Analysis of Hamming and Hausdorff 3D distance measures for complex pythagorean fuzzy sets and their applications in pattern recognition and medical diagnosis

Similarity measures are very effective and meaningful tool used for evaluating the closeness between any two attributes which are very important and valuable to manage awkward and complex information in real-life problems. Therefore, for better handing of fuzzy information in real life, Ullah et al. (Complex Intell Syst 6(1): 15–27, 2020) recently introduced the concept of complex Pythagorean fuzzy set (CPyFS) and also described valuable and dominant measures, called various types of distance measures (DisMs) based on CPyFSs. The theory of CPyFS is the essential modification of Pythagorean fuzzy set to handle awkward and complicated in real-life problems. Keeping the advantages of the CPyFS, in this paper, we first construct an example to illustrate that a DisM proposed by Ullah et al. does not satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Then, combining the 3D Hamming distance with the Hausdorff distance, we propose a new DisM for CPyFSs, which is proved to satisfy the axiomatic definition of complex Pythagorean fuzzy DisM. Moreover, similarly to some DisMs for intuitionistic fuzzy sets, we present some other new complex Pythagorean fuzzy DisMs. Finally, we apply our proposed DisMs to a building material recognition problem and a medical diagnosis problem to illustrate the effectiveness of our DisMs. Finally, we aim to compare the proposed work with some existing measures is to enhance the worth of the derived measures.


Introduction
Similarity and dissimilarity are important because they are used by a number of data mining techniques, such as clustering nearest neighbor classification and anomaly detection. The term proximity is used to refer to either similarity or dissimilarity. The similarity between two objects is a numeral measure of the degree to which the two objects are alike. Consequently, similarities are higher for pairs of objects that are more alike. Similarities are usually non-negative and are often between 0 (no similarity) and 1 (complete similar-ity). The dissimilarity between two objects is the numerical measure of the degree to which the two objects are different. Dissimilarity is lower for more similar pairs of objects. Frequently, the term distance is used as a synonym for dissimilarity. Dissimilarities sometimes fall in the interval [0, 1, but it is also common for them to range from 0 to ∞. Certain people have diagnosed the theory of similarity measures and distance measures for classical information and because of this reason, we loss a lot of information. For managing with such sort of issues, Zadeh [46] introduced the concept of fuzzy set (FS) by using a function from the universe of discourse to [0, 1], which was called the membership degree function, to describe the importance of an element in the universe of discourse. Then, Zadeh's fuzzy set theory constitutes the basis of fuzzy decision-making [10,11,19]. However, the FS can only deal with the situation containing two opposite responses. Therefore, it failed to deal with the situation with hesitant/neutral state of "this and also that". According to this, Atanassov [8] generalized Zadeh's fuzzy set by proposing the concept of intuitionistic fuzzy sets (IFSs), characterized by a membership function and a non-membership function meeting the condition that their sum at every point is less than or equal to 1. In the theory of IFSs, the condition that the sum of the membership degree and the non-membership degree is less than or equal to 1 induces some decision evaluation information that cannot be expressed effectively. Hence, the range of their applications is limited. To overcome this shortcoming, Yager [42][43][44] proposed the concepts of Pythagorean fuzzy sets (PFSs) and q-rung orthopair fuzzy sets (q-ROFSs). These sets satisfy the condition that the square sum or the qth power sum of the membership degree and the non-membership degree is less than or equal to 1. It is determined from the aforementioned in-depth research and DMPs that their use is restricted to handling only the data's uncertainty while failing to address its fluctuations at a particular point in time. But data derived from "medical research, a database for biometric and facial recognition" are constantly updated in tandem with time. Thus, a range of MD is expanded from a real subset to the unit disc of the complex plane to cope with these kinds of difficulties, which developed the idea of the complex fuzzy sets (CFSs) Ramot et al. [31]. Further, Alkouri and Salleh [7] introduced the concepts of complex intuitionistic fuzzy sets (CIFSs). To enlarge the representing domain, Ullah et al. [36] proposed the concept of complex Pythagorean fuzzy sets (CPyFSs), and Liu et al. [20] introduced the concept of complex qrung orthopair fuzzy sets (Cq-ROFSs). Since then, CIFSs, CPyFSs, and Cq-ROFSs have been widely applied to various fields, such as MCDM/MADM [1][2][3][4][5][6]12,15,[20][21][22]24,25,32,38,40], medical diagnosis [13,[27][28][29]32], pattern recognition [12,13,27,36], cluster analysis [12,47], and image processing [16].
It is necessary to gauge the degree of discriminating between the pairs of sets due to the intricate decision-making process. The most effective tools for this purpose are instant messengers. The decision-maker have the ability to assess the degree of discriminating between the sets among the many measures like entropy, similarity, inclusion, etc. The major goal of this work is to create some exponentialbased decision-makers to quantify the information, which is encouraged by the CPyFS model's characteristics and the quality of decision-maker. In order to achieve this, the data was designated under the CPyFS model to quantify the data using the suggested metric for resolving the decisionmaking procedures. The qualities of a few axioms are studied in detail. Later, an algorithm was developed based on the proposed investigation, to assess the differences for various types of complex fuzzy sets, the normalized distance measure (DisM) and the similarity measure (SimM), being a pair of dual concepts, are important tools for decisionmaking and pattern recognition under CPyFSs and CIFSs frameworks. For the CIFSs, Rani and Garg [32] presented a few two-dimensional (2D) CIFDisMs by using the Hamming, Euclidean, and Hausdorff distances. Then, Garg and Rani [12] proposed some new CIF information measures, including SimMs, DisMs, entropies, and inclusion measures) and obtained the transformation relationships among them. Meanwhile, they [12] developed a CIF clustering algorithm. For Cq-ROFSs, Garg et al. [13] gave the notion of Cq-ROF dice SimM and weighted Cq-ROF dice SimM and derived some new Cq-ROF dice SimMs. Liu et al. [21] proposed some cosine DisMs and SimMs for Cq-ROFSs and obtained developed a TOPSIS method under Cq-ROFS framework. To distinguish different Cq-ROFSs with high similarity, Mahmood and Ali [25] obtained some new SimMs for Cq-ROFSs. For CPyFSs, Aldring and Ajay [5] developed a MCGDM method by introducing a new CPyF projection measure between the alternatives and the relative CPyF ideal point. Based on the Hamming distance and the Hausdorff distance, Ullah et al. [36] developed two parametric DisMs for CPyFSs and applied to a building material recognition problem. However, because the different weights are assigned to the degrees of membership, non-membership, and hesitancy for the 3D DisM D 2 CPyFS of Ullah et al. [36], the may cause an unreasonable result that the DisM D 2 CPyFS does not satisfy the axiomatic definition of complex Pythagorean fuzzy DisM (see Example 1). The geometrical shape of the proposed work is described in the form of Fig. 1 [14,34,37,39,45], we propose some other new CPyFDisMs.

