A new distance measure of Pythagorean fuzzy sets based on matrix and and its application in medical diagnosis
AA new distance measure of Pythagorean fuzzy setsbased on matrix and and its application in medicaldiagnosis
Yuanpeng He
School of Computer and Information ScienceSouthwest University Chongqing400715 ChinaEmail: [email protected]
Fuyuan Xiao*
School of Computer and Information ScienceSouthwest University Chongqing400715 ChinaEmail: [email protected], [email protected]
Abstract —The pythagorean fuzzy set (PFS) which is developedbased on intuitionistic fuzzy set, is more efficient in elaboratingand disposing uncertainties in indeterminate situations, which is avery reason of that PFS is applied in various kinds of fields. Howto measure the distance between two pythagorean fuzzy sets isstill an open issue. Mnay kinds of methods have been proposed topresent the of the question in former reaserches. However, not allof existing methods can accurately manifest differences amongpythagorean fuzzy sets and satisfy the property of similarity.And some other kinds of methods neglect the relationship amongthree variables of pythagorean fuzzy set. To addrees the proplem,a new method of measuring distance is proposed which meetsthe requirements of axiom of distance measurement and is ableto indicate the degree of distinction of PFSs well. Then somenumerical examples are offered to to verify that the method ofmeasuring distances can avoid the situation that some counter-intuitive and irrational results are produced and is more effective,reasonable and advanced than other similar methods. Besides,the proposed method of measuring distances between PFSs isapplied in a real environment of application which is the medicaldiagnosis and is compared with other previous methods todemonstrate its superiority and efficiency. And the feasibility ofthe proposed method in handling uncertainties in practice is alsoproved at the same time.keywords: Pythagorean fuzzy set Similarity measureDistance measure Pattern recognition Medical diagnosis
I. I
NTRODUCTION
There is lots of uncertain things existing in the real world,and how to measure the level of uncertainty has attractedincremental attention and interests from researches all aroundthe world [1]–[5]. Therefore, many mathematic theory andmodels have been proposed to be applied into practical ap-plication, such as evidence theory [6]–[9], belief function[10]–[16], belief structure [17]–[20], probability assignment[21]–[26], entropy theory [27]–[33], Z numbers [34]–[37]and D numbers [38]–[43], which are palying important rolesin respects of people’s life. Except for these instrumentsmentioned above, an efficient tool to handle uncertainty calledfuzzy set is introduced by Zadeh [44], [45], which is moreoperable in seeking for useful information among uncertaintiesand becomes a key component of pattern recognition [13],[46]–[49] and decision making [50]–[53]. According to the original definition of fuzzy set, an universe of discourse U isgiven, a mapping from U to an unit range to is called afuzzy set and the value of the element in the mapping representthe membership. To some extent, the fuzzy set can manifestreal condition relatively well.However, when judging actual situations in real world,things are getting more complex, which indicates that a singlevalue can not reflect the essence of certain objects. Therefore,a further concept based on the fuzzy set is propposed whosename is intuitionistic fuzzy set. And it is developed byAtanassov [54] and defined to contain elements, namelymembership, non-membership and hesitance, which is obvi-ously convenient in expressing reality and more intuitive. Asa reinforced generation of fuzzy set, the intuitionistic fuzzy setis considered more efficient in handling ambiguity and deducesappropriate targets to be selected. And it is further developedin different fields, such as interval-valued intuitionistic fuzzyset [55], [56], quantum decision [57] and so on. Later on,a new extension is invented by Yager [58] which is calledthe pythagorean fuzzy set. The new concept derived fromintuitionistic fuzzy set is a quadratic form of the former fuzzyset, which means that the new modality of fuzzy set has alarger range of the change of variables and therefore has morepotential in indicating the probability of various objects. Thenew form of fuzzy set is also extended into different forms,such as interval-valued pythagorean fuzzy set [51], decisionmaking [59]–[61] and some other applications [62] To simplifyappellations of kinds of fuzzy set, intuitionistic fuzzy set iscalled IFS and pythagorean fuzzy set is named PFS.No matter which kind of fuzzy set is discussed about,how to measure the differences or distances among fuzzysets is an unavoidable problem. If the degree of differencescan be transformed in to the value of numbers, it is muchmore convenient to solving problems which occur in dscisionmaking, pattern recognition and medical diagnosis. In order tomanifest the distances properly, many methods of measuringdistances have been proposed and some of them have idealeffect in classification. The most widely used method ofmeasuring distances between IFSs are the Hamming distance a r X i v : . [ c s . A I] J a n RELIMINARIES
In this section, some concepts related to pythagorean fuzzysets (PFS) are introduced.
A. Intuitionistic fuzzy set
Let A denote an IFS in a finite universe of discourse whichis named X . And the mathematic form of IFS A can be definedas [54]: A = { ( x, µ ( x ) , ν ( x )) | x ∈ X } (1)Besides, the parameters satisfy µ ( x ) : X → [0 , (2)and ν ( x ) : X → [0 , (3) µ ( x ) is a representative of the degree of membership of x ∈ X and ν ( x ) is a representative of the degree of non-membershipof x ∈ X . And both of them meet the condition that: ≤ µ ( x ) + ν ( x ) ≤ (4) The hesitance function of an IFS A in X is defined as: π ( x ) = 1 − µ ( x ) − ν ( x ) (5)The value of π ( x ) can reflect the degree of hesitance of x ∈ X . B. Pythagorean fuzzy sets
Let A denote an PFS in a finite universe of discourse whichis named X . And the mathematic form of PFS A can bedefined as [58]–[60]: A = { ( x, A Y ( x ) , A N ( x )) | x ∈ X } (6)Besides, the parameters satisfy A Y ( x ) : X → [0 , (7)and A N ( x ) : X → [0 , (8) A Y ( x ) is a representative of the degree of membership of x ∈ X and ν ( x ) is a representative of the degree of non-membership of x ∈ X . And both of them meet the conditionthat: ≤ A Y ( x ) + A N ( x ) ≤ (9)The hesitance function of an PFS A in X is defined as: A H ( x ) = (cid:113) − A Y ( x ) − A N ( x ) (10)The value of A H ( x ) can reflect the degree of hesitance of x ∈ X . Property 1 :
Let B and C be two PFS of the finite universeof discourse X , then both of them satisfy [59], [60]: (1) B ⊆ C if ∀ x ∈ X B Y ( x ) ≤ C Y ( x ) and B N ( x ) ≥ C N ( x ) (2) B = C if ∀ x ∈ X B Y ( x ) = C Y ( x ) and B N ( x ) = C N ( x ) (3) B ∩ C = {(cid:104) x, min [ B Y ( x ) , C Y ( x )] , max [ B N ( x ) , C N ( x )] (cid:105)| x ∈ X } (4) B ∪ C = {(cid:104) x, max [ B Y ( x ) , C Y ( x )] , min [ B N ( x ) , C N ( x )] (cid:105)| x ∈ X } (5) B · C = {(cid:104) x, B Y ( x ) C Y ( x ) , ( B Y ( x ) + C Y ( x ) − B Y ( x ) C Y ( x )) (cid:105)| x ∈ X } (6) B n = {(cid:104) x, B nY ( x ) , (1 − (1 − B N ( x ) n )) (cid:105)| x ∈ X } In the next part, some separate methods of measuringthe distance between different IFSs and PFSs are brieflyintroduced. And some useful function is also mentioned, whichis helpful in judging the validity of IFS and even PFS in itsextended form. . Different methods of measuring distance related topythagorean fuzzy sets1.
