A Novel Fuzzy Parameterized Fuzzy Hypersoft Set and Riesz Summability Approach Based Decision Support System for Diagnosis of Heart Diseases
Abstract
:1. Introduction
1.1. Research Gap and Motivation
 The hypersoft setting, which demands the categorization of parameters into their relevant subclasses containing their subparametric values; such kind of classification can only be managed by employing maafunction, which takes the Cartesian product (Cproduct) of subparametricvalued classes as its domain and then approximates them for universal set.
 Riesz Summability setting, which is capable of tackling the sequential nature of data.
1.2. Significant Contributions
 An innovative model fuzzy parameterized fuzzy hypersoft set ($\Delta $set) is characterized and some of its axiomatic cum algebraic properties are investigated. This model employs maafunction with fuzzy parametric tuples as its domain and collection of fuzzy subsets as its codomain;
 The classical concept of Riesz mean is reviewed and modified for $hs$settings;
 The real attributes of CDset are analyzed for heartbased ailments analysis and only those of them are opted that have a pertinent role for the adopted model;
 In order to have their respective attribute values, the operational roles of all opted attributes are discussed along with description on their measuring units;
 The opted traits and their subvalues are changed to fuzzy values by employing a suitable algebraic technique;
 Two algorithms (one for aggregations of $\Delta $set and other for Riesz mean) are proposed and implemented in realworld scenario of medical diagnosis for heart diseases based on fuzzyvalued attributes of CDset.
2. Preliminaries
3. Fuzzy Parameterized Fuzzy Hypersoft Set ($\Delta $Set)
 It transforms to $fpfs$set if $hs$setting is replaced with ssetting.
 It takes the form of $fhs$set if fuzzy parameterization is omitted.
 It converts to $fs$set if fuzzy parameterization is ignored and $hs$setting is replaced with ssetting.
 It becomes sset if fuzzy parameterization is ignored, $hs$setting is replaced with ssetting and fuzzy grades are omitted.
 It converts to fset if fuzzy parameterization is ignored, $hs$setting is replaced with ssetting, and fuzzy approximations are ignored.
4. Methodology and Algorithms
4.1. Aggregations of $\Delta $Set
 Only select those parametric tuples that contain $\widehat{u}$ in their approximations, i.e., the value of ${\mathsf{\Gamma}}_{{\vartheta}_{\Delta}\left(\widehat{b}\right)}\left(\widehat{u}\right)$ will be equal to their corresponding fuzzy grades ${\vartheta}_{\Delta}\left(\widehat{b}\right)$.
 Compute the product of fuzzy parameterized value ${\delta}_{\Delta}\left(\widehat{b}\right)$ and the obtained value of ${\mathsf{\Gamma}}_{{\vartheta}_{\Delta}\left(\widehat{b}\right)}\left(\widehat{u}\right)$; then, determine the sum of these products.
 Lastly, divide the computed sum with cardinality $\left\mathbb{B}\right$ of $\mathbb{B}$.
 Divide the computed sum with the value ${\mathbb{X}}_{n}$ that is explained in Definition 12 and Example 1.
4.2. Cleveland Dataset
4.3. Salient Features of Opted Attributes
 Age. Aging is a selfdetermining menace aspect for heart ailments. Although this factor is reported higher in aged persons (more than 60 years), with the involvement of various supplementary reasons, adults can also be in danger. The cardiologists have classified the aging factor into four groups: (i) 20 years or less, (ii) 40 years or less, (iii) 60 years or less, (iv) more than 60 years.
