Novel concepts of m -polar spherical fuzzy sets and new correlation measures with application to pattern recognition and medical diagnosis

: In this paper, we introduce the notion of m -polar spherical fuzzy set ( m -PSFS) which is a hybrid notion of m -polar fuzzy set ( m -PFS) and spherical fuzzy set (SFS). The purpose of this hybrid structure is to express multipolar information in spherical fuzzy environment. An m -PSFS is a new approach towards computational intelligence and multi-criteria decision-making (MCDM) problems. We introduce the novel concepts of correlation measures and weighted correlation measures of m - PSFSs based on statistical notions of covariances and variances. Correlation measures estimate the linear relationship of the two quantitative objects. A correlation may be positive or negative depending on the direction of the relation between two objects and its value lies the interval [ − 1 , 1]. The same concept is carried out towards m -polar spherical fuzzy ( m -PSF) information. We investigate certain properties of covariances and the correlation measures to analyze that these concepts are extension of crisp correlation measures. The main advantage of proposed correlation measures is that these notions deal with uncertainty in the real-life problems e ﬃ ciently with the help of m -PSF information. We discuss applications of m -polar spherical fuzzy sets and their correlation measures in pattern recognition and medical diagnosis. To discuss the superiority and e ﬃ ciency of proposed correlation measures, we give a comparison analysis of proposed concepts with some existing concepts.