Pythagorean fuzzy set (PFS)
where the functions P P : X −→ [0, 1] and O P : X −→ [0, 1] define the degree of membership and the degree of nonmembership of the element ϑ ∈ X to the set P, respectively, and for every ϑ ∈ X, P 2 P (ϑ) + O 2 P (ϑ) ≤ 1. Moreover, the hesitancy degree H P (ϑ) of an element ϑ belonging to P is

Complex intuitionistic fuzzy set (CIFS)
where the functions P C : X −→ O C and O C : X −→ O C define the degree of membership and the degree of nonmembership of the element ϑ ∈ X to the set C, respectively, and for every ϑ ∈ X,

Definition 2.4 ([36, Definition 7]). A complex Pythagorean
where the functions P C : X −→ O C and O C : X −→ O C define the degree of membership and the degree of nonmembership of the element ϑ ∈ X to the set C, respectively, and for every ϑ ∈ X, Moreover, the hesitancy degree In [36], the pair ( [36] introduced the following basic operations for CPyFSs and CPyFNs.

Distance and similarity measures on CPyFSs
if it satisfies the following conditions: for any C 1 , C 2 , C 3 ∈ CPyFS(X), if it satisfies the following conditions: for any C 1 , C 2 , C 3 ∈ CPyFS(X),
The following example shows that Property 3 does not hold, and thus the DisM D 2 CPyFS does not satisfy the axiomatic definition of complex Pythagorean fuzzy DisM.
. This means that Property 3 does not hold because C 1 ⊆ C 2 ⊆ C 3 .

A novel DisM on C PyFN
be two CPyFNs. Define the DisM D Wu between C 1 and C 2 by and thus D Wu (C 1 , C 2 ) ≤ 1 by Eq. (8).

