# A Further Study on Multiperiod Health Diagnostics Methodology under a Single-Valued Neutrosophic Set

Authors: Jason Chih-sheng Chou; Yi-Fong Lin; Scott Shu-Cheng Lin
Journal: Computational and Mathematical Methods in Medicine
 (2020)

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## Abstract

Employing the concept and function of tangency with similarity measures and counterpart distances for reliable medical consultations has been extensively studied in the past decades and results in lots of isomorphic measures for application. We compared the majority of such isomorphic measures proposed by various researchers and classified them into (a) maximum norm and (b) one-norm categories. Moreover, we found that previous researchers used monotonic functions to transform an identity function and resulted in complicated expressions. In this study, we provide a theoretical foundation to explain the isomorphic nature of a newer measure proposed by the following research paper against its studied existing one in deriving the same pattern recognition results. Specifically, this study initially proposes two similarity measures using maximum norm, arithmetic mean, and aggregation operators and followed by a detailed discussion on their mathematical characteristics. Subsequently, a simplified version of such measures is presented for easy application. This study completely covers two previous methods to point out that the complex approaches used were unnecessary. The findings will help physicians, patients, and their family members to obtain a proper medical diagnosis during multiple examinations.


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## Body

## 1. Introduction

With the current state of medical consultations, a patient may present with several different inclinations of diseases, especially for multilevel diagnosis. This poses a critical question on how can a physician provide a reliable conclusion for the patient’s health status and extend to the patient and his or her family members. Zadeh [1] pioneered the development of fuzzy sets (FSs) to deal with complex, uncertain, vague, and incomplete problems in the physical world. There are many important extensions of the FSs such as the neutrosophic sets (NSs) that were proposed by Smarandache [2]. However, NSs pose to be difficult in actual applications such that Wang et al. [3, 4] developed single-valued neutrosophic sets (SVNSs) to provide a simplification of NSs. SVNSs have three membership functions: truth, indeterminacy, and falsity, which give sufficient information to analyze a patient’s condition and will be used to construct a systematic approach to determine the most probable disease. Many papers studied SVNSs: for example, Ye [5] constructed an operator using weighted averages under a hesitant fuzzy element (HFE) environment and a single-valued neutrosophic set. Another study is Ye [6] to consider similarity measures with the cosine function, and then Ye and Fu [7] referred to the similarity measures in Ye [6]. Ninety-three papers have cited Ye [6] in their references. We list them in the following: Peng and Dai [8], Ye [9], Biswas et al. [10], Broumi et al. [11], Chatterjee et al. [12, 13], Cui and Ye [14], Fan [15], Fan et al. [16], Fu et al. [17], Gou and Wang [18], Jha et al. [19, 20], Li et al. [21–23], Liu and Luo [24], Luo et al. [25], Nancy and Garg [26], Nguyen et al. [27], Peng [28], Peng and Dai [29], Rajangam and Annamalai [30], Şahin [31], Shao et al. [32], Shi and Yuan [33], Tian et al. [34], Wang et al. [35], Wei and Zhang [36], Xie et al. [37], Ye [38], Zhai et al. [39], Zhang et al. [40], Zhao et al. [41], Akram et al. [42], Ali et al. [43], Cai and Yang [44], Fan et al. [45], Garg and Nancy [46], Guan et al. [47], Karaaslan and Hayat [48], Khanet al. [49], Küçük and Şahin [50], Liu [51], Liu et al. [52], Liu et al. [53], Liu and You [54], Mondal et al. [55–57], Ngan et al. [58], Peng et al. [59], Pramanik et al. [60, 61], Şahin [62], Singh [63], Tian et al. [64], Tu et al. [65], Wei and Wei [66], Wu et al. [67], Wu et al. [68], Xiong and Cheng [69], Ye [70], Zeming et al. [71], Zhang et al. [72], Chen and Ye [73], Deli and Şubaş [74], Fan and Ye [75], Fu and Ye [76], Hu et al. [77], Lu and Ye [78], Ma et al. [79], Peng et al. [80–82], Pramanik et al. [83], Singh [84], Thanh et al. [85], Tian et al. [86], Wang and Zhang [87], Xu et al. [88], Zhou and Abdel Wahab [89], Chou [90], Liu and Tang [91], Meng et al. [92], Nancy and Garg [93], Shi [94], Son [95], Ye [96], Ye and Fu [7], Zhang et al. [97], Zhang et al. [98], and Zhang et al. [99]. Among those 93 papers, 92 papers only mentioned Ye [6] in their introduction, and then they concentrated on their new similarity measures and applied their new similarity measures to solve real application problems. Only one paper, Ye and Fu [7] have written down the similarity measures proposed by Ye [6]. However, Ye and Fu [7] did not provide any discussion for the similarity measures proposed by Ye [6] with cosine functions. Instead, Ye and Fu [7] developed new similarity measures with tangent functions. Hence, Ye and Fu [7] developed two tangent similarity measures and subsequently defined aggregation operators for multiperiod medical diagnosis. There are 49 papers that