Correlation Coefficients for Cubic Bipolar Fuzzy Sets With Applications to Pattern Recognition and Clustering Analysis

Cubic bipolar fuzzy set (CBFS) is a powerful model for dealing with bipolarity and vagueness altogether because it contains bipolar fuzzy information and interval-valued bipolar fuzzy information simultaneously. In this article, we define some new notions such as concentration, dilation, support and core of a CBFS. We introduce cubic bipolar fuzzy relations (CBFRs) and some of their types. As in statistics with real variables, we define variance and covariance between two CBFSs. Then, we propose correlation coefficients and their weighted extensions on the basis of variance and covariance of CBFSs. Later on, some properties of these correlation coefficients are discussed. We explore that their values lie in [−1,1]. Moreover, we discuss the applications of the proposed correlation coefficients in pattern recognition and clustering analysis. Numerical examples are provided for better understanding of the applicability and efficiency of proposed correlation coefficients.


I. INTRODUCTION
Zadeh [1] initiated the concept of fuzzy set (FS) theory which is a generalization of crisp set theory. This theory was developed to address the inexactness and uncertainty that arise in decision-making problems as a result of ambiguities in data and human judgments. Till now, this theory has been successfully applied in various fields including medical science, economics, computer science, physics, and chemistry. Later on, many researchers inaugurated different extensions of fuzzy sets such as interval-valued fuzzy set(IVFS) [2], intuitionistic fuzzy set (IFS) [3], hesitant fuzzy set (HFS) [26], pythagorean fuzzy set (PFS) [5], [6], q-rung orthopair fuzzy set(q-ROPFS) [7], neutrosophic set (NS) [27], and single-valued neutrosophic set (SVNS) [28], etc.
In many real life circumstances, human perception is based on bipolar or dual-sided thoughts. For instance, effects The associate editor coordinating the review of this manuscript and approving it for publication was Paolo Remagnino . and side effects, friendship and enmity, profit and loss are some examples of two-sided features of decision analysis. Zhang [8], [9] proposed the idea of bipolar fuzzy set (BFS) which unifies bipolarity and fuzziness. This set assigns each element a positive membership degree from [0,1] and a negative membership degree from [-1,0]. These degrees indicate the extent to which an element satisfies a property as well as its counter-property. Lee [10] gave the operations on bipolar-valued fuzzy sets. Lee and Hur [43] defined bipolar fuzzy relations. Wei et al. [11] introduced the idea of interval-valued bipolar fuzzy set(IVBFS) and discussed multi-criteria decision-making (MCDM) under interval-valued bipolar fuzzy information. Deli et al. [29] gave the abstraction of bipolar neutrosophic set (BNS) and applied it in MCDM problems. Ulucay et al. [30] explored some similarity measures on BNSs and applied them in multi-criteria decision making. Basset et al. [31] defined cosine similarity measures on BNSs and used them in the diagnosis of bipolar disorders. VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Jun et al. [4] brought out the concept of cubic set (CS) (hybrid of IVFS and FS) and defined the notions of P-union, P-intersection, R-union and R-intersection. They also defined internal cubic sets (ICSs) and external cubic sets (ECSs) and derived some useful results by taking into consideration both the ICSs and ECSs.
Correlation coefficient, an important notion in statistics, measures the linear relationship between two random variables. It is widely used in statistical analysis and engineering sciences. Since crisp set theory cannot tackle the ambiguities and uncertainties, therefore, the idea of correlation coefficient has been extended to FS theory for better results [45]. Later on, Gerstenkorn and Manko proposed a correlation coefficient for intuitionistic fuzzy sets whose values lie in [0,1]. Hung [33] adopted the statistical viewpoint to define a correlation coefficient for IFSs. Garg [16] proposed novel correlation coefficients for Pythagorean fuzzy sets and applied them in pattern recognition and medical diagnosis. Garg and Kaur [19] developed correlation coefficients for cubic intuitionistic fuzzy environment and discussed MCDM problems. Pramanik et al. [35] proposed a novel correlation coefficient for interval-valued bipolar neutrosophic set and applied it in multi-attribute decision making (MADM) problem. For more about pattern recognition and MADM, we refer to [36]- [40]. Riaz and Tehrim [14], Peng et al. [15], Basset et al. [31] initiated a novel model named as cubic bipolar fuzzy set (CBFS) which is a hybrid set of BFS and IVBFS. This model gives more precision and pliability as compared to the existing models because it accommodates bipolar and interval-valued bipolar fuzzy information simultaneously. As a result, this model offers maximum details about the occurrence of ratings, inexactness and bipolarity. They proposed some aggregation operators like cubic bipolar fuzzy weighted geometric (CBFWG) aggregation operators and cubic bipolar fuzzy weighted averaging (CBFWA) aggregation operators under P(R)-order and applied these operators in some multi-criteria group decision making (MCGDM) problems.
The main objectives and advantages of this article are listed below: 1) The first objective of this article is to handle vagueness and ambiguities more efficiently with the help of cubic bipolar fuzzy sets (CBFSs). 2) The second objective is to define new notions like concentration, dilation, support, core and binary relations for CBFSs. 3) The third objective is to develop correlation coefficients and their weighted versions for CBFSs. 4) The fourth objective is to propose new algorithms with the help of suggested correlation coefficients to solve complex problems of pattern recognition and clustering analysis under CBF environment. The usability and effectiveness of these algorithms is determined by numerical illustrations. The rest of the article is structured as follows: In section 2, we review some basic definitions of fuzzy sets, interval-valued fuzzy sets, bipolar fuzzy sets, interval-valued bipolar fuzzy sets. Then, we recall the definition and operations of CBFSs. In section 3, we propose concentration, dilation, support and core of a CBFS. Moreover, we inaugurate cubic bipolar fuzzy relations and some of their types. In section 4, we propose correlation measures and their weighted extensions and discuss their properties. In section 5, novel algorithms are presented for pattern recognition and clustering analysis on the basis of suggested correlation coefficients and the applicability of these algorithms is substantiated through numerical illustrations. Section 6 contains concluding remarks.