 Chest Pain Type. Chest pain is a significant factor that leads to the suffering of cardiac disorder. It may vary due to quality, span, area, and force. Its intensity may be sharp, distressing feeling, and deadly upset. The chest pain attached with heart diseases can be sorted as Typical Angina (TA), Atypical Angina (ATA), NonAnginal pain, and Asymptomatic (AM) (see [58]). The first two types are considered significant factors towards the suffering of heart diseases; the others are of less significance but cannot be ignored.
 Resting Blood Pressure. This pressure is produced due to blood flow in blood vessels on its walls. The narrowness of the blood vessels is reported due to this pressure. The medical experts have sorted it as systolic and diastolic. These are produced during active blood flow and relaxing state, respectively. Its measuring unit is mm Hg, in accordance with dataset. The standard values for systolic and diastolic are 120 and 80 mm Hg, respectively. More than 120 mm Hg and less than 80 mm Hg (see [59]) are considered abnormal values for systolic and diastolic, respectively.
 Serum Cholesterol. Cholesterol is a variety of fat, recognized as lipid, which is encapsulated in proteins bundles (lipoproteins) and flows in blood vessels and capillaries. The common types of cholesterol are LDL, HDL, and triglycerides. These cholesterols cause the narrowness of the blood vessels, which may lead to severe heart issue. The LDL and HDL are also regarded as bad cholesterol and good cholesterol, respectively. A particular lab test ”Lipid Profile Test (LPT)“ is used to assess the values of these cholesterols. Its measuring unit is mg/dL, which is used in the adopted dataset. The serum cholesterol depends upon these cholesterol collectively and its level is determined by summing up the values of HDL and LDL along with 20% of triglycerides. Its values lie in the interval [126, 564] (see [60]). The types of cholesterol and their ranges are provided in Figure 2.
 Fasting Blood Sugar. This is regarded as another authentic factor for the analysis of heart diseases. It is usually observed that heart patients have high glucose due to the ”tension reaction“. In other words, nondiabetic patients may also have its high ratio. The ranges for its usual observed values are presented in Figure 3. Its measuring unit is mg/dL, which is used in the adopted dataset. A value of 120 mg/dL (see [58]) is regarded as a typical value for healthy individual.
 Maximum Heart Rate Achieved. Heart rate is the number of hearts beats per minute (bpm) and is regarded as a reliable source to determine the oxygen utilization in heart patients. Its values lies in the interval of 71 bpm, 195 bpm (see [61]).
 Oldpeak and Slope. Oldpeak is usually meant for ShockToxicity depression (also known as STdepression), which is provoked by restbase work out. It is regarded as a trustworthy ECG (electrocardiogram) result for the analysis of disruptive coronary issues. Its measuring unit is mm, which can take values from the interval [0.0, 0.5]. Figure 4 presents its pictographic view. Its slope can be sorted into three types (see [58]): (i) Upsloping, (ii) Flat (Horizontal), (iii) Downsloping. The pictorial display of these categories is presented in Figure 5.
 Thal. This is a familiar turmoil of blood recognized as thalassemia, which can be sorted into four categories: (i) Null (i.e., no flow of blood at all) (ii) Fixed Defect (i.e., partial flow of blood in some sections of the heart), (iii) Normal Blood Flow, and (iv) Reversible Defect (i.e., observation of blood flow without normality). The corresponding values assigned by medical experts to these categories are 0, 3, 6, and 7, respectively (see [58]). In case of heart disease diagnosis, the category (i) is usually disregarded.
4.4. Determination of FuzzyValuesBased Ranges for Opted Parameters
4.5. Declaration of Problem
4.6. Proposed Algorithm Based on $\Delta $set and Its Implementation
Algorithm 1: Steps for the analysis of heartrelated diseases based on $\Delta $set. 