The definition of Hamming distance which measures dif-ference between IFSs D and E is written as [63]: d Hm ( D, E ) = 12 · ( | µ D ( x ) − µ E ( x ) | + | ν D ( x ) − ν E ( x ) | + | π D ( x ) − π E ( x ) | ) (11) The definition of normalized Hamming distance whichmeasures difference between IFSs D and E is written as [63]: (cid:101) d Hm ( D, E ) = 12 n · n (cid:88) i =1 ( | µ D ( x i ) − µ E ( x i ) | + | ν D ( x i ) − ν E ( x i ) | + | π D ( x i ) − π E ( x i ) | ) (12) The definition of Euclidean distance which measures dif-ference between IFSs D and E is written as [63]: d Eu ( D, E ) = ( 12 · (( µ D ( x ) − µ E ( x )) +( ν D ( x ) − ν E ( x )) + ( π D ( x ) − π E ( x )) )) (13) The definition of normalized Euclidean distance whichmeasures difference between IFSs D and E is written as [63]: (cid:101) d Eu ( D, E ) = ( 12 n · n (cid:88) i =1 (( µ D ( x i ) − µ E ( x i )) +( ν D ( x i ) − ν E ( x i )) + ( π D ( x i ) − π E ( x i )) )) (14) D. Methods of measuring the distance of IFS extended to PFSand existing method1.
The definition of Hamming distance which measures dif-ference between PFSs F and G is written as [69]: D Hm ( F, G ) = 12 · ( | F Y ( x ) − G Y ( x ) | + | F N ( x ) − G N ( x ) | + | F H ( x ) − G H ( x ) | ) (15) The definition of normalized Hamming distance whichmeasures difference between PFSs F and G is written as: (cid:101) D Hm ( F, G ) = 12 n · n (cid:88) i =1 ( | F Y ( x ) − G Y ( x ) | + | F N ( x ) − G N ( x ) | + | F H ( x ) − G H ( x ) | ) (16) The definition of Euclidean distance which measures dif-ference between PFSs F and G is written as [69]: D Eu ( F, G ) = ( 12 · (( F Y ( x ) − G Y ( x )) +( F N ( x ) − G N ( x )) + ( F H ( x ) − G H ( x ))) (17) The definition of normalized Euclidean distance whichmeasures difference between PFSs F and G is written as: (cid:101) D Eu ( F, G ) = ( 12 n · n (cid:88) i =1 (( F Y ( x ) − G Y ( x )) +( F N ( x ) − G N ( x )) + ( F H ( x ) − G H ( x )) )) (18) The definition of Chen’s distance which measures differencebetween PFSs F and G is written as [69]: D C ( F, G ) = [ 12 · ( | F Y ( x ) − G Y ( x ) | β + | F N ( x ) − G N ( x ) | β + | F H ( x ) − G H ( x ) | β )] β (19)The parameter β in Chen’s distance which satisfies the condi-tion that β ≥ . When β = 1, the method of Chen degeneratesto the method of Hamming distance. Besides, when β = 2,the method of Chen degenerates to the method of Euclideandistance. The definition of Chen’s distance which measures differencebetween PFSs F and G is written as: (cid:101) D C ( F, G ) = [ 12 n · n (cid:88) i =1 ( | F Y ( x ) − G Y ( x ) | β + | F N ( x ) − G N ( x ) | β + | F H ( x ) − G H ( x ) | β )] β (20)The parameter β in Chen’s distance which satisfies the condi-tion that β ≥ . When β = 1, the method of Chen degeneratesto the method of Hamming distance. Besides, when β = 2,the method of Chen degenerates to the method of Euclideandistance. Notion :
On the base of the method of measuring the distancebetween IFSs, all of the methods offered above have beenextended to adapt to the situation that the distance between2 PFSs is supposed to be measured. More Importantly, theextended method of measuring the distance between PFSs stillconforms to the concept of normalization.
E. Score function on IFS and its extension A score function of IFS is initialized by Chen andTan and it is utilized in the solution of problems which areproduced in multi-attribute decision using intuitionistic sets.Let A be an intuitionistc fuzzy set in the finite universe ofdiscourse X, which is { x , x , ...x n } . And IFS A is givenas A = {(cid:104) x, µ ( x ) , ν ( x ) } . Besides, the score function S A isdefined as [70]: S A ( x i ) = µ ( x ) − ν ( x ) The value of the formula illustrates a degree that whether theintuitionistic fuzzy sets is comfortable enough for decisionmakers to have a clear and straightforward expectation toactual situations. Obviously, the value of µ ( x ) gets larger, thesmaller the value of ν ( x ) is going to be and the reliability of x ∈ X gets greater, which means the value of S A ( x i ) getsbigger. Therefore, the value of score function S A ( x i ) can beregarded as a kind of level of support about the element x ∈ X .When S A ( x i ) > , it can concluded that it is more believableto classify x as a component of X . When S A ( x i ) < , it canconcluded that it is more believable to classify x as a exceptionof X . More than that, the absolute value of the score functioncan manifest the situation of the degree of certainty of IFS.If the mass is getting bigger, then the IFS is more certain. Ifthe mass is getting smaller, then the IFS is more uncertain.For example, let A and B be two PFSs and they are definedespectively as A = {(cid:104) x, . , . (cid:105)} and B = {(cid:104) x, . , . (cid:105)} .What should be pointed out is that the degree of hesitance inthese two IFSs is exactly identical. However, the IFS B ismore valuable and useful, because it offers more informationand can clearly illustrate actual situations. And obviously, thevalues of | S A ( x ) = 0 | and | S B ( x ) = 0 . | is exactly consistentwith the conclusion mentionedd above. As a consequence theabsolute value of S A ( x ) can be utlized as an efficient methodto measure the level of certainty of an IFS. On the base of the definition of score function which isdeveloped on the concept of intuitionistic fuzzy set, a newscore function SP is on the notion of pythagorean fuzzy setwhich is defined as: SF A ( x i ) = A Y ( x i ) − A N ( x i ) The extended formula of PFS plays a similar role in handlingPFSs, like the role of the degraded score function based onIFS which has.
