i ) - i Q ( x i ) | ∨ | f P ( x i ) - f Q ( x i ) | ∨ | r P ( x i ) - r Q ( x i ) | ) ] - 1 }(3) S 4 ( P , Q ) = 1 m ∑ l = 1 m { 2 [ 1 - 1 2 ( | t P ( x i ) - t Q ( x i ) | + | i P ( x i ) - i Q ( x i ) | + | f P ( x i ) - f Q ( x i ) | + | r P ( x i ) - r Q ( x i ) | ) ] - 1 }(4)

Where ∨ is the maximum operator, r _P  (x _i ) = 1 - (t _P  (x _i ) + i _P  (x _i ) + f _P  (x _i )) and r _Q  (x _i ) = 1 - (t _Q  (x _i ) + i _Q  (x _i ) + f _Q  (x _i )) act as the RG of the element x _i  ∈ X in the set P and Q respectively.

Following are the some important properties of the S₁ (P, Q) , S₂ (P, Q) , S₃ (P, Q) , S₄ (P, Q).

Theorem 1. [28] The S₁ (P, Q) have the following properties:

0 ⩽ S₁ (P, Q) ⩽ 1

Theorem 2. [28] The S₂ (P, Q) have the following properties:

0 ⩽ S₂ (P, Q) ⩽ 1

Theorem 3. [28] The S₃ (P, Q) have the following properties:

0 ⩽ S₃ (P, Q) ⩽ 1

Theorem 4. [28] The S₄ (P, Q) have the following properties:

0 ⩽ S₄ (P, Q) ⩽ 1

S TS 1 ( P , Q ) = 1 m ∑ l = 1 m { 2 [ 1 - ( | t P n ( x i ) - t Q n ( x i ) | ∨ | i P n ( x i ) - i Q n ( x i ) | ∨ | f P n ( x i ) - f Q n ( x i ) | ) ] - 1 }(5)

To overcome the limitations of existing fuzzy structures Mahmood et al. [19] proposed a new framework of spherical fuzzy set (SFS). Further, in generalization of FSs, IFSs, PyFSs, PFSs and SFS a novel and most helpful tool in fuzziness was proposed by Mahmood et al. [20] known as TSFS which has no limitations at all.

Definition 5. [19] A SFS is of the type I ={ x, t (x) , i (x) , f (x) |x ∈ X } where t (x) : X → [0, 1], i (x) : X → [0, 1] and f (x) : X → [0, 1] denotes the MG, HG and NMG of x respectively with a restriction 0 ⩽ t² (x) + i² (x) + f² (x) ⩽ 1 and r ( x ) = 1 - ( t 2 ( x ) + i 2 ( x ) + f 2 ( x ) ) is the RG of x in X. Further, t² (x) , i² (x) , f² (x) is a SF number (SFN).

Definition 6. [20] A TSFS is of the type I ={ x, t (x) , i (x) , f (x) } where t (x) : X → [0, 1], i (x) : X → [0, 1] and f (x) : X → [0, 1] denotes the MG, HG and NMG of x respectively with a restriction 0 ⩽ t ⁿ  (x) + i ⁿ  (x) + f ⁿ  (x) ⩽ 1 for n ∈ ℤ and r ( x ) = 1 - ( t n ( x ) + i n ( x ) + f n ( x ) ) n is the RG of x in X. Further, (t ⁿ  (x) , i ⁿ  (x) , f ⁿ  (x)) is a TSF number (TSFN).

Remark 1. [20] Consider the Definition 5,

For n = 2, TSFS becomes SFS.

Hence, TSFS is generalized from all existing structures.

Since there exists limitations in PFSs and PFSMs as we are not independent to indicate the values in PFSs. To overcome the problem here some SMs are initiated in the context of TSFSs. Further, it is proved that the fresh SMs are the generalizations of existing (Eq . (1) -- (4)). It can also implement to the areas where current SMs are not kept to mind.

Definition 7. For TSFNs P = (t _P , i _P , f _P ) and Q = (t _Q , i _Q , f _Q ), the TSFSMs can be defined as:

For TSFNs P = (t _P , i _P , f _P ), Q = (t _Q , i _Q , f _Q ), and R = (t _R , i _R , f _R ), the S_TS1 (P, Q) has the below features:

0 ⩽ S_TS1 (P, Q) ⩽ 1

Proof.