4.7. Proposed Algorithm Based on Riesz Summability
Algorithm 2: Analysis of Heartrelated Diseases through the concept of Riesz Summability. 

5. Discussion and Comparison Analysis
 1.
 The setting when parameters and their subparametricvaluesbased tuples are ambiguous, i.e., decision makers are not sure about their preferencebased selection. In other words, the parameters and their subparametricvaluesbased tuples are uncertain for decisionmakers.
 2.
 The setting where it is necessary to categorize the parameters into their related disjoint subclasses having their subparametric values. This setting demands the entitlement of multiargument approximate function, which has the capability to cope with such subparametricvalued disjoint classes. Its domain is the Cproduct of these classes and range is the subsets of initial universe.
Merits of Proposed Study
 The presented approach took the importance of inspiration of fuzzyparameterization associated by $\Delta $set to manage modernday DM issues. The assignment of parameterized fuzzy grade imitates the possibility of recognition level; in this way, it has incredible prospective in the real description within the scope of computational scenarios.
 Real attributes of CDset are converted to fuzzy membership by using algebraic technique.
 The sequential nature of approximate values of $\Delta $set is managed by employing classical concept of Riesz Summability and analogous results have been achieved.
 Since the presented model put emphasis on comprehensive study of parameters (i.e., additional classification of parameters) more willingly than focusing on parameters merely, consequently, it enables decisionmakers to have better and more reliable decisions.
 The two proposed algorithms have ranked the patients with analogous and consistent results by considering a smaller number of attributes.
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Ordering by Scrutiny  Ordering by CDSet  Parameters (Short Names)  Parameters (Full Names) 

1  3  age  Age in years 
2  4  sex  Sex (male/female) 
3  9  cp  Chest pain type) 
4  10  trestpbs  Resting blood pressure (mm Hg) 
5  12  chol  Serum cholesterol (mg/dL) 
6  16  fbs  Fasting blood sugar (120 mg/dL) 
7  19  restecg  Resting electrocardiographic results 
8  32  Thalach  Maximum heart rate achieved 
9  38  Exang  Exerciseinduced angina 
10  40  Oldpeak  ST depression induced by exercise relative to rest 
11  41  slope  The slope of the peak exercise ST segment 
12  44  ca  Number of major vessels (0–3) colored by fluoroscopy 
13  51  thal  3 = normal; 6 = fixed defect; 7 = reversible defect 
14  58  num  Diagnosis of heart disease (angiographic disease status) 
Ordering by Scrutiny  Ordering by CDSet  Parameters (Short Names)  Parameters (Full Names)  Values related to Parameters in CDSet 

1  3  age  Age in years  0–20, 21–40, 41–60, Above 60 
3  9  cp  Chest pain type  1. Typical angina, 2. atypical angina, 3. nonanginal pain, 4. asymptomatic 
4  10  trestpbs  Resting blood pressure (mm Hg)  90–200 mm Hg 
5  12  chol  Serum cholesterol (mg/dL)  126–564 mg/dL 
6  16  fbs  Fasting blood sugar (120 mg/dL)  120 mg/dL 
8  32  Thalach  Maximum heart rate achieved  71–195 
10  40  Oldpeak  ST depression induced by exercise relative to rest  0.0–5.6 
11  41  slope  The slope of the peak exercise ST segment  1. upsloping, 2. flat, 3. downsloping 
13  51  thal  3 = normal; 6 = fixed defect; 7 = reversible defect  1. normal, 2. fixed defect, 3. reversible defect 
Selected Parameters  Relevant Values in CDSet  Transformed Fuzzy Membership Grades 

Age  0–20, 21–40, 41–60, 61–80  0–0.25, 0.2625–0.50, 0.5125–0.75, 0.7625–1.00 
Chest pain type)  1, 2, 3, 4  0.25, 0.50, 0.75, 1.00 
Resting blood pressure  90–200  0.45–1.00 
Serum cholesterol  126–564  0.2234–1.0000 
Fasting blood sugar  0, 120  0,1 
Maximum heart rate achieved  71–195  0.3641–1.0000 
Oldpeak  0.0–5.6  0–1 
Slope  1, 2, 3  0.33, 0.66, 1.00 
Thal  3, 6, 7  0.43, 0.86, 1.00 
${\ddot{\mathit{\partial}}}^{\mathbf{ij}}$  $\mathit{\mu}\left({\ddot{\mathit{\partial}}}^{\mathbf{ij}}\right)$  ${\ddot{\mathit{\partial}}}^{\mathbf{ij}}$  $\mathit{\mu}\left({\ddot{\mathit{\partial}}}^{\mathbf{ij}}\right)$ 