F. Propeties of Chen’s method and the proof of them
Some of the propoties of Chen’s distance can be interpretedas follows:
Property 2 :
Let H and I be two PFSs in the finite universeof discourse X, then both of them satisfy: (1) (cid:101) D C ( H, I ) = 0 if A = B and A ∈ X, B ∈ X (2) (cid:101) D C ( H, I ) = (cid:101) D C ( I, H ) and A ∈ X, B ∈ X (3) ≤ (cid:101) D C ( H, I ) ≤ and A ∈ X, B ∈ X As what have been mentioned above, Hamming distance andEuclidean distance are special cases of normalized Chen’sdistance. Therefore, both of them are expected to meet thecondition which Chen’s distance requires.And the demonstration of properties of normalized Chen’sdistance is offered as follows:
Proof 1 :
Let A and B be two PFSs in the finite universe ofdiscourse X , both of them are supposed to satisfy: (cid:101) D C ( A, B ) = 0
As a result, the attribute of (cid:101) D C ( A, B ) should be consideredin the definition of Chen’s distance that [ · ( | F Y ( x ) − G Y ( x ) | β + | F N ( x ) − G N ( x ) | β + | F H ( x ) − G H ( x ) | β )] β = 0 Therefore, it can be concluded that | F Y ( x ) − G Y ( x ) | β = 0 | F N ( x ) − G N ( x ) | β = 0 | F H ( x ) − G H ( x ) | β = 0 From the equations, the relationships between every twoparameters can be obtained F Y ( x ) = G Y ( x ) F N ( x ) = G N ( x ) F H ( x ) = G H ( x ) Hence, it comes to a conclusion that when (cid:101) D C ( A, B ) = 0 ,PFS A is equal to PFS B . Proof 2 :
Let A and B be two PFSs in the finite universe ofdiscourse X , both of them are supposed to satisfy:The distance between PFS A and B is presented as: (cid:101) D C ( A, B ) = [ n · n (cid:80) i =1 ( | A Y ( x ) − B Y ( x ) | β + | A N ( x ) − B N ( x ) | β + | A H ( x ) − B H ( x ) | β )] β Then the distance between PFS B and A is presented as: (cid:101) D C ( B, A ) = [ n · n (cid:80) i =1 ( | B Y ( x ) − A Y ( x ) | β + | B N ( x ) − A N ( x ) | β + | B H ( x ) − A H ( x ) | β )] β After comparing these two equations, it can be summarizedthat (cid:101) D C ( A, B ) = (cid:101) D C ( B, A ) because | A Y ( x ) − B Y ( x ) | β = | B Y ( x ) − A Y ( x ) | β | A N ( x ) − B N ( x ) | β = | B N ( x ) − A N ( x ) | β | A H ( x ) − B H ( x ) | β = | B H ( x ) − A H ( x ) | β Proof 3 :
Let A and B be two PFSs in the finite universe ofdiscourse X , both of them are supposed to satisfy: (cid:101) D C ( A, B )= [ n · n (cid:80) i =1 ( | A Y ( x ) − B Y ( x ) | β + | A N ( x ) − B N ( x ) | β + | A H ( x ) − B H ( x ) | β )] β = [ n · n (cid:80) i =1 ( | A Y ( x ) − B Y ( x ) | β + | A N ( x ) − B N ( x ) | β + | (1 − A Y ( x ) − A N ( x )) − (1 − B Y ( x ) − B N ( x )) | β )] β = [ n · n (cid:80) i =1 ( | A Y ( x ) − B Y ( x ) | β + | A N ( x ) − B N ( x ) | β + | A Y ( x ) − B Y ( x ) + A N ( x ) − B N ( x ) | β )] β ≤ [ n · n (cid:80) i =1 ( | A Y ( x ) − B Y ( x ) + A N ( x ) − B N ( x ) | β + | A Y ( x ) − B Y ( x ) + A N ( x ) − B N ( x ) | β )] β = [ n · n (cid:80) i =1 ( | A Y ( x ) − B Y ( x ) + A N ( x ) − B N ( x ) | β )] β From the definition of pythagorean fuzzy sets, it can besummed up that ≤ A Y ( x ) ≤ ≤ B Y ( x ) ≤ ≤ A N ( x ) ≤ ≤ B N ( x ) ≤ ≤ ( A Y ( x ) + A N ( x )) ≤ ≤ ( B Y ( x ) + B N ( x )) ≤ As a result, it can be easily calculated that ≤ ( | A Y ( x ) − B Y ( x ) + A N ( x ) − B N ( x ) | β ) ≤ According to the equation above, a further conclusion can bemade ≤ [ n · n (cid:80) i =1 ( | A Y ( x ) − B Y ( x ) + A N ( x ) − B N ( x ) | β )] β ≤ Therefore, it can be obtained that ≤ (cid:101) D C ( A, B ) ≤ All of the properties of Chen’s method of measuring distanceand the ones which degenerates from it have been proven.III. P
ROPOSED METHOD
How to measure the differences between PFSs is still anopen issue which plays a crucial part in target recognition. Inthis part, a new proposed method of measuring the distancebetween PFSs which is developed by a method of measuringthe distance between IFSs based on a similarity matrix. Be-sides, the proposed method not only takes the advantage of theprevious method which handles IFSs, but also further improvethe performance of the similarity matrix in managing the massof membership, non-membership and the hesitance. Then,the properties of the new proposed method are inferred andproven. In numerical examples, the new proposed method ofmeasuring distance between PFSs satifies the distance measureaxiom and is more efficient in producing intuitive and rationalresults.