Proposition 4.3 For
Proof It follows directly from Eq. (8) and triangle inequality.
, we consider the following four cases: These, together with T 2 And thus, 1 4 Similarly, it can be verified that 1 4 · (|W 2 |}. Therefore, similarly to the proof of (i), it can be verified that and These, together with T 2 And thus, it can be similarly verified that Meanwhile, by (i) and (ii), we have and These, together with F 2 And thus, it can be similarly verified that Summing Propositions 4.1-4.4, we have the following result.
(2) If we consider the imaginary part as zero in Eq. (8) and replace the constraint P 2 + O 2 ≤ 1 by P + O ≤ 1, then the distance D Wu reduces to IF environment. By Theorem 4.1, we know that d P is a DisM on .

A novel DisM on CPyFSs
where ω = (ω 1 , ω 2 , . . . , ω n ) is the weight vector of ω j ( j = 1, 2, . . . , ) with ω j ∈ (0, 1] and j=1 ω j = 1. By Theorem 4.1, we have the following result.   Table 1. Given an unknown building material M , which is expressed by CPyFNs for the above 7 attributes in Table 1. We want to determine to which building material the unknown material M belongs. If we take the weight vector ω of 7 attributes as ω = (0.11, 0.14, 0.1, 0.18, 0.21, 0.10, 0.16) , the SimMs calculated by Eq. (16) are given in Table 2. By the principle of the maximum degree of SimMs, the unknown building material belongs to the class M 4 -polyvinyl chloride flooring.

Building material recognition problem
The pattern classification results by using different DisMs/SimMs are listed in Table 3. Observing from Table 3, we know that (1) our result listed in Table 1 is consistent with the results obtained by the DisM WD 1 CPyFS in [36] and DisMs defined by Eqs. (9)-(15); (2) because the DisM WD 2 CPyFS in [36] does not satisfy the axiomatic definition of complex Pythagorean fuzzy DisM, the pattern classification result obtained by this DisM is M 1 -sealant, which is unreasonable.
Observing from Table 3, we notice that the existing measures are provided the different ranking results such as M 1 and M 4 . But, we also notice that the proposed all measures are given the same ranking measures. Further, we aim to consider some measures which was proposed by different scholars, for this, Rani and Garg [32] presented a few two-dimensional (2D) CIFDisMs by using the Hamming, Euclidean, and Hausdorff distances. Then, Garg and Rani [12] proposed some new CIF information measures, including SimMs, DisMs, entropies, and inclusion measures and obtained the transformation relationships among them. Meanwhile, they [12] developed a CIF clustering algorithm. But these all measures are not able to resolve our selected information because of their limitations, where the computed measures of Rani and Garg [32] have been failed because these measures are the particular case of the proposed measures.        CIFDisMs by using the Hamming, Euclidean, and Hausdorff distances. Then, Garg and Rani [12] proposed some new CIF information measures, including SimMs, DisMs, entropies, and inclusion measures) and obtained the transformation relationships among them. Meanwhile, they [12] developed a CIF clustering algorithm. But these all measures are not able to resolve our selected information because of their limitations, where the computed measures of Rani and Garg [32] have been failed because these measures are the particular case of the proposed measures.

Conclusion
Complex Pythagorean fuzzy set is very effective and realistic for evaluating awkward and complex information in real-life problems. Keeping the advantages of the CPyFS, the main theme of this analysis is described below: (1) Ullah et al. [36] introduced the notion of CPyFS to deal with uncertain information and proposed a 2D CPyFDisM D 1 CPyFS and a 3D CPyFDisM D 2 CPyFS . (2) The DisM D 2 CPyFS does not satisfy the axiomatic definition of CPyFDisM (see Example 1).
(3) To overcome the drawback of the DisM D 2 CPyFS in [36], we constructed a novel 3D DisMD Wu for CPyFSs based on the 3D Hamming distance and the Hausdorff distance for IFSs and show that it satisfies the axiomatic definition of CPyFDisM. (4) Based on the DisMs for IFSs in [14,34,37,39,45], we proposed some other new CPyFDisMs. (5) We illustrated the effectiveness of our DisMs, we give the comparative analysis by using our proposed DisMs to a building material recognition problem and a medical diagnosis problem.
In the future, we aim to develop some new ideas such as averaging or geometric aggregation operators [26], Maclaurin symmetric mean operators [35], Aczel-Alsina operators [18], information measures [33], similarity measures [23], Einstein aggregation operators, Hamacher aggregation operators, improved Dombi aggregation operators [17], interaction aggregation operators, similarity measures, distance measures, many techniques based on CPyFSs and try to justify it with the help of some suitable applications such as road signals, computer networks, game theory, artificial intelligence, and decision-making theory.