${\ddot{\partial}}^{12}$  0.5  ${\ddot{\partial}}^{13}$  0.7 
${\ddot{\partial}}^{21}$  0.25  ${\ddot{\partial}}^{22}$  0.50 
${\ddot{\partial}}^{32}$  0.75  ${\ddot{\partial}}^{42}$  0.57 
${\ddot{\partial}}^{51}$  1.00  ${\ddot{\partial}}^{61}$  0.42 
${\ddot{\partial}}^{62}$  0.72  ${\ddot{\partial}}^{72}$  0.66 
${\ddot{\partial}}^{83}$  1.00  ${\ddot{\partial}}^{92}$  0.86 
${\ddot{\mathit{\hslash}}}_{\mathit{i}}$  ${\ddot{\mathit{\hslash}}}_{1}$  ${\ddot{\mathit{\hslash}}}_{2}$  ${\ddot{\mathit{\hslash}}}_{3}$  ${\ddot{\mathit{\hslash}}}_{4}$  ${\ddot{\mathit{\hslash}}}_{5}$  ${\ddot{\mathit{\hslash}}}_{6}$  ${\ddot{\mathit{\hslash}}}_{7}$  ${\ddot{\mathit{\hslash}}}_{8}$ 

$\mu \left({\ddot{\partial}}^{12}\right)$  0.5  0.5  0.5  0.5  
$\mu \left({\ddot{\partial}}^{13}\right)$  0.7  0.7  0.7  0.7  
$\mu \left({\ddot{\partial}}^{21}\right)$  0.25  0.25  0.25  0.25  
$\mu \left({\ddot{\partial}}^{22}\right)$  0.5  0.5  0.5  0.5  
$\mu \left({\ddot{\partial}}^{32}\right)$  0.75  0.75  0.75  0.75  0.75  0.75  0.75  0.75 
$\mu \left({\ddot{\partial}}^{42}\right)$  0.57  0.57  0.57  0.57  0.57  0.57  0.57  0.57 
$\mu \left({\ddot{\partial}}^{51}\right)$  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0 
$\mu \left({\ddot{\partial}}^{61}\right)$  0.42  0.42  0.42  0.42  
$\mu \left({\ddot{\partial}}^{62}\right)$  0.72  0.72  0.72  0.72  
$\mu \left({\ddot{\partial}}^{72}\right)$  0.66  0.66  0.66  0.66  0.66  0.66  0.66  0.66 
$\mu \left({\ddot{\partial}}^{83}\right)$  1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0 
$\mu \left({\ddot{\partial}}^{92}\right)$  0.86  0.86  0.86  0.86  0.86  0.86  0.86  0.86 
$\mu \left({\ddot{\hslash}}_{i}\right)$  0.667  0.701  0.695  0.729  0.690  0.723  0.717  0.751 
$\frac{{\ddot{\mathit{\hslash}}}_{\mathit{i}}}{\mathit{\mu}\left({\ddot{\mathit{\hslash}}}_{\mathit{i}}\right)}\setminus {\widehat{\mathit{p}}}_{\mathit{i}}$  ${\widehat{\mathit{p}}}_{1}$  ${\widehat{\mathit{p}}}_{2}$  ${\widehat{\mathit{p}}}_{24}$  ${\widehat{\mathit{p}}}_{25}$  ${\widehat{\mathit{p}}}_{75}$  ${\widehat{\mathit{p}}}_{303}$ 