A. The new method of measuring distance
Let A and B be two PFSs in the finite universe of discourseX, the new formula which produces distances is defined asfollows: D N ( A, B ) = (cid:118)(cid:117)(cid:117)(cid:116) n n (cid:88) i =1 (cid:126)m i M ( (cid:126)m i M ) T AB Y + AB N + AB H (21) AB Y = A Y ( x i ) + B Y ( x i ) (22) AB N = A N ( x i ) + B N ( x i ) (23) AB H = A H ( x i ) + B H ( x i ) (24) (cid:126)m i = ( A Y ( x i ) − B Y ( x i ) , A N ( x i ) − B N ( x i ) , A H ( x i ) − B H ( x i )) (25)However, the definition of the matrix M is still not clear.The most important efficacy of the matrix is to adjust andoptimize the mass of membership, non-membership and thehesitance. Adopting the index of hesitance as a parameter togenerating distance between different PFSs is not straight-forward and concise, because the index of hesitance itselfis a kind of uncertainty, which is very difficult to clarifythe relationship between different hesitance. And coincidently,there is some kind of relationship between the hesitance andmembership and non-membership. For example, let A and B be two PFSs in the finite universe of discourse X, where PFS A = {(cid:104) , , (cid:105)} and PFS B = {(cid:104) . , . , . (cid:105)} . In PFS A , the index of hesitance is equal to 1, which indicates that PFS A is totally uncertain. Besides, on the contrary, the indexof hesitance of PFS B is 0, it means there is no uncertaintyor vagueness commonly. However, the mass of membershipand non-membership is so close that it is very difficult tomake reasonable decisions. But one thing can be pointed out,the index of hesitance is not independent of the other twoparameters, namely membership and non-membership, whichmeans all of them have a similarity under certain relationshipand is already sufficiently demonstrated in examples provided.Therefore, considering the relationship among the parameters,a new matrix is developed which is based on the previouslydefined matrix [71].And the definition of the new matrix M is written as follows: M = Y N H (26) Y = ( A Y ( x i ) + B Y ( x i ))( A Y ( x i ) + B Y ( x i ) + A N ( x i ) + B N ( x i )) (27) N = ( A N ( x i ) + B N ( x i ))( A Y ( x i ) + B Y ( x i ) + A N ( x i ) + B N ( x i )) (28) H = (cid:112) − Y − N (29)The main idea of the definition is that because uncertaintycan not accurately show the real situation, then the massof index is supposed to be distributed to membership andnon-membership according to their mass in different PFSsto strengthen their ability of identification and not to changethe original conditions, which enlarges the useful amount ofinformation of pythagorean fuzzy sets and is helpful in targetrecognition. Due to the distribution to membership and non-membership, an adjustment of the index of hesitance shouldalso be considered. Therefore, considering the mathematicform of PFS, it is proposed that H = √ − Y − N isadopted as the remaining amount of information after the dis-tribution to membership and non-membership. The operationof reduction in the index of hesitance improves the degree ofidentification, which is very helpful in measuring the distanceof PFSs. Specific details about matrix :
The new distance proposedin this parper is based on the transformation of vectors from 3parameters in PFSs. However, thr role of the vector and matrixis not just showing the original mass of 3 parameters. Whatcan be concluded is that membership and non-membership isindependent of each other, so both of the parameters is consid-ered as orthogonal. Though membership and non-membershipare treated equally, the relationship between hesitance andthe other 2 paramerters is not just orthogonal, but presentsa kind of proportion in membership and non-membership. Italso conforms to the operation of some methods which handlesthe uncertainties of multiple elements propositions in evidecetheory and is closely related to fuzzy sets. For example, twoPFS A = {(cid:104) . , . , . (cid:105)} and PFS B = {(cid:104) . , . , . (cid:105)} is given to explain the process of the proposed vector and thenew matrix which adjust the distribution of 3 parameters. Therocess is presented as follows: (cid:126)m i = ( A Y ( x i ) − B Y ( x i ) , A N ( x i ) − B N ( x i ) , A H ( x i ) − B H ( x i )) = (0 . , . , − . Y = ( A Y ( x i )+ B Y ( x i ))( A Y ( x i )+ B Y ( x i )+ A N ( x i )+ B N ( x i )) = . +0 . . +0 . ) = 0 . N = ( A N ( x i )+ B N ( x i ))( A Y ( x i )+ B Y ( x i )+ A N ( x i )+ B N ( x i )) = . +0 . . +0 . ) = 0 . H = √ − Y − N = 0 . (cid:126)m i × Y N H = (0 . , . , − . × . . . = (0 , , . Then the construction of a crucial part of the numerator iscompleted.
B. Demonstration of Properties of proposed method
In this part, many properties of the proposed method isverified, like symmetry and triangle inequality.
Proof 4 :
The commutative property is verified in this proof.Let A and B be two PFSs in the finite universe of discourseX. Considering D N ( A, B ) and D N ( B, A ) , two equations canbe obtained, one is D N ( A, B ) = (cid:115) n n (cid:80) i =1 (cid:126)m i M ( (cid:126)m i M ) T AB Y + AB N + AB H AB Y = A Y ( x i ) + B Y ( x i ) AB N = A N ( x i ) + B N ( x i ) AB H = A H ( x i ) + B H ( x i ) (cid:126)m i = ( A Y ( x i ) − B Y ( x i ) , A N ( x i ) − B N ( x i ) , A H ( x i ) − B H ( x i )) M = Y N H Y = ( A Y ( x i )+ B Y ( x i ))( A Y ( x i )+ B Y ( x i )+ A N ( x i )+ B N ( x i )) N = ( A N ( x i )+ B N ( x i ))( A Y ( x i )+ B Y ( x i )+ A N ( x i )+ B N ( x i )) the other one is D N ( B, A ) = (cid:115) n n (cid:80) i =1 (cid:126)m i M ( (cid:126)m i M ) T BA Y + BA N + BA H BA Y = B Y ( x i ) + A Y ( x i ) BA N = B N ( x i ) + A N ( x i ) BA H = B H ( x i ) + A H ( x i ) (cid:126)m i = ( B Y ( x i ) − A Y ( x i ) , B N ( x i ) − A N ( x i ) , B H ( x i ) − A H ( x i )) M = Y N H Y = B Y ( x i )+ A Y ( x