Introduction
Crisp set theory and logic are key factors in the foundation of science and engineering. A crisp set describes the belongingness of an element in a digital manner, i.e., an element either belongs to a analytic hierarchy process. Gundogdu et al. [23] presented the analysis of usability test parameters affecting the mobile application designs by using spherical fuzzy sets. Shishavan et al. [24] proposed novel similarity measures in spherical fuzzy environment and their applications. Rafiq et al. [25] established new cosine similarity measures of spherical fuzzy sets and their applications in decisionmaking. Deli and Cagman [26] introduced spherical fuzzy numbers and spherical triangular fuzzy numbers and MCDM applications.
Garg and Arora [27] proposed extended TOPSIS method based on correlation coefficient for MCDM with IFSS information. Ashraf and Abdullah [28] also independently proposed the idea of spherical fuzzy set. Mahmood et al. [29] introduced some elementary operations on SFSs. Sitara et al. [30] decision-making analysis based on q-rung picture fuzzy graph structures. Akram et al. [31] proposed a hybrid decision-making framework under complex spherical fuzzy prioritized weighted aggregation operators. Recently, Akram et al. [32] introduced Extensions of Dombi aggregation operators with MCDM applications. Riaz and Hashmi [33] introduced a new extension of fuzzy sets named as linear Diophantine fuzzy set (LDFS). Kamaci [34] presented linear Diophantine fuzzy algebraic structures and their application to coding theory. Ayub et al. [35] introduced linear Diophantine fuzzy relations and their algebraic structures with decision making application. Shaheen et al. [36] analyzed risk analysis with generalized hesitant fuzzy rough sets (GHFRS).
Zhang [37] introduced bipolar fuzzy set (BFS) as an extension of FS. This set is used to deal with the data which is bipolar in nature that describes information and with its counter property. Lee [38] introduced bipolar-valued fuzzy sets and their elementary operations. Chen et al. [39] introduced mpolar fuzzy sets (m-PFS) as an extension of bipolar fuzzy sets. An m-PFS deals with multipolarity by assigning m degrees to each element of a crisp set. Recently, some hybrid structures of m-PFSs have been introduced for modeling uncertainties in multi-criteria decision-making (MCDM) problems. Naeem et al. [40] introduced the notion of Pythagorean m-polar fuzzy set (P-m-PFS) as a hybrid model of PFS and m-PFS. Riaz et al. [41] proposed Pythagorean m-polar fuzzy soft set (P-m-PFSS) as a hybrid model of PFS, m-PFS, and SS.
Correlation is a property that evaluates the mutual or reciprocal relation between two objects in connection to each other, while similarity measurement tests the proximity of two objects. In engineering and statistics, correlation have significant applications. Statistics have used probability approaches successfully to deal with many real space problems that rely on random data analysis. Similarly, the same probability theory has also tackled several mathematical issues in the scientific fields. These techniques, though, have certain inconveniences, as these procedures are dependent on a random choice of samples with a certain degree of confidence. If we gather data on a vast scale, it involves massive fluidity, which makes it impossible to refine idealism. Highlighting these challenges is how to achieve idealistic performance. A big question? It is very difficult to address this query. The first thing to note is that confusion still takes a position in every real situation, regardless of the technicality, medical or administrative problem. Correlation in statistics is one of our most essential topics of different subjects like business, finance, engineering, management, etc. The dilemma, though, is that this kind of correlation can only address the crisp data and we have resolved several real issues without considering complexity. Many ambiguous structures have been added to deal with uncertainty since the birth of the fuzzy set theory. Many extensions of fuzzy sets have been developed to express vague information until now. Consequently, correlation measures must be introduced in almost all extensions of fuzzy sets. Some correlation measures for IFSs was proposed by Ejegwa et. al. [42], Xu et. al. [43], Gerstenkorn and Manko [44], and Szmidt and Kacprzyk [45]. Later, Garg [46] suggested correlation measures in PFS environment. Liu et. al. [47] and Thao [48] observed the restrictions of the earlier defined correlations and proposed some new directional correlations for PFS environment. Correlation measures studied by Mahmood and Ali [49] to define an extended TOPSIS method based on PFS environment. Zulqarnain et al. [50] defined correlated TOPSIS for mask selection procedure. Joshi [51] defined some weighted correlations for MCDM. 1. In many real-life problems, we have to deal with the hybrid situations where multi-polarity of each of three index (membership, non-membership and neutral-membership degree) of the alternatives are required but with the help of existing models we can't handle this situations. To deal with this type of problems, we introduce the hybrid structure of m-polar fuzzy set and spherical fuzzy set named as m-polar spherical fuzzy set (m-PSFS). We introduce the idea of m-polar spherical fuzzy numbers (m-PSFNs). These concepts are more efficient to express vague and uncertain information in a realistic manner. 2. We introduce some basic operational laws of m-PSFSs and m-PSFNs. Based on these operational laws, we analyze certain properties of m-PSFSs and m-PSFNs. 3. We define the score function and certainty functions for comparison of mPSFNs and for finding the ranking of feasible alternatives. 4. We introduce new correlation measures and weighted correlation measures for m-PSFSs. 5. We present a MCDM method based on these correlation measures for medical diagnosis and pattern recognition.
This article is arranged as follows. In Section 2, we discuss basic concepts of m-PFS and SFS. In Section 3, the notion of m-PSFS is introduced which is a robust fusion of m-PFS and SFS. Some basic operations on m-PSFSs are introduced to analyze key properties of m-PSFSs. We also defined several novel features of m-PSFSs. In Section 4, the correlation measures and weighted correlation measures for m-PSFSs are proposed. The proposed correlation measures have their value in the interval [−1, 1]. In Section 5, we give applications of proposed correlation measures for m-PSFSs to pattern recognition and medical diagnosis. Lastly, the conclusion od current research work is summarized in Section 6.

m-Polar spherical fuzzy set
In this section, first we introduce the notion of m-PSFS which is a robust fusion of m-PFS and SFS. Then we define some fundamental operations on m-PSFSs and their related key properties. Several novel features of m-PSFSs are also investigated.
It is interesting to note that information given by hesitancy part of m-PSFS is uncertain. It does not give exact information as given by membership and non-membership grades. Thus the h-complement of m-PSFS is denoted as C * and defined as Definition 3.6. The union of two m-PSFSs S 1 and S 2 defined over the same universal set X is defined as Some novel features of m-polar spherical fuzzy set be two m-PSFNs and λ > 0, then Proof. The proof is obvious.
and λ > 0, then following results hold The rest of 7 possibilities may be discussed similarly.
3 By definition Thus 4 can be proved analogously.
The rest of possibilities may be considered accordingly. 3 The rest of possibilities may be considered accordingly.