$\frac{{\ddot{\hslash}}_{1}}{0.667}$  0.2  0.3  0.0  0.4  0.6  0.7 
$\frac{{\ddot{\hslash}}_{2}}{0.701}$  0.0  0.4  0.5  0.6  0.7  0.8 
$\frac{{\ddot{\hslash}}_{3}}{0.695}$  0.3  0.5  0.3  0.0  0.4  0.5 
$\frac{{\ddot{\hslash}}_{4}}{0.729}$  0.5  0.4  0.3  0.2  0.0  0.1 
$\frac{{\ddot{\hslash}}_{5}}{0.690}$  0.0  0.2  0.3  0.4  0.5  0.6 
$\frac{{\ddot{\hslash}}_{6}}{0.723}$  0.4  0.4  0.5  0.6  0.8  0.0 
$\frac{{\ddot{\hslash}}_{7}}{0.717}$  0.3  0.6  0.4  0.4  0.5  0.2 
$\frac{{\ddot{\hslash}}_{8}}{0.751}$  0.7  0.5  0.3  0.5  0.4  0.3 
${\widehat{p}}_{1}$  $(0.2,\frac{{\ddot{\mathit{\hslash}}}_{1}}{0.667})$, $(0.3,\frac{{\ddot{\hslash}}_{3}}{0.695})$, $(0.5,\frac{{\ddot{\hslash}}_{4}}{0.729})$, $(0.4,\frac{{\ddot{\hslash}}_{6}}{0.723})$, $(0.3,\frac{{\ddot{\hslash}}_{7}}{0.717})$, $(0.7,\frac{{\ddot{\hslash}}_{8}}{0.751})$ 
${\widehat{p}}_{2}$  $(0.3,\frac{{\ddot{\hslash}}_{1}}{0.667})$, $(0.4,\frac{{\ddot{\hslash}}_{2}}{0.701})$, $(0.5,\frac{{\ddot{\hslash}}_{3}}{0.695})$, $(0.4,\frac{{\ddot{\hslash}}_{4}}{0.729})$, $(0.2,\frac{{\ddot{\hslash}}_{5}}{0.690})$, $(0.4,\frac{{\ddot{\hslash}}_{6}}{0.723})$, $(0.6,\frac{{\ddot{\hslash}}_{7}}{0.717})$, $(0.5,\frac{{\ddot{\hslash}}_{8}}{0.751})$ 
${\widehat{p}}_{24}$  $(0.5,\frac{{\ddot{\hslash}}_{2}}{0.701})$, $(0.3,\frac{{\ddot{\hslash}}_{3}}{0.695})$, $(0.3,\frac{{\ddot{\hslash}}_{4}}{0.729})$, $(0.3,\frac{{\ddot{\hslash}}_{5}}{0.690})$, $(0.5,\frac{{\ddot{\hslash}}_{6}}{0.723})$, $(0.4,\frac{{\ddot{\hslash}}_{7}}{0.717})$, $(0.3,\frac{{\ddot{\hslash}}_{8}}{0.751})$ 
${\widehat{p}}_{25}$  $(0.4,\frac{{\ddot{\hslash}}_{1}}{0.667})$, $(0.6,\frac{{\ddot{\hslash}}_{2}}{0.701})$, $(0.2,\frac{{\ddot{\hslash}}_{4}}{0.729})$, $(0.4,\frac{{\ddot{\hslash}}_{5}}{0.690})$, $(0.6,\frac{{\ddot{\hslash}}_{6}}{0.723})$, $(0.4,\frac{{\ddot{\hslash}}_{7}}{0.717})$, $(0.5,\frac{{\ddot{\hslash}}_{8}}{0.751})$ 
${\widehat{p}}_{75}$  $(0.6,\frac{{\ddot{\hslash}}_{1}}{0.667})$, $(0.7,\frac{{\ddot{\hslash}}_{2}}{0.701})$, $(0.4,\frac{{\ddot{\hslash}}_{3}}{0.695})$, $(0.5,\frac{{\ddot{\hslash}}_{5}}{0.690})$, $(0.8,\frac{{\ddot{\hslash}}_{6}}{0.723})$, $(0.5,\frac{{\ddot{\hslash}}_{7}}{0.717})$, $(0.4,\frac{{\ddot{\hslash}}_{8}}{0.751})$ 
${\widehat{p}}_{303}$  $(0.7,\frac{{\ddot{\hslash}}_{1}}{0.667})$, $(0.8,\frac{{\ddot{\hslash}}_{2}}{0.701})$, $(0.5,\frac{{\ddot{\hslash}}_{3}}{0.695})$, $(0.1,\frac{{\ddot{\hslash}}_{4}}{0.729})$, $(0.6,\frac{{\ddot{\hslash}}_{5}}{0.690})$, $(0.2,\frac{{\ddot{\hslash}}_{7}}{0.717})$, $(0.3,\frac{{\ddot{\hslash}}_{8}}{0.751})$ 
${\widehat{\mathit{p}}}_{\mathit{i}}$  ${\mathit{\zeta}}_{\mathit{\Delta}}^{{\mathbb{D}}_{1}}\left({\widehat{\mathit{p}}}_{\mathit{i}}\right)$ 