i )( B Y ( x i )+ A Y ( x i )+ B N ( x i )+ A N ( x i )) N = B N ( x i )+ A N ( x i )( B Y ( x i )+ A Y ( x i )+ B N ( x i )+ A N ( x i )) It can be easily concluded that, when PFS A and B interchangetheir places, the attributes AB Y , AB N , AB H , matrix M , Y and N do not change their values. But how the numerator (cid:126)m i M ( (cid:126)m i M ) T change is not very clear. Nevertheless, it canbe also easily proved like this: (cid:126)m i M ( (cid:126)m i M ) T = ( A Y ( x i ) − B Y ( x i ) + Y ( A H ( x i ) − B H ( x i ))) + ( A N ( x i ) − B N ( x i ) + N ( A H ( x i ) − B H ( x i ))) + (( B H ( x i ) − A H ( x i ))( √ − Y − N )) = ( B Y ( x i ) − A Y ( x i ) + Y ( B H ( x i ) − A H ( x i ))) + ( B N ( x i ) − A N ( x i ) + N ( B H ( x i ) − A H ( x i ))) + (( A H ( x i ) − B H ( x i ))( √ − Y − N )) Therefore, it is proved that the proposed method of measuringdistance between PFSs satisfy the property that D N ( A, B ) = D N ( B, A ) . Proof 5 :
A simple example is offered to verify a basicattirbute that is when 2 PFSs are exactly the same, theirdistance is coincidently 0. Let A and B become two identicalPFSs, namely {(cid:104) . , . , . (cid:105)} , their distance is supposed tobe 0 intuitively. And according to the definition of the vector (cid:126)m i , every component in the vector is equal to 0, which meansthe numerator (cid:126)m i M ( (cid:126)m i M ) T of the method of measuringdistance is exactly 0 and the value of the whole formula isalso 0. Therefore,the distance of the proposed method satisfiesthe property that D N ( A, B ) = 0 if and only if A = B . Proof 6 :
In this proof, the distance between 2 PFSs isdemonstrated that ≤ D N ( A, B ) ≤ . Let A and B betwo PFSs in the finite universe of discourse X, then it canbe obtained that (cid:126)m i M = ( A Y ( x i ) − B Y ( x i ) + Y ( A H ( x i ) − B H ( x i )) ,A N ( x i ) − B N ( x i ) + N ( A H ( x i ) − B H ( x i )) , ( B H ( x i ) − A H ( x i ))( √ − Y − N ) m i M ( (cid:126)m i M ) T = ( A Y ( x i ) − B Y ( x i ) + Y ( A H ( x i ) − B H ( x i ))) + ( A N ( x i ) − B N ( x i ) + N ( A H ( x i ) − B H ( x i ))) + (( B H ( x i ) − A H ( x i ))( √ − Y − N )) = ( A Y ( x i ) − B Y ( x i )) + ( A N ( x i ) − B N ( x i )) + ( A H ( x i ) − B H ( x i )) + 2 Y ( A Y ( x i ) − B Y ( x i ))( A H ( x i ) − B H ( x i ))+ 2 N ( A N ( x i ) − B N ( x i ))( A H ( x i ) − B H ( x i ))= A Y ( x i ) + B Y ( x i ) + A N ( x i ) + B N ( x i ) + A H ( x i ) + B H ( x i ) − A Y ( x i ) B Y ( x i ) − A N ( x i ) B N ( x i ) − A H ( x i ) B H ( x i )+ ( A H ( x i ) − B H ( x i ))((2 Y A Y ( x i ) + 2 N A N ( x i )) − (2 Y B Y ( x i ) + 2 N B N ( x i ))) ≤ A Y ( x i ) + B Y ( x i ) + A N ( x i ) + B N ( x i ) + A H ( x i ) + B H ( x i ) Obviously, when the entirety of Y A Y ( x i ) + 2 N A N ( x i ) islarger than Y B Y ( x i ) + 2 N B N ( x i ) ’s, A H ( x i ) − B H ( x i ) isdefinitely negative, according to the defintion of pythagoreanfuzzy set. Vice versa. Therefore, it can be inferred that ≤ (cid:126)m i M ( (cid:126)m i M ) T A Y ( x i )+ B Y ( x i )+ A N ( x i )+ B N ( x i )+ A H ( x i )+ B H ( x i ) ≤ ≤ n n (cid:80) i =1 (cid:126)m i M ( (cid:126)m i M ) T A Y ( x i )+ B Y ( x i )+ A N ( x i )+ B N ( x i )+ A H ( x i )+ B H ( x i ) ≤ Let T = n n (cid:80) i =1 (cid:126)m i M ( (cid:126)m i M ) T A Y ( x i )+ B Y ( x i )+ A N ( x i )+ B N ( x i )+ A H ( x i )+ B H ( x i ) Therefore, ≤ √ T ≤ Because √ T has the same significance with D n ( A, B ) . Then,it can be concluded that ≤ D n ( A, B ) ≤ . Proof 7 :
The trangle inequality is going to be proven in thispart. Some assumptions are given as:
Assumption A Y ( x ) ≤ B Y ( x ) ≤ C Y ( x ) Assumption C Y ( x ) ≤ B Y ( x ) ≤ A Y ( x ) Assumption B Y ( x ) ≤ min { A Y ( x ) , C Y ( x ) } Assumption B Y ( x ) ≥ max { A Y ( x ) , C Y ( x ) } It can be easily demonstrated that the inequality | A Y ( x ) − C Y ( x ) | ≤ | A Y ( x ) − B Y ( x ) | + | B Y ( x ) − C Y ( x ) | which meets condtions of Assumption 1 and Assumption 2.On the base of Assumption 3, it can be concluded that A Y ( x ) ≥ B Y ( x ) C Y ( x ) ≥ B Y ( x ) Therefore, it can be obtained that | A Y ( x ) − B Y ( x ) | + | B Y ( x ) − C Y ( x ) | − | A Y ( x ) − C Y ( x ) | = f ( x ) = A Y ( x ) − B Y ( x ) + C Y ( x ) − B Y ( x ) − A Y ( x )+ C Y ( x ) if A Y ( x ) ≥ C Y ( x ) A Y ( x ) − B Y ( x ) + C Y ( x ) − B Y ( x ) + A Y ( x ) − C Y ( x ) if A Y ( x ) ≤ C Y ( x )= 2 · ( min { A Y ( x ) , C Y ( x ) } − B Y ( x )) ≥ Homoplastically, on the base Assumption 4, it can be summa-rized that | A Y ( x ) − B Y ( x ) | + | B Y ( x ) − C Y ( x ) | − | A Y ( x ) − C Y ( x ) | = f ( x ) = B Y ( x ) − A Y ( x ) + B Y ( x ) − C Y ( x ) − A Y ( x )+ C Y ( x ) if A Y ( x ) ≥ C Y ( x ) B Y ( x ) − A Y ( x ) + B Y ( x ) − C Y ( x ) + A Y ( x ) − C Y ( x ) if A Y ( x ) ≤ C Y ( x )= 2 · ( B Y ( x ) − max { A Y ( x ) , C Y ( x ) } ) ≥ Hence, the property of inequality | A Y ( x ) − C Y ( x ) | ≤ | A Y ( x ) − B Y ( x ) | + | B Y ( x ) − C Y ( x ) | is also satisfied under the condition provided by Assumption3 and Assumption 4. And because every component in themethod of measuring distance is correspondingly propor-tionable to membership, non-membership and the index ofhesitance. As a result, the triangle inequality has been proved.It can be concluded that D N ( A, B ) + D N ( B, C ) ≥ D N ( A, C ) All of the required properties of the proposed which areexpected to be satisfied have been demonstrated. In the nextpart, some examples are offered to further verify the validityof proposed method.IV. N
UMERICAL EXAMPLES AND APPLICATION
In this section, lots of numerical examples and actualapplications are utilized to illustrate the effectiveness of themethod proposed in this paper of measuring distances betweenPFSs. And the accuracy and superiority of this method is alsotestified in this part.