It’s obvious.

Definition 8. For TSFNs P = (t _P , i _P , f _P ) and Q = (t _Q , i _Q , f _Q ), the TSF SMs can be defined as:

S TS 2 ( P , Q ) = 1 m ∑ l = 1 m { 2 [ 1 - 1 2 ( | t P n ( x i ) - t Q n ( x i ) | + | i P n ( x i ) - i Q n ( x i ) | + | f P n ( x i ) - f Q n ( x i ) | ) ] - 1 }(6)

For TSFNs P = (t _P , i _P , f _P ), Q = (t _Q , i _Q , f _Q ), and R = (t _R , i _R , f _R ), the S_TS2 (P, Q) have the following features:

0 ⩽ S_TS2 (P, Q) ⩽ 1

Definition 9. For TSFNs P = (t _P , i _P , f _P ) and Q = (t _Q , i _Q , f _Q ) the TSF SMs can be defined as:

S TS 3 ( P , Q ) = 1 m ∑ l = 1 m { 2 [ 1 - ( | t P n ( x i ) - t Q n ( x i ) | ∨ | i P n ( x i ) - i Q n ( x i ) | ∨ | f P n ( x i ) - f Q n ( x i ) | ∨ | r P n ( x i ) - r Q n ( x i ) | ) ] - 1 }(7) where, r n ( x i ) = 1 - ( t n + i n + f n ) n denotes the RG.

For TSFNs P = (t _P , i _P , f _P ), Q = (t _Q , i _Q , f _Q ), and R = (t _R , i _R , f _R ), the S_TS3 (P, Q) have the following features:

0 ⩽ S_TS3 (P, Q) ⩽ 1

Definition 10. For TSFNs P = (t _P , i _P , f _P ) and Q = (t _Q , i _Q , f _Q ), the TSF SMs can be defined as:

S TS 4 ( P , Q ) = 1 m ∑ l = 1 m { 2 [ 1 - 1 2 ( | t P n ( x i ) - t Q n ( x i ) | + | i P n ( x i ) - i Q n ( x i ) | + | f P n ( x i ) - f Q n ( x i ) | + | r P n ( x i ) - r Q n ( x i ) | ) ] - 1 }(8) where, r n ( x i ) = 1 - ( t n + i n + f n ) n denotes the RG.

For TSFNs P = (t _P , i _P , f _P ), Q = (t _Q , i _Q , f _Q ), and R = (t _R , i _R , f _R ), the S_TS4 (P, Q) have the features:

0 ⩽ S_TS4 (P, Q) ⩽ 1

Example 1. In Table 1 the information about the alternatives. We are to apply newly proposed SMs to find the highly similar data. Here we are given the data in the context of TSFNs. Let P₁, P₂ and P₃ be three TSFNs in domain X ={ x₁, x₂, x₃ } given in Table 1. We are to find highly similar to Q.

By applying S_TS1 (P, Q), S_TS2 (P, Q), S_TS3 (P, Q) and S_TS4 (P, Q) on TSFNs for n = 3 and presented in Table 2.

Here, we will discuss the application of fresh initiated SMs given in equation (5) - (8)in PR and clustering analysis.

4.3 Clustering analysis

Clustering analysis gives a set of objects; we create categories to categorize different groups, such as items in the same group are more similar than items in other groups. To analyse clusters in TSF context, the following algorithm is used.

Algorithm:

Step1: Consider {P₁, P₂, … , P _n  }a set of TSFNs in X. Develop a TSF- similarity matrix S = [s _ij ] _n×n, s _ij  = (P _i , P _j ) with the help of fresh SMs.

Step 2: Sketch the TSF-graph T = (E, V) with P _i  (i = 1, 2, …, n) as nodes and s _ij as edges.

Step 3: Sketch MST of the TSF-graph T = (E, V) using Prim (1957) method or Kruskal (1956) method.

Align s _ij decreasing order by their attributes.

Elect s _ij with top attributes.

Now, from remaining choose s _ij with maximum weight and do not form a circuit with pre-selected.

Repeat (c) till all s _ij selected. This is how we get the MST of the TSF-graph.

Step 4: By deleting all s _ij having weight less than α, cluster all the P _i into groups.

Example 3. Make P ={ P₁, P₂, P₃, P₄, P₅, P₆ } a set of vehicles classified on the basis of c₁ saifty, c₂ price, c₃ and comfort. The characteristics are given in the form of TSFNs in Table 5.