II. PRELIMINARIES
We devote this section to discuss some fundamental concepts related to cubic bipolar fuzzy sets that are helpful throughout this article.
Definition 1: [1] Let M be an initial universe. A fuzzy set F on M is defined as Definition 6: [13] A cubic bipolar fuzzy set A over the initial universe M can be defined as where S is an IVBFS and B is a BFS on M. Thus, cubic bipolar fuzzy set can also be written as : m ∈ M} be two CBFSs on M and λ > 0. Then, the operations on CBFSs under P-order are given below: Definition 8: (m)} : m ∈ M} be two CBFSs on M and λ > 0. Then, the operations on CBFSs under R-order are given below: Definition 9:

III. SOME NOVEL CONCEPTS OF CBFSs
In this section, we propose some important concepts including concentration, dilation, support and core of a cubic bipolar fuzzy set. Moreover, we introduce cubic bipolar fuzzy relations and discuss some of their types. Definition 10: initial universe M. Referring to definition 7(part v) and taking λ = 2 in it, we obtain A (m))) 2 } : m ∈ M Then, A 2 is termed as P-concentration of A and denoted by CON P (A).  be a CBFS on M. Then, P-concentration of A is given as follows Example 2: For the CBFS A given in example 1, the Rconcentration is given as follows 2 is called P-dilation of A and denoted by DIL P (A). Example 3: Consider the same CBFS A given in example 1. Then, P-dilation of A is given as follows Example 4: The R-dilation of CBFS A taken from example 1 is given as follows The support of A, denoted by Supp (A), can be defined by taking into consideration both the positive and negative membership degrees separately. Therefore, it is expressed as union of positive support (Supp + (A)) and negative support (Supp − (A)), i.e., , 0} for all m ∈ M is called null cubic bipolar fuzzy set. It is denoted by .
Definition 17: Let M be initial universe. A cubic bipo- for all m ∈ M is called absolute cubic bipolar fuzzy set. It is denoted byÃ.
Remark 1: The two well-known laws of crisp set theory, named as law of contradiction and law of excluded middle, do not hold in cubic bipolar fuzzy theory. That is, If we replace P-order by R-order in the above example, we can see that law of contradiction and excluded middle still do not hold.
. Example 9: Consider the CBFR given in example 8. Then, R −1 is given as in Table 1.
Definition 23: A CBFR R ∈ CBFR(M) is called an equivalence relation on M if it is reflexive, symmetric and transitive. Example 10: A CBFR on M = {m 1 , m 2 , m 3 } is given in Table 1. It is easy to check that this CBFR is an equivalence relation.

