Computational and Mathematical Methods in Medicine
Volume 2020 (2020), Article ID 6093545, 11 pages
https://doi.org/10.1155/2020/6093545

Review Article

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

Jason Chih-sheng Chou,¹ Yi-Fong Lin,² and Scott Shu-Cheng Lin³

¹The MITRE Corporation, McLean, VA 22102, USA
²Department of Fashion Industry Management, Hsing Wu University, Taiwan
³Department of Hotel Management, Lee-Ming Institute of Technology, Taiwan

Correspondence should be addressed to Yi-Fong Lin; daniellyf@mail.lit.edu.tw

Received 30 March 2020; Accepted 6 July 2020; Published 1 August 2020

Academic Editor: Markos G. Tsipouras

Copyright © 2020 Jason Chih-sheng Chou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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, ) 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 , then

The strictly decreasing property of cosine implies . In this example, Ye [6] would create a new similarity measure as

such that

In fact, the similarity measure proposed by Ye [6] preserves the original ordering of Equation (1). The strictly increasing property of tangent implies . As an example of the following study, Ye and Fu [7] would create a new similarity measure as

such that

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. Section 2 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 , such that is a universe of discourse with the three membership functions: truth , indeterminacy , and falsity that satisfy , , and .

For and which are different SVNSs with a universal set , 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”

where

Dice [127] defined the “Dice similarity measure”:

Salton and McGill [128] presented that

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]:

Using the concept of tangency, Ye and Fu [7] developed two new equations as follows:

and

Finally, Ye and Fu [7] constructed two weighted measures that are still built on tangency:

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]:

Assign as the group of different symptoms, as a series of multiple periods, and as a group of possible diagnoses. Let the weight of period be denoted as with and . For a patient with various symptoms, Ye and Fu [7] used to denote the SVNS for this patient throughout the period for symptoms and , and to represent the SVNS for disease related to the symptom . The algorithm is cited as follows:

Step 1. Ye and Fu [7] applied of Equation (15) or of Equation (13) to compute the similarity between and as or .

Remark 1. Ye and Fu [7] used where the relation of the patient’s disease and the period is not indicated clearly. Hence, the proposed change of the expression is from to .

Step 2. To derive the weighted aggregation value, for the patient related to the disease :

or

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

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

and thereby deriving that

From the definition of , , and of Equations (7)–(9), respectively, the following is obtained:

such that

Theorem 2 below summarizes the findings.

Theorem 2. Using two SVNSs and together with the universe of discourse , is confirmed.

4. Our Proposed Similarity Measures

Using the two SVNSs A and B with a universal set as above, it is assumed that:

with

and

with

such that is related to the maximum norm, and is related to the arithmetic mean.

Our proposed weighted aggregation value for a patient, , to a disease, , is computed as

where is the SVNS for the patient , with the symptom , at period , and is the SVNS for the disease and the symptom . On the other hand, we provide a second aggregated value by our proposed second similarity measure of Equation (27) as

5. Diagnoses Illustration

The example from Ye and Fu [7] is used, wherein a patient group presents with an array of diseases,  = {viral fever, malaria, typhoid, gastritis, stenocardia}, the list of symptoms as,  = {temperature, headache, stomach pain, cough, chest pain}, and a set of four patients, . The characteristic values of the patients and symptoms for a certain period are listed in Table 1. For example, is the SVNS for the patient with the symptom at period .

Table 1: (Reproduction of Table 5 of Ye and Fu [7]) Characteristic values related to the patients and their symptoms for three periods.

(temperature)
(headache)
(stomach pain)
(cough)
(chest pain)

(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)

(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)

(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)

(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)

(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)

(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)

(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)

(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)

(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)

(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)

(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)

(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)

Table 2 continues the list between (a) probable diseases and (b) their corresponding symptoms. For example, is the SVNS for disease with the symptom .

Table 2: (Reproduction of Table 4 of Ye and Fu [7]) Characteristic values related to five diseases and five symptoms.

(temperature)
(headache)
(stomach pain)
(cough)
(chest pain)

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)

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)

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)

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)

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 of and , for and with weights for periods , , and and weights for symptoms , for , the results are presented in Table 3.

Table 3: Study findings of and .

Diagnosis result

0.6730
0.5990
0.6560
0.5260
0.5270
viral fever

0.7970
0.8730
0.6780
0.5510
0.5330
malaria

0.5860
05040
0.5540
0.5300
0.4520
viral fever

0.5210
0.4750
0.5640
0.6020
0.8910
stenocardia

0.7770
0.7323
0.7483
0.6810
0/6510
viral fever

0.8683
0.9250
0.7897
0.6883
0.6670
malaria

0.7573
0.6983
0.6960
0.7133
0.6573
viral fever

0.6917
0.6617
0.7170
0.7577
0.9443
stenocardia

Maximum values and thus, the most probable diagnosis.

From Table 3, by the aggregation operator of with the maximum norm or the aggregation operator with the arithmetic mean, the same results are derived for patients and suffering from viral fever, the patient suffering malaria, and the patient suffering stenocardia. The study’s derivations are the same as that of Yu and Fu [7] with the aggregation operator of Equation (18) and the aggregation operator 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, ) to complicated expressions. Based on the abbreviations of and , it is possible to rewrite of Equation (16) and of Equation (17) proposed by Yu and Fu [7] to simplify their expressions as follows

Comparing Equation (25) with Equation (31), the following general expression is implied:

such that in Equation (25), for , , and in Equation (31), for , .

By the same argument, comparing Equation (27) with Equation (32) obtains a general expression:

such that in Equation (27), for , , and in Equation (32), for , .

Consequently, we can further simplify of Equation (12) and of Equation (13) proposed by Ye [6] as follows

Hence, Equations (25), (31), and (35) are used in comparison to determine their relationship with of Equation (26). By the same observation, it is found that Equations (27), (32), and (36) are related to of Equation (28).

In Equations (25) and (27), the most natural approach is applied by us. Meanwhile, in Equations (31) and (32), Ye and Fu [7] used , and in Equations (35) and (36), Ye [6] used .

and are both increasing functions from to . On the other hand, is a decreasing function from to . All of them are monotonic functions.

For a similarity measure, say , , and are ideal, such as in Equations (25), (27), (31), and (32) and then abstractly expressed in Equations (33) and (34), researchers use or .

There are infinite increasing functions that satisfy and . However, Ye and Fu [7] did not provide any explanation of why was selected by them.

Moreover, there are infinite decreasing functions that satisfy and . Ye [6] and Ye and Fu [7] also did not explain why was used.

Since our proposed approach to use the simplest form can still derive the desired diagnosis, the study recommends not to apply the complicated method proposed by Ye [6] with and Ye and Fu [7] with . Instead, using our simplest form 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 of Equation (18) was used with tangent similarity measure of Equation (15) and the aggregation operator of Equation (19) with tangent similarity measure 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 of Equation (29) and aggregation operator of Equation (30).

In Ye and Fu [7], the improved similarity measure of Ye [6] was used and repeated three times, then combined with the weighted mean such that the improved similarity measure of the latter study can be used to solve multiperiod medical diagnosis problems. Ye and Fu [7] already successfully applied 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].

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

This research was partially supported by the Ministry of Science and Technology, with grant no. MOST 107-2410-H-241-001.

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