A. Examples and discussions
Example 1:
Let A and B be two PFSs in the finite universe ofdiscourse X, which are defined as A = { ( x, A Y ( x ) , A N ( x )) } and B = { ( x, B Y ( x ) , B N ( x )) } . Besides, a further limitationis defined as: ABLE ID
ISTANCES GENERATED BY E UCLIDEAN ’ S METHOD AND PROPOSEDMETHOD
T he change of δ (cid:48) s value DistancesEuclidean proposed method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TABLE IIT
HE CHANGE OF THE VALUE OF SCORE FUNCTION δ . . . . . . . . . S ( A ) − . − . − . − . . . . . S ( B ) − − . − . − . − . . . . . | S ( A ) | . . . . . . . . | S ( B ) | . . . . . . . . A Y ( x ) = δA N ( x ) = 1 − δA Y ( x ) = δ − . A N ( x ) = 1 . − δA H ( x ) = 0 = B H ( x ) When setting the index of hesitance to 0, the differencebetween 2 PFSs is only about the values of membershipand non-membership. The operation is carried to simplifythe process of calculation. And of course, any parametercan be modified in this way, as long as the mass of eachparameter satisfies the properties of pythagorean fuzzy set.And the differece of classic Euclidean distance and the newproposed is going to be discussed. All the results generatedby the two methods are shown in table 1. And the valuesobtained by score function are also shown in table 2 with theconsistent increase of the value of parameter δ which indicatesa corresponding change in the distance of IFSs.It can be easily told that the distance generated by classicEuclidean’s method is a fixed value. However, this phenon-menon is conflicting with the intuitive judgements. With thechange of the δ , if the results have not changed, then the results The change of D i s t an c e Proposed methodEuclidean distance
Fig. 1. |S(A)||S(B)|
Fig. 2. produced should be regarded irrational and counter-intuitive,which also indicates that the classic Euclidean distance is notsensitive to tiny variation among different PFSs. It may notaccurately reveal potential changes in the relation of PFSswhen they vary correspondingly but not symmetrically, whichis crucial about whether a method of measuring distancecan make reasonable judgements. Besides, when coming toanalyse the variation of the variable δ , the rationality andcorrectness of the proposed method is highlighted. With theincrease of parameter δ , when δ is in the range . to . , the level of vagueness in these PFSs is getting higher,so the distance between two PFSs reaches its zenith. Thatis because the new proposed method can better detect thechanges of different PFSs and present the difference in theresults produced, which improved the sensitivity of distancemeasure compared to classic Euclidean distance. And the trendof the changes of PFSs can also be shown by the variation ofthe value of score function. While the parameter gradually ABLE IIIE
XAMPLES OF
PFS S P F Ss Case Case A i {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} B i {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} P F Ss Case Case A i {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} B i {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} P F Ss Case Case A i {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} B i {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} P F Ss Case Case A i {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} B i {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} TABLE IVD
ISTANCES GENERATED BY DIFFERENT METHODS
M ethods Case Case Case Case Case Case Case Case (cid:101) d Hm . . . . . . . . (cid:101) d Eu . . . . . . . . (cid:101) D Hm . . . . . . . . (cid:101) D Eu . . . . . . . . (cid:101) D C ( β = 1) 0 . . . . . . . . (cid:101) D C ( β = 2) 0 . . . . . . . . (cid:101) D N . . . . . . . . reached the value of . , both of the absolute values ofthe score function of PFSs A and B are becoming smaller,which indicates the level of ambiguity of both PFSs is alsogetting higher. Because the PFSs are getting more uncertain,the distance between them is becoming bigger to indicatetheir distance can not be properly measured. Therefore, thenew proposed method reflects this factor in the results andcan better embody underlying relationship between or amongPFSs. Example 2:
In this part, some numerical examples are givento illustrate the proority of new proposed method comparedto other previous methods. Let A i and B i be two PFSs in the finite universe of discourse X = { x , x } and all of thedocimastic cases are presented in table 3. And the distancesgenerated by different methods are shown in table 4.When coming to analysing the examples given in table 3, it canbe easily concluded that A = A = A , but B (cid:54) = B (cid:54) = B ; A = A , but B (cid:54) = B ; A = A , but B (cid:54) = B .However, the results produced by different methods showa completely different figure among provided methods ofmeasuring distances. After inspecting all of results, it can besummarized that: After comparing all of the method of measuring distancesbetween different PFSs, every method can accurately detecthe differences under the condition in
Case and Case . However, the results produced by (cid:101) D Hm , (cid:101) D Eu , (cid:101) D C ( β = 1) and (cid:101) D C ( β = 2) seem not satisfying. They produce theexactly the same results in completely different cases, whichis counter-intuitive and irrational. Besides, when checking the results produced by (cid:101) d Hm and (cid:101) d Eu in Case and Case , they are also unreasonable due tothe same results in different cases. More than what have been mentioned in point , (cid:101) d Hm also dose not perform well in Case and Case . Becausethe method produces another counter-intuitive results underdifferent conditions. All in all, when comparing all the cases mentioned above,it is very easy to distinguish that the distance of the newproposed method is more sensitive to the changes in PFSsand significantly indicates the discrimination level of differentPFSs, which is much more efficient than any other methods.It is more rational and closer to actual situations. A more important point should be noticed is that the newproposed method performs well under any cases given in thetable 3 in producing distances between PFSs, which other onesgenerate irrational and counter-intuitive results in the samecases.
Notice:
It can be inferred from table 5 that except for the
TABLE VP
ROPERTIES DIFFERENT METHODS SATISFY
Methods PropertiesNon − degeneracy Symmetry Triangleinequality Boundness (cid:101) dHm √ √ √ √ (cid:101) dEu √ √ √ √ (cid:102) DHm √ √ √ √ (cid:102)
DEu √ √ √ √ (cid:102) DC ( β = 1) √ √ √ √ (cid:102) DC ( β = 2) √ √ √ √ (cid:102) DN √ √ √ √ normalized Chen’s distance measure (cid:101) D C , other methods ofmeasuring distances like (cid:101) d Hm , (cid:101) d Eu , (cid:101) D Hm , (cid:101) D Eu , (cid:101) D N satisfyall the properties which the methods of distance measure aresupposed to have. Moreover, the reason for the new proposedmethod can generate more rational and intuitive results is thatthe new proposed method considers discrepancies in PFSs andenlarges their influences in producing distances, which plays asignificant role in manifesting differences. As a result, the newproposed method is more acceptable and conforms to acltualsituations. B. Applications and discussions
In this section, a new algorithm which is developed on thebase of the new proposed method is designed for medicalpattern recognition problems.