IV. CORRELATION COEFFICIENTS OF CBFSs
In this section, we propose some correlation coefficients for any two cubic bipolar fuzzy sets (CBFSs), which determine the strength of relationship between them. Moreover, an advanced feature of these correlation coefficients is their ability to determine whether two CBFSs are positively or negatively correlated.
be two CBFNs, then the deviation ofα andβ can be computed by using the following expressioñ The deviation consists of two parts: the deviation of positive membership degrees and that of negative membership degrees. Since, cubic bipolar fuzzy information deals with a property as well as its counter-property so it sounds reasonable to add up the deviation of positive and negative membership degrees.
The deviation of CBFNs satisfies the following properties The mean value of A is given by Let M = {m 1 , m 2 , . . . , m n } be initial universe and A = The variance of A can be defined as 2 (1) The covariance between A and B is given by Clearly, covariance satisfies the following properties: Definition 28: For two CBFSs A and B, the correlation coefficient is defined by Theorem 4: Let A and B be two CBFSs on initial universe M = {m 1 , m 2 , . . . , m n }. Then, the correlation coefficient given in Eq.(3) satisfies the following conditions: Proof: (i) Straightforward. (ii) To prove this, we utilize Cauchy-Schwarz inequality which states that ( (iv) For a CBFS A, the complement of A is given by , as shown at the bottom of the page. Thus, Proof: We omit the proof. In various multi-attribute decision making (MADM) scenarios, different attributes are assigned different weights by the decision experts. Therefore, weights of the elements m i ∈ M (i = 1, 2, . . . , n) should be taken into consideration. For this purpose, the above-defined correlation coefficients θ 1 (A, B) and θ 2 (A, B) can be extended to weighted correlation coefficients as follows: Definition 30: Let w = (w 1 , w 2 , . . . , w n ) be the weight vector of the elements m i ∈ M (i = 1, 2, . . . , n) with the conditions that w i ≥ 0 and n i=1 w i = 1. Then, for any two CBFSs A and B on M, the weighted correlation coefficient is defined by where Proof: We only prove (ii) and the remaining parts are straightforward.
(ii) By using Cauchy-Schwarz inequality, we have Step 1. Consider some known patterns L 1 , L 2 , . . . , L m in the form of CBFSs in a finite initial universe M.
Step 2. Construct an unknown pattern Q in the form of CBFS in M, this pattern is to be recognized.
Step 3. Find the correlation coefficients of Q and L j (j = 1, 2, . . . , m) by using Eqs. (3)or (4). If the elements of the initial universe M own some weights, then weighted correlation coefficients given in Eqs. (5) or (6) can be utilized.
Step 4. The pattern Q belongs to the pattern L j for which the value of correlation coefficient is maximum.

V. APPLICATIONS IN PATTERN RECOGNITION AND CLUSTERING ANALYSIS
This section provides applications of our proposed correlation coefficients to pattern recognition and clustering analysis under cubic bipolar fuzzy environment.