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Table 5

Car data

	c 1	c 2	c 3
P 1	(0.4, 0.3, 0.2)	(0.8, 0.1, 0.0)	(0.5, 0.1, 0.2)
P 2	(0.1, 0.3, 0.5)	(0.4, 0.3, 0.2)	(0.6, 0.1, 0.2)
P 3	(0.4, 0.4, 0.2)	(0.3, 0.6, 0.1)	(0.4, 0.3, 0.2)
P 4	(0.3, 0.3, 0.1)	(0.1, 0.0, 0.0)	(0.8, 0.0, 0.2)
P 5	(0.6, 0.2, 0.2)	(0.9, 0.1, 0.0)	(0.0, 0.1, 0.9)
P 6	(0.4, 0.1, 0.2)	(0.3, 0.1, 0.2)	(0.3, 0.3, 0.4)

By fresh SMs, S_TS1, S_TS2, S_TS3, S_TS4 we construct TSF-similarity matrix for n = 6.

M S TS 1 = ( 1.000 0.729 0.765 0.627 0.578 0.754 0.729 1.000 0.799 0.793 0.399 0.849 0.765 0.799 1.000 0.713 0.414 0.854 0.627 0.793 0.712 1.000 0.392 0.781 0.578 0.399 0.414 0.392 1.000 0.430 0.754 0.849 0.854 0.781 0.430 1.000 ) M S TS 2 = ( 1.000 0.931 0.929 0.887 0.816 0.938 0.931 1.000 0.967 0.947 0.753 0.983 0.929 0.967 1.000 0.930 0.755 0.987 0.887 0.947 0.930 1.000 0.717 0.940 0.816 0.753 0.755 0.717 1.000 0.767 0.938 0.983 0.987 0.940 0.767 1.000 ) M S TS 3 = ( 1.000 0.757 0.833 0.585 0.600 0.765 0.757 1.000 0.812 0.705 0.547 0.828 0.833 0.812 1.000 0.597 0.523 0.850 0.585 0.705 0.597 1.000 0.400 0.729 0.600 0.547 0.523 0.400 1.000 0.484 0.765 0.828 0.850 0.729 0.484 1.000 ) M S TS 4 = ( 1.000 0.818 0.860 0.678 0.682 0.821 0.818 1.000 0.878 0.800 0.591 0.896 0.860 0.878 1.000 0.703 0.582 0.909 0.678 0.800 0.703 1.000 0.508 0.805 0.682 0.591 0.582 0.508 1.000 0.534 0.821 0.896 0.909 0.805 0.534 1.000 )

Consider M _{S TS1} , we have construct the TSF-graph T = (E, V) as shown in Fig. 1. The MST of constructed TSF-graph is shown in Fig. 2.

By arranging the edges in descending order as: e 63 > e 62 > e 32 > e 42 > e 64 > e 31 > e 61 > e 21 > e 43 > e 41 > e 51 > e 65 > e 53 > e 52 > e 54 .

The edge with maximum weight is e₆₃ between P₆ and P₂.

From the rest we have to elect the edge with less weight i.e. e₆₂ between P₆ and P₂.

Repeat till c) till last edge, by this way we get the MST in Fig. 2 as:

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Fig. 1

TSF graph.

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Fig. 2

MST corresponding to S_TS1.

We group all the TSFSs into clusters for different values of α by selecting α and deleting edges that weight less.

In Tables 6, 8 and in Table 9, the clusters answering to M_STS1, M_STS2, M_STS3 and M_STS4 respectively are depicted that shows the results, such as objects in the same group are more similar than objects in other groups.

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Table 6

Clusters answering to M _{S TS1}

Class	Threshold (α)	Clusters
1	0.7809	{P1} , { P5 } , { P2, P3, P4, P6 }
2	0.7929	{P1} , { P5 } , { P2, P3, P4, P6 }
3	0.7995	{P1} , { P4 } , { P5 } , { P2, P3, P6 }
4	0.8493	{P1} , { P4, } , { P5 } , { P2, P3, P6 }
5	0.8538	{P1} , { P2 } , { P4, } , { P5 } , { P3, P6 }
6	1.0000	{P1} , { P2 } , { P3 } , { P4, } , { P5 } , { P6 }