have cited Ye and Fu [7] in their references. We list them in the following: Liu et al. [100], Peng and Dai [8], Awang et al. [101], Chatterjee et al. [12, 13], Cui and Ye [14], Fu et al. [17], Gou and Wang [18], Hu et al. [102], Li et al. [21], Long et al. [103], Luo et al. [25], Nancy and Garg [26], Nguyen et al. [27], Rajangam and Annamalai [30], Ren et al. [104], Shen et al. [105], Shi and Yuan [33], Smith [106], Xie et al. [37], Akram et al. [107], Ali et al. [43], Hu et al. [108], Ji et al. [109–111], Karaaslan [112], Karaaslan and Hayat [48], Khan et al. [113], Liu [51], Mondal et al. [55–57], Naz et al. [114], Peng and Dai [115], Tu et al. [65], Uluçay et al. [116], Wu et al. [67], Ye [70], Ashraf et al. [117], Chen and Ye [73], Fu and Ye [76], Guan et al. [118], Guo et al. [119], Hu et al. [77], Lu and Ye [78], Naz et al. [120], Peng and Dai [121], and Thanh et al. [85]. We have examined those 49 papers to find out that they only review Ye and Fu [7] in their literature study without any further discussions concerning Ye and Fu [7]. There are many similarity measures developed by researchers. However, there are only a few studies that conduct comprehensive examinations for previously published papers. For example, Julian et al. [122] improved three questionable results of Mitchell [123]. Deng and Chao [124] and Tung et al. [125] further revised two questionable findings of Julian et al. [122]. We follow this trend to examine the findings of Ye [6] and Ye and Fu [7] with a detailed analysis.Prior studies generally repeated in proposing a newer similarity measure and been evaluated against an existing one (i.e., a studied measure) in deriving the same pattern recognition results for medical diagnosis problems. Thus, they result in lots of isomorphic measures. Many researchers as mentioned above claimed that their new similarity measures are supported by existing ones. We took a detailed examination for some similarity measures and found that their counterpart distances (that is, similarity measure plus distance is the unity) can be classified into (a) maximum norm and (b) one-norm categories. Additionally, many researchers used monotonic functions to transform the identity function (that is,fx=x) into complicated expressions. These review comments trigger us to study the similarity measures and point out that they are isomorphic when compared to those two categories. Hence, we provide a theoretical foundation to explain why a newly proposed measure by the following paper will derive the same results as its studied one (i.e., an existing measure).Initially, we will use a simple example to explain the isomorphic property of this research for ordinary readers. We concentrate on Ye [6] and Ye and Fu [7], and they are representing an existing measure (i.e., Ye [6]) and the following measure (i.e., Ye and Fu [7]). Ye [6] used the cosine function, and Ye and Fu [7] applied the tangent function, where cosine is a strictly decreasing function, and tangent is a strictly increasing function. The isomorphic property can be understood by assuming 1>dista,x>dista,y, then
(1)Sima,x<Sima,y.The strictly decreasing property of cosine impliescosdista,x<cosdista,y. In this example, Ye [6] would create a new similarity measure as
(2)Sim6a,x=cosdista,x,such that
(3)Sim6a,x<Sim6a,y.In fact, the similarity measure proposed by Ye [6] preserves the original ordering of Equation (1). The strictly increasing property of tangent implies tandista,x>tandista,y. As an example of the following study, Ye and Fu [7] would create a new similarity measure as
(4)Sim7a,x=1−tandista,x,such that
(5)Sim7a,x<Sim7a,y.The new similarity measure proposed by Ye and Fu [7] preserves the original ordering of Equation (1) as well. In Ye and Fu [7], they claimed that they used the tangent function to develop a new similarity measure. We point out that Ye and Fu [7] applied “one minus tangent function” to create a new similarity measure. Based on the isomorphic property, Ye [6] and Ye and Fu [7] will preserve the same orderings as those in previously published papers that were constructed with different monotonic functions. Therefore, we decide to provide a reasonable explanation for researching the reason why a newer similarity measure will still derive the same results as those in already published papers.The rest of the paper is organized as follows. Section2 of this paper introduced some previous results related to Ye and Fu [7] in terms of the tangent similarity measures and aggregation operators. Section 3 highlighted the theoretical results of Jaccard [126], Dice [127], and Salton and McGill [128]. Our theoretical development of two similarity measures using maximum norm, arithmetic mean, and two aggregation operators is presented in Section 4. Section 5 showed how this study used the subject example of Ye and Fu [7] as a comparison for the proposed two aggregation operators and thereby showing that it has the same diagnosis results. A detailed analysis is presented in Section 6 about the tangent and improved cosine similarity measures and how there are transformations of the proposed revision thereby showing that these complicated computations of Ye [6] and Ye and Fu [7] are redundant. The discussion is concluded in Section 7.