Correlation coefficient for m-PSFSs
The correlation coefficient is a statistical analysis of how strong an association exists between two variables' variations. The scope of values is -1.0 to 1.0. There was an error in the correlation calculation if the computed number was greater than 1.0 or less than -1.0. A negative correlation of -1.0 indicates a complete negative correlation, while a positive correlation of 1.0 indicates a complete positive correlation. A correlation of 0.0 means that the changes of the two variables have no linear relationship. In this section, we define correlation coefficient in m-PSFSs. We analyze their usefulness and address some of their characteristics.
: ς ∈ X be a m-PSFS over a universal set X.
Then the average or mean of S is defined as If we take X={ς 1 , ς 2 , ς 3 , ς 4 , ς 5 } and 3PSFSs as then the average or mean of S is given by , where be m-PSFSs over X. Then covariance of S A and S B can be defined as Where, for j=1,2,3,...,m. Where, for j=1,2,3,...,m.
Proposition 4.4. The covariance and variance for m-PSFSs of S A and S B follow the following properties.
Proof. (a) and (b) are simple to prove. (c) We will use Cauchy-Schwarz inequality to prove our result. Cauchy-Schwarz inequality states that The other two are simple to prove.
Example 4.6. Let Table 2 Proof. (a) It is easy to prove.
Definition 4.8. Let S A and S B be m-PSFSs over X as mentioned in definition 4.2. Then correlation coefficient of S A and S B can also be defined as Where, Proof. (a) It is easy to prove.
Two main aspects of most real-world issues should be addressed: complexity and weights. Uncertainty is a major factor that can influence our decisions and analyses when dealing with such issues. Furthermore, weights have an effect on the majority of unknown mathematical models. We describe weighted Correlation coefficients (WCCs) to deal with such issues. Definition 4.10. Let S A and S B be m-PSFSs defined over X as mentioned in definition 4.2. The alternatives are effected by ς 1 , ς 2 , · · · , ς n are affected by the weights ω 1 , ω 2 , · · · , ω n with the condition n i=1 ω i = 1 and 0 ≤ ω i ≤ 1. Then weighted covariance of S A and S B can be defined as Where, for j=1,2,3,...,m.
Definition 4.11. Let S A be a m-PSFS on X as mentioned in definition 4.2. Then weighted variance of S A can be defined as for j=1,2,3,...,m.
Proposition 4.12. The covariance and variance for m-PSFSs of S A and S B follow the following properties.
Proof. (a) and (b) are simple to prove. (c) We will use Cauchy-Schwarz inequality to prove our result. Cauchy-Schwarz inequality states that The other two are simple to prove. Definition 4.13. Let S A and S B be m-PSFSs over X as mentioned in definition 4.2. Then weighted correlation coefficient of S A and S B can be defined by Where, for j=1,2,3,...,m.
Theorem 4.14. For the m-PSFSs S A and S B , we have (a) The weighted correlation ω ρ is symmetrical, that is, ω ρ(S A , S B ) = ω ρ(S B , S A ) (b) The weighted correlation ω ρ lies between −1 and 1, that is, Proof. (a) It is easy to prove.
Definition 4.15. Let S A and S B be m-PSFSs over X as mentioned in definition 4.2. Then weighted correlation coefficient of S A and S B can also be defined as and ω ( j) Theorem 4.16. For the m-PSFSs S A and S B , we have (a) The correlation ω is symmetrical, that is, ω (S A , S B ) = ω (S B , S A ) (b) The correlation lies between −1 and 1, that is, Proof. (a) It is easy to prove.
The weights ( 1 n , 1 n , · · · , 1 n ) is changeless in correlation. So, once such weights occur, we can just ignore them and proceed unweightedly. In other words, the weighted alternatives can be considered uneven. We may observe that and

Applications of proposed correlation measures
In statistics and engineering the correlation plays a major role. The cumulative relation between two variables can be analysed using the level of stability of the two variables by means of correlation analysis. Correlation actions are prevalent indicators and appropriate methodologies to MCDM problems. Several intellectuals have used correlations in different uncertain conditions to solve these problems quickly and productively. The correlations defined in this article will be used in this section to precisely recognize patterns and medical diagnosis.
Pattern recognition is automatic identification of collection and aggregation in statistics. It is usually classified based on the type of learning experience for producing the value of output. Pattern recognition methods usually help to identify all the independent variables moderately and to make "most likely" the insights to match, taking their statistical variance into consideration. Pattern recognition is usually divided into the classifications of learning process of creating new value of the output.

Application to pattern recognition
Now we develop Algorithm 1 for pattern recognition with the help of proposed correlation based on m-PSF information.