A. PATTERN RECOGNITION
Pattern recognition is a data analysis technique that employs machine learning algorithms to identify patterns and regularities in data. Pattern recognition has a broad range of applications, including image processing, aerial photo interpretation, speech and fingerprint recognition, optical character recognition in scanned documents such as contracts and photographs, and even medical imaging and diagnosis.
To identify an unknown pattern from the known ones under cubic bipolar fuzzy data, we adopt the following steps: The flow diagram of the proposed algorithm 1 is presented in Figure 1. We want to know which pattern Q belongs to? For this purpose, we determine the correlation coefficients of Q and L j , j = 1, 2, 3 by using Eqs. (3) and (4). The results are presented in Table 2. It is evident from the above table that Q belongs to the pattern L 3 . Now, suppose that the elements of M carry weights and their weights are 0.37, 0.28 and 0.35, respectively. Then, the weighted correlation coefficients by using Eqs. (5) and (6) can be calculated between unknown and known patterns. The results are summarized in Table 3. Again, the pattern Q belongs to the pattern L 3 .

B. CLUSTERING ANALYSIS
Clustering refers to a process that divides a set of data points into clusters such that the data points in the same cluster have more similar traits than those in different clusters. In what follows, we propose a novel clustering algorithm under cubic bipolar fuzzy environment. Before this, we discuss some basic ideas.
where w i , i = 1, 2, . . . , n are the weights assigned to the attributes by decision experts.
Step 2. Find F 2 and check whether F 2 ⊆ F. If this holds, then F is the equivalent correlation matrix, otherwise, construct the equivalent correlation matrix F 2 κ : Step 3. For a confidence level ρ, find a ρ-cutting matrix F ρ = (f The flow chart diagram of the proposed algorithm 2 is given in Figure 2.

1) NUMERICAL EXAMPLE
A robot is a re programmable multi-functional manipulator that can move material, parts, equipment, or specialized devices using variable programmed motion to perform a wide range of tasks. Due to recent development in information technology and engineering sciences, the utilization of robots has been increased in different advanced manufacturing systems. Robots are capable of performing repetitive, challenging, and dangerous tasks with great accuracy. Therefore, a variety of industrial applications such as automated assembly, material handling, machine loading, spray painting and welding, are proficiently performed by the robots.
To demonstrate the differences of the opinions of different experts, we present the evaluation information in the form of CBFSs which is given in Table 4. Now, we apply the clustering algorithm to cluster the robots.
Step 1: Calculate the correlation coefficients of L j (j = 1, 2, . . . , 5) by using Eq.(5) and construct the correlation matrix:   Step 2: Calculate Since F 2 F, so we construct the equivalent correlation matrix by calculating further compositions as follows: Step 3: For a confidence level ρ, apply the definition 35 on the entries of equivalent correlation matrix F 16 to obtain a ρ-cutting matrix. It is obvious that different values of ρ will give different ρ-cutting matrices.
Step 4: We discuss a sensitivity analysis on the basis of different values of confidence level ρ, and acquire all possible clusters of the 5 robots. The results are summarized in Table 5.

VI. CONCLUSION
CBFS is a generalization of bipolar fuzzy set which handles the two-sided approach of decision analysis and inexactness of the data by taking into consideration both IVBFS and BFS simultaneously. In this article, we defined some well-known terminologies for CBFSs which include concentration, dilation, support and core of a CBFS. We developed cubic bipolar fuzzy relations along with their certain types. In statistics, correlation coefficient examines the strength of relationship between two variables. It also determines whether two variables tend to behave in the same manner (positive correlation) or in the opposite manner (negative correlation). The correlation coefficients for CBFSs should also possess these characteristics. Therefore, as in statistics with real variables, we defined variance and covariance for two CBFSs and then, developed correlation coefficients and their weighted extensions on the basis of variance and covariance of CBFSs. These correlation coefficients range in [-1,1] which make them superior to some of the existing correlation coefficients that lie in [0,1]. We investigated some properties of these correlation coefficients. We applied these correlation coefficients to pattern recognition and clustering analysis and demonstrated their usability and effectiveness through numerical illustrations. In the future, we will study distance and similarity measures [36] and knowledge measures [40] for CBFS.