## 2. Review of Related Results for Ye and Fu [7]

Wang et al. [4] defined an SVNS as A=x,TAx,IAx,FAxx∈X, such that X is a universe of discourse with the three membership functions: truth TAx, indeterminacy IAx, and falsity FAx that satisfy TAx, IAx, and FAx∈0,1.ForA=xj,TAxj,IAxj,FAxjxj∈X and B=xj,TBxj,IBxj,FBxjxj∈X which are different SVNSs with a universal set X=x1,x2,⋯,xn, Ye and Fu [7] reviewed three similarity measures: Jaccard [126], Dice [127], and Salton and McGill [128] as follows.Jaccard [126] defined the “Jaccard index”
(6)SJA,B=1n∑j=1nΩjΔAj+ΔBj−Ωj,where
(7)Ωj=TAxjTBxj+IAxjIBxj+FAxjFBxj,(8)ΔAj=TA2xj+IA2xj+FA2xj,(9)ΔBj=TB2xj+IB2xj+FB2xj.Dice [127] defined the “Dice similarity measure”:
(10)SDA,B=1n∑j=1n2ΩjΔAj+ΔBj.Salton and McGill [128] presented that
(11)SCA,B=1n∑j=1nΩjΔAjΔBj,which is based on the inner product of two vectors and then normalizes the result.Next, Ye and Fu [7] claimed that Equation (11) has some disadvantages such that they presented two improved cosine similarity measures as follows in Ye [6]:
(12)C1A,B=1n∑j=1ncosπmaxTAxj−TBxj,IAxj−IBxj,FAxj−FBxj2,(13)C2A,B=1n∑j=1ncosπTAxj−TBxj+IAxj−IBxj+FAxj−FBxj6.Using the concept of tangency, Ye and Fu [7] developed two new equations as follows:
(14)T1A,B=1−1n∑j=1ntanπmaxTAxj−TBxj,IAxj−IBxj,FAxj−FBxj4,and
(15)T2A,B=1−1n∑j=1ntanπTAxj−TBxj+IAxj−IBxj+FAxj−FBxj12.Finally, Ye and Fu [7] constructed two weighted measures that are still built on tangency:
(16)Tw1A,B=1−∑j=1nwjtanπmaxTAxj−TBxj,IAxj−IBxj,FAxj−FBxj4,(17)Tw2A,B=1−∑j=1nwjtanπTAxj−TBxj+IAxj−IBxj+FAxj−FBxj12.During a multiple period diagnosis, Ye and Fu [7] used tangent and improved cosine measures to find out the most probable disease suffered by the patient. We recall the decision algorithm proposed by Ye and Fu [7]:AssignS=S1,S2,⋯,Sm as the group of different symptoms, T=t1,t2,⋯,tq as a series of multiple periods, and D=D1,D2,⋯,Dn as a group of possible diagnoses. Let the weight of period tk be denoted as wtk with wtk>0 and ∑k=1qwtk=1. For a patient P with various symptoms, Ye and Fu [7] used Sjtk to denote the SVNS for this patient throughout the period tk for symptoms Sj and Cij, and to represent the SVNS for disease Di related to the symptom Sj. The algorithm is cited as follows:Step 1.
Ye and Fu [7] applied T2 of Equation (15) or C2 of Equation (13) to compute the similarity between Sjtk and Cij as T2P,Di,tk or C2P,Di,tk.Remark 1.
Ye and Fu [7] used TwiP,tk where the relation of the patient’s disease Di and the period tk is not indicated clearly. Hence, the proposed change of the expression is from TwiP,tk to T2P,Di,tk.Step 2.
To derive the weighted aggregation value,MP,Di for the patient related to the disease Di:
(18)MT2P,Di=∑k=1qwtkT2P,Di,tk,or
(19)MC2P,Di=∑k=1qwtkC2P,Di,tk.Step 3.
Present the most probable diagnosis to the patient according to the highest amount among weighted aggregations.Step 4.
Finish.