Algorithm 1 (Pattern Recognition)
Step 1: Read the pattern to be recognized as P and express in the form of m-PSFS.
Step 3 (ii): If the universe X for the m-PSF model contains weighted alternatives, find the respective weighted correlation coefficients using the formula (4.3) or (4.4).
Step 4: Pattern P belongs to P (i) , if any pattern P (i) finds the greatest correlation with P.
Example 5.1. Pattern recognition is one of the most important and fundamental concern in robotics, machine learning and security management. We consider a perticular problem of pattern recognition, namely, facial recognition. Facial recognition is a mechanism through which an individual's identification can be confirmed by his face. In photos, videos and even in real time, facial recognition systems can be used to identify people. One of the main category of biometric safety is facial identification. Suppose a machine, which is already feeded with three patterns P (1) , P (2) and P (3) of three different persons in the form of m-PSFSs, has to detect a face of an unknown person. (We are considering only three persons in this example because our aim is to give a mathematical model only. However, a certain machine can be feeded with hundreds and thousands of the patterns of different persons at a time). The three patterns are expressed in in terms of 3-PSFS in Table 4, Table 5, Table 6, respectively. The machine first detects the face of an unknown person, analyzes it and converts it into a m-PSF model, say P, which is given in Table 7 as follows. It, then, finds the correlations of the unknown pattern from the known ones and decides that which pattern is closest to the unknown pattern. This whole facial recognition process can be performed through the following Algorithm 1, that is, the machine can be feeded with the proposed algorithm in order to detect the person.
All of the above results show that the pattern P finds the greatest correlation from P (3) . Therefore, the facial recognition machine detects the person P as P (3) , and then the examiners has to decide the physical existence of the unknown person.

Medical diagnosis
Now we develop Algorithm 2 for medical diagnosis with the help of proposed correlation for m-PSF information.
Step 4: The disease β (i) showing highest correlation with patient P is the actual disease. Example 5.2. In this example, we use the Algorithm 2 to analyze the actual disease of a patient. Suppose a patient has to be diagnosed in a healthcare center and healthcare experts diagnosis the patient with the following symptoms as follows: ς 1 = headache, ς 2 = fatigue, ς 3 = chest pain, ς 4 = breathing difficulty and the possible diseases as follows: The assessment matrices for β (1) , β (2) ,β (3) and β (4) are expressed in Table 9, Table 10, Table 11, Table  12, respectively. Table 9. Assessment matrix for β (1) .  Assume the expert opinion for the patient and his symptoms is evaluated in terms 3PSFS as given in Table 13. Table 13. Expert opinion in terms of 3-PSFS. In order to check the actual disease of patient, we find the correlation coefficients of β from β (1) , β (2) using formula (4.1) and (4.2) respectively as follows.

Conclusions
MCDM has been studied by a large number of scholars and researchers. The methods designed for this purpose are generally influenced by the judgment operational framework used. The most of its key issues are attached to vague, imprecise, and multi-polar data that can't be adequately explained by fuzzy set alone. An m-polar fuzzy set (m-PFS) has the ability to deal with vagueness by multi-polarity and the spherical fuzzy set (SFS) deal with uncertainty by using three independent grades (membership degree, neutral-membership degree, and non-membership degree).
In order to deal with real-life vague circumstances when decision makers require a new mathematical model to deal with multi-polarity as well as three independent spherical index, we introduced a new hybrid model named as m-polar spherical fuzzy set (m-PSFS) as a robust fusion of SFS and m-PFS. The existing models namely m-PFS and SFS are the special cases of suggested hybrid m-PSFS. To ensure the algebraic structures of this robust extension, we developed fundamental operations on m-PSFSs and investigated their related results. A suitable numbers of illustrations are presented to explain suggested notions and results. We introduced the correlation measures and weighted correlation measures for m-PSFSs. Certain properties of covariances and the correlation measures are proposed to analyze that suggested concepts are novel extension of crisp correlation measures. The main advantage of proposed correlation measures is that these notions deal with vagueness and uncertainty in the real-life problems efficiently with the help of m-PSF information. We discussed applications of m-PSFSs and their correlation measures in pattern recognition and medical diagnosis. To discuss the superiority and efficiency of proposed correlation measures, we give a comparison analysis of proposed concepts with some existing concepts.