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Table 7

Clusters answering to M _{S TS2}

Class	Threshold (α)	Clusters
1	0.9375	{P5} , { P1, P2, P3, P4, P6 }
2	0.9401	{P1} , { P5 } , { P2, P3, P4, P6 }
3	0.9672	{P1} , { P4 } , { P5 } , { P2, P3, P6 }
4	0.9828	{P1} , { P4, } , { P5 } , { P2, P3, P6 }
5	0.9867	{P1} , { P2 } , { P4, } , { P5 } , { P3, P6 }
6	1.0000	{P1} , { P2 } , { P3 } , { P4, } , { P5 } , { P6 }

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Table 8

Clusters answering to M _{S TS3}

Class	Threshold (α)	Clusters
1	0.7654	{P4} , { P5 } , { P1, P2, P3, P6 }
2	0.8118	{P1} , { P4, } , { P5 } , { P2, P3, P6 }
3	0.8283	{P1} , { P4, } , { P5 } , { P2, P3, P6 }
4	0.8328	{P1} , { P2 } , { P4, } , { P5 } , { P3, P6 }
5	0.8497	{P1} , { P2 } , { P4, } , { P5 } , { P3, P6 }
6	1.0000	{P1} , { P2 } , { P3 } , { P4, } , { P5 } , { P6 }

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Table 9

Clusters answering to M _{S TS4}

Class	Threshold (α)	Clusters
1	0.8209	{P4} , { P5 } , { P1, P2, P3, P6 }
2	0.8604	{P4} , { P5 } , { P1, P2, P3, P6 }
3	0.8780	{P1} , { P4, } , { P5 } , { P2, P3, P6 }
4	0.8957	{P1} , { P4, } , { P5 } , { P2, P3, P6 }
5	0.9093	{P1} , { P2 } , { P4, } , { P5 } , { P3, P6 }
6	1.0000	{P1} , { P2 } , { P3 } , { P4, } , { P5 } , { P6 }

The fresh SMs are the generalizations of SMs of proposed in [28], proved by the following remarks.

Remark 2. By placing n = 2, Eq. (6) reduces to SMs of SFSs and by placing n = 1, Eq. (6) reduces to SMs of PFSs [28]. Similarly, placement of i _P  = i _Q  = 0 and n = 1 reduces Eq. (6) to SMs for IFSs, showing that the fresh SMs are generalized.

Remark 3. By placing n = 2 Eq. (8) reduces to SMs of SFSs and by placing n = 1 Eq. (7) and (8) reduces to SMs of PFSs [28]. Similarly, placement of i _P  = i _Q  = r _P  = r _Q  = 0 and n = 1 reduces Eq. (8) SMs of IFSs, showing that the fresh SMs are generalized.

For more comparative study, considering the example 1, from Table 2 it is clear that the sum of all grades of the data given in Table 10 exceeds from the unit interval [0, 1] for n = 1, so the information is not in picture fuzzy form. By observing Table 11 it is clear that the sum of all grades of the data given in Table 1 is from the unit interval [0, 1] for n = 2, so the information is in spherical fuzzy and from Table 12 it is observed that the data is in T-spherical fuzzy form for n = 3. From all above discussion it is clear that TSFSMs also deals with the data that is not in the form of PFNs.

Further, the main features of the SMs proposed in the context of TSFSs are that these SMs can cope with the information given in [28].

This composition depicted the foundation of IFSs, PyFSs and PFSs in detail for noticing the restricted idea of their constructions. It is examined how the deficiencies are improved by utilizing the system of SFSs and TSFSs with the assistance of mathematical models. At that point, some similar proportions of IFSs and PFSs have been noticed and their constraints were examined. To sum up these similitude gauges, some new SMs were created. The speculation of new SMs was talked about with the assistance of a few comments and models showing its assorted construction. The new SMs of TSFSs was applied to pattern recognition and clustering, the data was given in type of PFNs and the outcomes got was discovered to be like already existing outcomes. Another issue in the climate of TSFSs was tackled utilizing new SMs and it was examined that the existing SMs could not deal with this kind of information. The possibility of SFSs and TSFSs is novel furthermore; some different devices of likeness and distance measures, relationship coefficients could be created in not so distant future. Further, some collection administrators could be produced for TSFSs and applied in dynamic issues, bunching issues, and so forth.

Authors–contributions

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Declarations conflict of interest

The authors announce, no conflicts of interest are there.