## 3. Our Theoretical Results for Similarity Measures of Jaccard, Dice, and Cosine

This section demonstrated that(20)SCA,B≥SDA,B≥SJA,B.Hence, we will prove that relations among the three similarity measures proposed by Jaccard [126], Dice [127], and Ye [6].Building on the concept of the arithmetic average is greater than or equal to the geometric average, it is found that(21)ΔAj+ΔBj2≥ΔAjΔBj,and thereby deriving that
(22)SCA,B≥SDA,B.From the definition ofΔAj, ΔBj, and Ωj of Equations (7)–(9), respectively, the following is obtained:
(23)ΔAj+ΔBj−2Ωj=TAxj−TAxj2+IAxj−IAxj2+FAxj−FAxj2≥0,such that
(24)2ΔAj+ΔBj≥1ΔAj+ΔBj−Ωj,SDA,B≥SJA,B.Theorem2 below summarizes the findings.Theorem 2.
Using two SVNSsA=xj,TAxj,IAxj,FAxjxj∈X and B=xj,TBxj,IBxj,FBxjxj∈X together with the universe of discourse X=x1,x2,⋯,xn, SCA,B≥SDA,B≥SJA,B is confirmed.


## 4. Our Proposed Similarity Measures

Using the two SVNSs A and B with a universal setX=x1,x2,⋯,xn as above, it is assumed that:
(25)Mw1A,B=1−∑j=1nwjαj,with
(26)αj=maxTAxj−TBxj,IAxj−IBxj,FAxj−FBxj,and(27)Mw2A,B=1−∑j=1nwjβj,with
(28)βj=TAxj−TBxj+IAxj−IBxj+FAxj−FBxj3,such that Mw1 is related to the maximum norm, and Mw2 is related to the arithmetic mean.Our proposed weighted aggregation value for a patient,P, to a disease, D, is computed as
(29)Mw1P,Di=∑k=1qwtkMw1SjPtk,Cij,where SjPtk is the SVNS for the patient P, with the symptom Sj, at period tk, and Cij is the SVNS for the disease Di and the symptom Sj. On the other hand, we provide a second aggregated value by our proposed second similarity measure of Equation (27) as
(30)Mw2P,Di=∑k=1qwtkMw2SjPtk,Cij.