Problem narration :
Assume there are a finite universe ofdiscourse X = { x , x , x , ..., x n } , existing medical patterns P = { P , P , P , ..., P k } consisting of n elements in the formof PFSs, expressed as P j = {(cid:104) x i , P jY ( x i ) , P jN ( x i ) } (1 ≤ j ≤ k ) in the finite universe of discourse X . And several examples E = { E , E , E , ..., E r } which is composed of r samples isgiven to be rocognized and testify the correctness of the newalgorithm. And all of the elements in example E is denotedas the form of PFS and the whole example is written as E u = {(cid:104) x i , E uY ( x i ) , E uN ( x i ) } (1 ≤ u ≤ r ) . All in all, whatare expected to be achieved is to decide or classify everyelement in example E u whether belongs to the pattern P j .The algorithm is designed as: Step 1:
For every element in E u , the new proposed method ofmeasuring distances is utlized to produce the distance between P j and E u . D N ( P j , E u ) = (cid:113) n (cid:80) ni =1 (cid:126)m i M ( (cid:126)m i M ) T P j E u Y + P j E u N + P j E u H P j E u Y = P j Y ( x i ) + E u Y ( x i ) P j E u N = P j N ( x i ) + E u N ( x i ) P j E u H = P j H ( x i ) + E u H ( x i ) (cid:126)m i = ( P j Y ( x i ) − E u Y ( x i ) , P j N ( x i ) − E u N ( x i ) , P j H ( x i ) − E u H ( x i )) M = Y N H Y = ( P j Y ( x i )+ E u Y ( x i ))( P j Y ( x i )+ E u Y ( x i )+ P j N ( x i )+ E u N ( x i )) N = ( P j N ( x i )+ E u N ( x i ))( P j Y ( x i )+ E u Y ( x i )+ P j N ( x i )+ E u N ( x i )) H = √ − Y − N Step 2 :
After calculating the distance of every pair of P j and E u , the smallest value of the distances between two PFSs P j and E u is selected, which is written as: (cid:101) D choosenN = min ≤ u ≤ r (cid:101) D N ( P j , E u ) Step 3 :
According to the results generated by step 2, anelement is classified into a pattern P β , which is written as β = arg min ≤ u ≤ r { (cid:101) D N ( P j , E u ) } E u ← P β In order to make the process of classifying more straightfor-ward, a flow chart is offered as follows.Except for that, the corresponding pseudocode is given in
Algorithm . And it can be easily concluded that the complex-ity of time of Algorithm is O ( n ) , where n is a magnitudeof certain problems. Example :
To clarify the specific process of the algorithm, asimple is offered to illustrate details in handling data. Assumethere are three medical patterns P , P and P which areexpressed in the form of PFS in the finite universe of discourse ABLE VIS
YMPTOMS EXPRESSED IN THE FORM OF
PFS
OF PATIENTS IN APPLICATION P atients Symptom Symptom Symptom Symptom Symptom P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) Algorithm 1 :The details of the proposed algorithm
Input:
The sets of every pattern P = { P , P , P , ..., P k } The sets of every sample E = { E , E , E , ..., E r } Output:
The results of classification of samples E u for j = 1; j ≤ k do —– for u = 1; u ≤ r do ———–Generate the distance (cid:101) D N ( P j , E u ) between differentPFSs by using the new proposed method—– end /- Step 1 -/Choose the minimum value of (cid:101) D N ( P j , E u ) as the final dis-tance /- Step 2 -/Classify the tested sample E u into the corresponding pattern P j /- Step 3 -/ end X = { x , x , x } and the details of the three medical patternsare written as follows: P = {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} P = {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} P = {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} Besides, two medical samples S and S are expressed in theform of PFS in the finite unicerse of discourse and defined as: S = {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} S = {(cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105) , (cid:104) x , . , . (cid:105)} What should be done is to categorize sample S and S into coincident classes. According to the procedure introducedabove, the process of achieving the final outcomes is writtenas:Step 1: Generate the distance among P , P , P and S , S by using the new proposed method respectively. The resultsare written as: (cid:101) D N ( P , S ) = 0 . (cid:101) D N ( P , S ) = 0 . (cid:101) D N ( P , S ) = 0 . (cid:101) D N ( P , S ) = 0 . (cid:101) D N ( P , S ) = 0 . (cid:101) D N ( P , S ) = 0 . Step 2: Choose the smallest value of the distances generatedby the new proposed method. And according to the rugulation,the process is written as: (cid:101) D choosenN = (cid:101) D N ( P , S ) = 0 . (cid:101) D choosenN = (cid:101) D N ( P , S ) = 0 . Step 3: Classify the specific samples into corresponding pat-terns: S ← P S ← P And this is the full process of the new algorithm.
Application 1 :
Suppose there are four patients, namely Al , Bob , Joe and
T ed , who are denoted as P = { P , P , P , P } .Besides, five attibutes which are symptoms in fact areintrodeced as T emperature , Headache , Stomach pain , Cough and
Chest pain , which are denoted as A = { a , a , a , a , a } . More than that, diagnostic results aredivided into five categories, V iralf ever , M alaria , T yphoid , Stomach problem and
Chest , which are also denoted as D = { D , D , D , D , D } . With the help of the conceptof PFS, all of the examples are presented in the form ofPFS, which is effective in judging proper patterns. And everydetails of the patients are shown in table . Additionaly, thediagnoses in the form of PFS are also presented in table .By using the new proposed method of measuring distancebetweent PFSs, all the results generated by the new proposedmethod are shown in table . According the standards raisedin the algorithm, all of final judgements are presented intable . And it can be concluded that Al ( P ) is diagnosedthat he suffers from M alaria ( D ) , Bob ( P ) is diagnosedthat he suffers from Stomach problem ( D ) , Joe ( P ) isdiagnosed that she suffers from T yphoid ( D ) and T ed ( P ) is diagnosed that he suffers from V iral f ever ( D ) . And allthe judgements produced by other methods and the results ABLE VIIS
YMPTOMS EXPRESSED IN THE FORM OF
PFS
OF DIAGNOSES IN APPLICATION Diagnoses Symptom Symptom Symptom Symptom Symptom D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) TABLE VIIID
ISTANCES GENERATED BY PROPOSED METHOD IN APPLICATION P atients DistancesD D D D D P . . . . . P . . . . . P . . . . . P . . . . . TABLE IXJ
UDGEMENTS GENERATED BY DIFFERENT METHODS IN APPLICATION Methods DiagnosesP P P P De et al. [66]
Malaria Stomachproblem T yphoid Malaria
Own [65]
Malaria Stomachproblem Malaria Malaria
Szmidt et al. [67]
Malaria Stomachproblem T yphoid V iralfever
Mondal et al. [68]
Malaria Stomachproblem T yphoid V iralfever
Wei et al. [55]
Malaria Stomachproblem T yphoid V iralfever
Proposed method
Malaria Stomachproblem T yphoid V iralfever generated by the new proposed method are placed together intable to verify the correctness of the latter one. In the chart,what is the most obvious is that all of the methods have reachan agreement that Al ( P ) is diagnosed with M alaria ( D ) and Bob ( P ) is diagnosed with Stomach problem ( D ) ,which is satisfying and rational. However, when comingto judge the situation of patient Joe ( P ) , the diagonosesvary among different methods. Five of them give judgementsthat Joe ( P ) is suffering from T yphoid ( D ) and onlyone of them considers that this patient is suffering from M alaria ( D ) . Additionally, with respect to T ed ( P ) , fourof the methods choose V iral f ever ( D ) as the diagnosis of Fig. 3.
