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

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.

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 coun-terpart 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, f ðxÞ = 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 > dist ða, xÞ > distða, yÞ, then The strictly decreasing property of cosine implies cos ð dist ða, xÞÞ < cos ðdist ða, yÞÞ. 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 tan ðdist ða, xÞÞ > tan ðdist ða, yÞÞ. 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.

Computational and Mathematical Methods in Medicine
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.

Review of Related Results for Ye and Fu [7]
Wang et al. [4] defined an SVNS as A = fhx, T A ðxÞ, I A ðxÞ, F A ðxÞijx ∈ Xg, such that X is a universe of discourse with the three membership functions: truth T A ðxÞ, indeterminacy I A ðxÞ, and falsity F A ðxÞ that satisfy T A ðxÞ, I A ðxÞ, and F A ðxÞ ∈ ½0, 1. For SVNSs with a universal set X = fx 1 , x 2 , ⋯, x n g, 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 3 Computational and Mathematical Methods in Medicine 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 S = fS 1 , S 2 , ⋯, S m g as the group of different symptoms, T = ft 1 , t 2 , ⋯, t q g as a series of multiple periods, and D = fD 1 , D 2 , ⋯, D n g as a group of possible diagnoses. Let the weight of period t k be denoted as wðt k Þ with wðt k Þ > 0 and ∑ q k=1 wðt k Þ = 1. For a patient P with various symptoms, Ye and Fu [7] used S j ðt k Þ to denote the SVNS for this patient throughout the period t k for symptoms S j and C i j , and to represent the SVNS for disease D i related to the symptom S j . The algorithm is cited as follows: Step 1. Ye and Fu [7] applied T 2 of Equation (15) or C 2 of Equation (13) to compute the similarity between S j ðt k Þ and C i j as T 2 ðP, D i , t k Þ or C 2 ðP, D i , t k Þ. Remark 1. Ye and Fu [7] used T w i ðP, t k Þ where the relation of the patient's disease D i and the period t k is not indicated clearly. Hence, the proposed change of the expression is from T w i ðP, t k Þ to T 2 ðP, D i , t k Þ.
Step 2. To derive the weighted aggregation value, MðP, D i Þ for the patient related to the disease D i : or Step 3. Present the most probable diagnosis to the patient according to the highest amount among weighted aggregations.

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 Δ A j , Δ B j , and Ω j of Equations (7)-(9), respectively, the following is obtained: such that Theorem 2 below summarizes the findings.

Our Proposed Similarity Measures
Using the two SVNSs A and B with a universal set X = fx 1 , x 2 , ⋯, x n g as above, it is assumed that: with and with such that M w 1 is related to the maximum norm, and M w 2 is related to the arithmetic mean. Our proposed weighted aggregation value for a patient, P, to a disease, D, is computed as where S P j ðt k Þ is the SVNS for the patient P, with the symptom S j , at period t k , and C i j is the SVNS for the disease D i and the symptom S j . On the other hand, we provide a second aggregated value by our proposed second similarity measure of Equation (27) as

Diagnoses Illustration
The example from Ye and Fu [7] is used, wherein a patient group presents with an array of diseases, D = fD 1 , D 2 , ⋯, D 5 g = {viral fever, malaria, typhoid, gastritis, stenocardia}, the list of symptoms as, S = fS 1 , S 2 , ⋯, S 5 g = {temperature, headache, stomach pain, cough, chest pain}, and a set of four patients, fP 1 , P 2 , P 3 , P 4 g. The characteristic values of the patients and symptoms for a certain period are listed in Table 1. For example, S P 1 1 ðt 1 Þ is the SVNS for the patient P 1 with the symptom S 1 at period t 1 = ð0:8, 0:6, 0:5Þ. Table 2 continues the list between (a) probable diseases and (b) their corresponding symptoms. For example, C 1 2 = ð0:3, 0:2, 0:5Þ is the SVNS for disease D 1 with the symptom S 2 .
From Table 3, by the aggregation operator of M w 1 with the maximum norm or the aggregation operator M w 2 with the arithmetic mean, the same results are derived for patients P 1 and P 3 suffering from viral fever, the patient P 2 suffering malaria, and the patient P 4 suffering stenocardia. The study's derivations are the same as that of Yu and Fu [7] with the aggregation operator M T 2 of Equation (18) and the aggregation operator M C 2 of Equation (19). Table 1: (Reproduction of Table 5 of Ye and Fu [7]) Characteristic values related to the patients and their symptoms for three periods.

Further Discussion for Aggregation Operators
In this section, we will explain that previous researchers used monotonic functions to transform the identity function (that is, f ðxÞ = x) to complicated expressions. Based on the abbreviations of α j and β j , it is possible to rewrite T w 1 of Equation (16) and T w 2 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 M w 1 , f ðxÞ = x, and in Equation (31), for T w 1 , f ðxÞ = tan ðπ x/4Þ. By the same argument, comparing Equation (27) with Equation (32) obtains a general expression: such that in Equation (27), for M w 2 , f ðxÞ = x, and in Equation (32), for T w 2 , f ðxÞ = tan ðπ x/4Þ. Consequently, we can further simplify C 1 ðA, BÞ of Equation (12) and C 2 ðA, BÞ of Equation (13) proposed by Ye [6] as follows Hence, Equations (25), (31), and (35) are used in comparison to determine their relationship with α j of Equation (26). By the same observation, it is found that Equations (27), (32), and (36) are related to β j of Equation (28).
In Equations (25) and (27), the most natural approach f ðxÞ = x is applied by us. Meanwhile, in Equations (31)  For a similarity measure, say Sim, Simð0Þ = 1, and Simð1Þ = 0 are ideal, such as in Equations (25), (27), (31), and (32) and then abstractly expressed in Equations (33) and (34) There are infinite increasing functions that satisfy f ð0Þ = 0 and f ð1Þ = 1. However, Ye and Fu [7] did not provide Table 2: (Reproduction of Table 4 of Ye and Fu [7]) Characteristic values related to five diseases and five symptoms.  Moreover, there are infinite decreasing functions that satisfy f ð0Þ = 1 and f ð1Þ = 0. Ye [6] and Ye and Fu [7] also did not explain why f ðxÞ = cos ðπ x/2Þ was used.
Since our proposed approach to use the simplest form f ðxÞ = x can still derive the desired diagnosis, the study recommends not to apply the complicated method proposed by Ye [6] with f ðxÞ = cos ðπ x/2Þ and Ye and Fu [7] with f ðxÞ = tan ðπ x/4Þ. Instead, using our simplest form f ðxÞ = 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 M T 2 of Equation (18) was used with tangent similarity measure T 2 of Equation (15) and the aggregation operator M C 2 of Equation (19) with tangent similarity measure C 2 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 M w 1 of Equation (29) and aggregation operator M w 2 of Equation (30).
In Ye and Fu [7], the improved similarity measure C 2 of Ye [6] was used and repeated three times, then combined with the weighted mean such that the improved similarity measure C 2 of the latter study can be used to solve multiperiod medical diagnosis problems. Ye and Fu [7] already successfully applied C 2 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.

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.