## 5. Diagnoses Illustration

The example from Ye and Fu [7] is used, wherein a patient group presents with an array of diseases, D=D1,D2,⋯,D5 = {viral fever, malaria, typhoid, gastritis, stenocardia}, the list of symptoms as, S=S1,S2,⋯,S5 = {temperature, headache, stomach pain, cough, chest pain}, and a set of four patients, P1,P2,P3,P4. The characteristic values of the patients and symptoms for a certain period are listed in Table 1. For example, S1P1t1 is the SVNS for the patient P1 with the symptom S1 at period t1=0.8,0.6,0.5.Table 1
(Reproduction of Table 5 of Ye and Fu [7]) Characteristic values related to the patients and their symptoms for three periods.
tkS1 (temperature)S2 (headache)S3 (stomach pain)S4 (cough)S5 (chest pain)P1t1(0.8,0.6,0.5)(0.5,0.4,0.3)(0.2,0.1,0.3)(0.7,0.6,0.3)(0.4,0.3,0.2)t2(0.7,0.3,0.2)(0.6,0.3,0.2)(0.3,0.2,0.4)(0.6,0.5,0.2)(0.6,0.5,0.3)t3(0.5,0.2,0.4)(0.6,0.3,0.4)(0.3,0.3,0.5)(0.4,0.3,0.2)(0.6,0.4,0.4)P2t1(0.6,0.6,0.1)(0.1,0.2,0.6)(0.3,0.2,0.8)(0.6,0.2,0.3)(0.2,0.3,0.7)t2(0.5,0.4,0.2)(0.2,0.2,0.6)(0.2,0.1,0.7)(0.8,0.3,0.1)(0.1,0.1,0.8)t3(0.8,0.3,0.1)(0.2,0.1,0.5)(0.1,0.1,0.9)(0.7,0.2,0.0)(0.1,0.1,0.8)P3t1(0.3,0.1,0.2)(0.3,0.2,0.2)(0.7,0.6,0.7)(0.3,0.2,0.2)(0.4,0.4,0.3)t2(0.4,0.2,0.2)(0.5,0.1,0.3)(0.4,0.2,0.2)(0.5,0.3,0.3)(0.6,0.3,0.2)t3(0.8,0.7,0.6)(0.7,0.5,0.5)(0.4,0.1,0.1)(0.7,0.3,0.4)(0.7,0.4,0.5)P4t1(0.2,0.1,0.7)(0.2,0.3,0.7)(0.2,0.2,0.7)(0.2,0.1,0.8)(0.8,0.2,0.1)t2(0.1,0.1,0.6)(0.1,0.2,0.8)(0.2,0.1,0.8)(0.3,0.0,0.9)(0.7,0.1,0.2)t3(0.1,0.1,0.8)(0.1,0.2,0.7)(0.3,0.1,0.8)(0.2,0.1,0.9)(0.9,0.1,0.1)Table2 continues the list between (a) probable diseases and (b) their corresponding symptoms. For example, C12=0.3,0.2,0.5 is the SVNS for disease D1 with the symptom S2.Table 2
(Reproduction of Table 4 of Ye and Fu [7]) Characteristic values related to five diseases and five symptoms.
S1 (temperature)S2 (headache)S3 (stomach pain)S4 (cough)S5 (chest pain)D1 viral fever(0.4,0.6,0.0)(0.3,0.2,0.5)(0.1,0.3,0.7)(0.4,0.3,0.3)(0.1,0.2,0.7)D2 malaria(0.7,0.3,0.0)(0.2,0.2,0.6)(0.0,0.1,0.9)(0.7,0.3,0.0)(0.1,0.1,0.8)D3 typhoid(0.3,0.4,0.3)(0.6,0.3,0.1)(0.2,0.1,0.7)(0.2,0.2,0.6)(0.1,0.0,0.9)D4 gastritis(0.1,0.2,0.7)(0.2,0.4,0.4)(0.8,0.2,0.0)(0.2,0.1,0.7)(0.2,0.1,0.7)D5 stenocardia(0.1,0.1,0.8)(0.0,0.2,0.8)(0.2,0.0,0.8)(0.2,0.0,0.8)(0.8,0.1,0.1)With the proposed method ofMw1Ps,Di and Mw2Ps,Di, for s=1,2,3,4 and i=1,2,⋯,5 with weights for periods wt1=0.25, wt2=0.35, and wt3=0.4 and weights for symptoms wj=1/5, for j=1,2,⋯,5, the results are presented in Table 3.Table 3
Study findings ofMw1Ps,Di and Mw2Ps,Di.
D1D2D3D4D5Diagnosis resultMw1P1,Di0.6730∗0.59900.65600.52600.5270D1 viral feverMw1P2,Di0.79700.8730∗0.67800.55100.5330D2 malariaMw1P3,Di0.5860∗050400.55400.53000.4520D1 viral feverMw1P4,Di0.52100.47500.56400.60200.8910∗D5 stenocardiaMw2P1,Di0.7770∗0.73230.74830.68100/6510D1 viral feverMw2P2,Di0.86830.9250∗0.78970.68830.6670D2 malariaMw2P3,Di0.7573∗0.69830.69600.71330.6573D1 viral feverMw2P4,Di0.69170.66170.71700.75770.9443∗D5 stenocardia∗Maximum values and thus, the most probable diagnosis.From Table3, by the aggregation operator of Mw1 with the maximum norm or the aggregation operator Mw2 with the arithmetic mean, the same results are derived for patients P1 and P3 suffering from viral fever, the patient P2 suffering malaria, and the patient P4 suffering stenocardia. The study’s derivations are the same as that of Yu and Fu [7] with the aggregation operator MT2 of Equation (18) and the aggregation operator MC2 of Equation (19).