T ed ( P ) and the other tqo methods regard that T ed ( P ) issuffering from M alaria ( D ) . All of the results demonstratethat it is very difficult to diagnose T ed ( P ) , because theremay be a potential relationship between V iral f ever ( D ) and M alaria ( D ) leading to a conflicting stage when comparingthe results of different methods. Anyway, the results producedby Szmidt et al.’s method, Mondal et al.’s method and Weiet al.’s method conform to the ones produced by the newproposed method, which proves that the accuracy and validityof the new proposed method in real application and thefeasibility of the new proposed method in practical usage. Application 2 :
Suppose there are four patients, namely
Ram , M ari , Sugu and
Somu , who are denoted as P = { P , P , P , P } . Besides, five attibutes which aresymptoms in fact are introdeced as T emperature , Headache , Stomach pain , Cough and
Chest pain , which are denotedas A = { a , a , a , a , a } . More than that, diagnostic resultsare divided into five categories, V iralf ever , M alaria , T yphoid , Stomach problem and
Chest , which are alsodenoted as D = { D , D , D , D , D } . With the help ofthe concept of PFS, all of the examples are presented in theform of PFS, which is effective in judging proper patterns.And every details of the patients are shown in table . ABLE XS
YMPTOMS EXPRESSED IN THE FORM OF
PFS
OF PATIENTS IN APPLICATION P atients Symptom Symptom Symptom Symptom Symptom P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) TABLE XIS
YMPTOMS EXPRESSED IN THE FORM OF
PFS
OF DIAGNOSES IN APPLICATION Diagnoses Symptom Symptom Symptom Symptom Symptom D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) TABLE XIID
ISTANCES GENERATED BY PROPOSED METHOD IN APPLICATION P atients DistancesD D D D D P . . . . . P . . . . . P . . . . . P . . . . . TABLE XIIIJ
UDGEMENTS GENERATED BY DIFFERENT METHODS IN APPLICATION Methods DiagnosesP P P P Ngan et al. [72]
Malaria Stomach problem T yphoid Malaria
Proposed method
Malaria V iral fever Stomach problem Malaria
Additionaly, the diagnoses in the form of PFS are alsopresented in table . By using the new proposed method ofmeasuring distance betweent PFSs, all the results generated Fig. 4. by the new proposed method are shown in table . Accordingthe standards raised in the algorithm and other methods, allof final judgements are presented in table .Although the results produced by two methods are somekind of different, it can be easily inferred that the newproposed method conforms to the actual situation. Withrespect to patient and patient , two methods have reacheda agreement, which is satisfying. But when coming to judgingpatient and patient , the situation is becoming morecomplex. The referenced method considers M ari ( P ) is ABLE XIVS
YMPTOMS EXPRESSED IN THE FORM OF
PFS
OF PATIENTS IN APPLICATION P atients Symptom Symptom Symptom Symptom Symptom P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) P (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) TABLE XVS
YMPTOMS EXPRESSED IN THE FORM OF
PFS
OF DIAGNOSES IN APPLICATION Diagnoses Symptom Symptom Symptom Symptom Symptom D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) D (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) (cid:104) a , . , . (cid:105) TABLE XVID
ISTANCES GENERATED BY PROPOSED METHOD IN APPLICATION P atients DistancesD D D D D P . . . . . P . . . . . P . . . . . P . . . . . TABLE XVIIJ
UDGEMENTS GENERATED BY DIFFERENT METHODS IN APPLICATION Methods DiagnosesP P P P Smuel and Rajakumar [73]
Stress Spinal problem V ision problem Stress
Xiao [74]
Stress Spinal problem V ision problem Stress
Proposed method
Stress Spinal problem V ision problem Stress suffering from
Stomach problem , while the new proposedmethod illustrates that
M ari ( P ) is diagnosed with V iral f ever . Compared with the characteristics of patient Fig. 5. and pattern and , it can be concluded that the situation ofpatient conforms to every symptoms contained in pattern while the situation of patient is conflicting with pattern in symptoms and , which indicates that there is abigger probability of patient suffering from V iral f ever instead of
Stomach problem . Similar circumstances occurin the judgment of patient . The referenced methodconsiders Sugu ( P ) is suffering from T yphoid , whilethe new proposed method illustrates that
Sugu ( P ) isdiagnosed with Stomach problem . Patient is conflictingith pattern in symptoms and while the situationof patient is only discordant with the symptom , whichindicates that the judgements given by the new proposedmethod are more rational and intuitive according to the dataoffered in table and . As a result, the new proposedmethod has much better performance in handling actual cases. Application 3 :
Suppose there are four patients, namely
Ragu , M athi , V elu and
Karthi , who are denoted as P = { P , P , P , P } . Besides, five attibutes which are symptomsin fact are introdeced as Headache , Acidity , Burning eyes , Back pain and
Depression , which are denoted as A = { a , a , a , a , a } . More than that, diagnostic results aredivided into five categories, Stress , U lcer , V ision problem , Spinal problem and
Bloodpressure , which are also denotedas D = { D , D , D , D , D } . With the help of the conceptof PFS, all of the examples are presented in the form ofPFS, which is effective in judging proper patterns. And everydetails of the patients are shown in table . Additionaly, thediagnoses in the form of PFS are also presented in table .By using the new proposed method of measuring distancebetweent PFSs, all the results generated by the new proposedmethod are shown in table . According the standards raisedin the algorithm and other methods, all of final judgementsare presented in table .After checking the results generated by the new propsoedmethods, P has the least value of (cid:101) D N = 0 . ; P hasthe least value of (cid:101) D N = 0 . ; P has the least value of (cid:101) D N = 0 . ; P has the least value of (cid:101) D N = 0 . .All in all, the results produced by Sumuel and Rajakumar’smethod and Xiao’s method conform to the ones produced bythe new proposed method, which proves that the accuracy andvalidity of the new proposed method in real application andthe feasibility of the new proposed method in practical usage.V. C ONCLUSION
In the theory of pythagorean fuzzy sets, how to accuratelyand properly measure the distance between PFSs is still anopen issue, which may lead to chaos in pattern recognition.To solve this problem, in this paper, a completely new methodof measuring distance between different PFSs is proposed,which statisfy all of the properties required by the axiom ofmeasurement. The mainadvantage of the new proposed methodis that it considers the indec of hesitance and distribute its massto membership and non-membership in a reasonable way tostrengthen the role which membership and non-membershipplays in generating distances betweent PFSs. Besides, therudction in the index of hesitance is also very important inalleviating the vagueness in PFSs and helpful in recognizingcorresponding targets or patterns. Owing to this operation, theaccuracy of the new proposed method is further improved.All in all, the proposeed method produces much more rationalresults than some previous methods, which is more closer toactual situations and conforms to intuitive judgements. Whencompared with other methods, the results produced by theproposed method also reached an agreement with the ones generated by other methods, which demonstrates the feasibilityof the new proposed method in practical application. In thispaper, the algorithm which is developed on the basis of thenew proposed method offers a promising and reliable solutionto address the recognition problems in medical diagnosis. Andin future, this new proposed method can be used to measurevagueness, correlation and so on in corresponding researchareas. R
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