## 6. Further Discussion for Aggregation Operators

In this section, we will explain that previous researchers used monotonic functions to transform the identity function (that is,fx=x) to complicated expressions. Based on the abbreviations of αj and βj, it is possible to rewrite Tw1 of Equation (16) and Tw2 of Equation (17) proposed by Yu and Fu [7] to simplify their expressions as follows
(31)Tw1A,B=1−∑j=1nwjtanπαj/4,(32)Tw2A,B=1−∑j=1nwjtanπβj/4.Comparing Equation (25) with Equation (31), the following general expression is implied:
(33)Simw1A,B=1−∑j=1nwjfαj,such that in Equation (25), for Mw1, fx=x, and in Equation (31), for Tw1, fx=tanπx/4.By the same argument, comparing Equation (27) with Equation (32) obtains a general expression:
(34)Simw2A,B=1−∑j=1nwjfβj,such that in Equation (27), for Mw2, fx=x, and in Equation (32), for Tw2, fx=tanπx/4.Consequently, we can further simplifyC1A,B of Equation (12) and C2A,B of Equation (13) proposed by Ye [6] as follows
(35)C1A,B=1n∑j=1ncosπαj/2,(36)C2A,B=1n∑j=1ncosπβj/2.Hence, Equations (25), (31), and (35) are used in comparison to determine their relationship with αj of Equation (26). By the same observation, it is found that Equations (27), (32), and (36) are related to βj of Equation (28).In Equations (25) and (27), the most natural approach fx=x is applied by us. Meanwhile, in Equations (31) and (32), Ye and Fu [7] used fx=tanπx/4, and in Equations (35) and (36), Ye [6] used fx=cosπx/2.fx=x and fx=tanπx/4 are both increasing functions from f0=0 to f1=1. On the other hand, fx=cosπx/2 is a decreasing function from f0=1 to f1=0. All of them are monotonic functions.For a similarity measure, saySim, Sim0=1, and Sim1=0 are ideal, such as in Equations (25), (27), (31), and (32) and then abstractly expressed in Equations (33) and (34), researchers use 1−∑j=1mwjfαj or 1−∑j=1mwjfβj.There are infinite increasing functions that satisfyf0=0 and f1=1. However, Ye and Fu [7] did not provide any explanation of why fx=tanπx/4 was selected by them.Moreover, there are infinite decreasing functions that satisfyf0=1 and f1=0. Ye [6] and Ye and Fu [7] also did not explain why fx=cosπx/2 was used.Since our proposed approach to use the simplest formfx=x can still derive the desired diagnosis, the study recommends not to apply the complicated method proposed by Ye [6] with fx=cosπx/2 and Ye and Fu [7] with fx=tanπx/4. Instead, using our simplest form fx=x will help researchers solve their similarity measure problems and also reduce the complicated computation caused by Ye [6] and Ye and Fu [7].For completeness, we point out that Section 5.3 of Ye and Fu [7] shows the comparative analysis of Ye and Fu [7]. It is mentioned that the multiple period method is better than a single period of Ye [6] since the latter poses more difficulty in giving a suitable diagnosis for a specific patient presenting with a specific disease. Recalling the example of Section 5 of Ye and Fu [7], it showed that the aggregation operator MT2 of Equation (18) was used with tangent similarity measure T2 of Equation (15) and the aggregation operator MC2 of Equation (19) with tangent similarity measure C2 of Equation (13) which was proposed by Ye [6]. Both operators derived the same diagnosis results for four patients as obtained in Table 3 by aggregation operator Mw1 of Equation (29) and aggregation operator Mw2 of Equation (30).In Ye and Fu [7], the improved similarity measure C2 of Ye [6] was used and repeated three times, then combined with the weighted mean such that the improved similarity measure C2 of the latter study can be used to solve multiperiod medical diagnosis problems. Ye and Fu [7] already successfully applied C2 in their algorithm as mentioned that using the tangent function and cosine measure, the diagnoses are similar and therefore proving that their proposed multiperiod method was indeed effective. The reference implies the possibility to use the cosine measure from Ye [6] for medical diagnoses over multiple periods. Ye and Fu [7] first used the improved cosine similarity measure of Equation (13) proposed by Ye [6] to support their tangent similarity measure of Equation (15), since aggregation approach by the two measures derive identical diagnosis results. On the contrary, in the same section, Ye and Fu [7] claimed that the improved cosine similarity measure of Equation (13) proposed by Ye [6] is unable to solve multiperiod medical diagnosis problems. Hence, the study points out that Section 5.2 or the comparative analysis of Ye and Fu [7] contained questionable results.


## 7. Conclusion

In this paper, we tried to claim that there is an isomorphism between (a) our simplest similarity measures and (b) those complicated similarity measures. Hence, to construct new similarity measures that can be explained by isomorphism is tedious and unnecessary. This study provided a detailed analysis of the improved cosine similarity measure proposed by Ye [6] and the tangent similarity measure proposed by Ye and Fu [7] to point out that these are transformations of the maximum norm and the arithmetic mean. As a recommendation, researchers can directly apply the well-known distance: the maximum norm and the arithmetic mean, to simplify the complicated computations in Ye and Fu [7].



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