${\widehat{p}}_{1}$  0.217050 
${\widehat{p}}_{2}$  0.294063 
${\widehat{p}}_{24}$  0.232288 
${\widehat{p}}_{25}$  0.275663 
${\widehat{p}}_{75}$  0.343900 
${\widehat{p}}_{303}$  0.278850 
${\widehat{\mathit{p}}}_{\mathit{i}}$  ${\mathit{\zeta}}_{\mathit{\Delta}}^{{\mathbb{D}}_{2}}\left({\widehat{\mathit{p}}}_{\mathit{i}}\right)$ 

${\widehat{p}}_{1}$  0.306081 
${\widehat{p}}_{2}$  0.414684 
${\widehat{p}}_{24}$  0.327569 
${\widehat{p}}_{25}$  0.388736 
${\widehat{p}}_{75}$  0.484964 
${\widehat{p}}_{303}$  0.393231 
Authors  Structures  Focus on Attributes  Focus on Subattributive Values  Data Set  Proper Criteria for Fuzzification of Fuzzy Parameters  Riesz Summability 

Ça$\stackrel{\u02d8}{g}$ man et al. [39]  $fpfs$set  Yes  Ignored  Hypothetical  N/A  N/A 
Yılmaz et al. [40]  $fpfs$set  Yes  Ignored  Hypothetical  N/A  Yes 
Kirişci [41,42]  $fpfs$set  Yes  Ignored  CDset  N/A  N/A 
Riaz et al. [43]  $fpfs$set  Yes  Ignored  Hypothetical  N/A  N/A 
Zhu et al. [44]  $fpfs$set  Yes  Ignored  Hypothetical  N/A  N/A 
Rahman et al. [48]  $fpfhs$set  Yes  Yes  Hypothetical  N/A  N/A 
Proposed Study  $fpfhs$set  Yes  Yes  CDset  Adopted  Yes 
Authors  Structures  NOA  NOP  Ranking Based on Riesz Summability Method  Ranking Based on Other Adopted Method  Remarks 

Kirişci [41]  $fpfs$set  11  06  N/A  ${\widehat{p}}_{1}\succ {\widehat{p}}_{2}\succ {\widehat{p}}_{24}\succ {\widehat{p}}_{75}\succ {\widehat{p}}_{25}\succ {\widehat{p}}_{303}$  subattributive values are ignored. 
Kirişci [42]  $fpfs$set  11  06  N/A  ${\widehat{p}}_{75}\succ {\widehat{p}}_{24}\succ {\widehat{p}}_{25}\succ {\widehat{p}}_{1}\succ {\widehat{p}}_{2}\succ {\widehat{p}}_{303}$  subattributive values are ignored. 
Proposed Study  $fpfhs$set  09  06  ${\widehat{p}}_{75}\succ {\widehat{p}}_{2}\succ {\widehat{p}}_{303}\succ {\widehat{p}}_{25}\succ {\widehat{p}}_{24}\succ {\widehat{p}}_{1}$  ${\widehat{p}}_{75}\succ {\widehat{p}}_{2}\succ {\widehat{p}}_{303}\succ {\widehat{p}}_{25}\succ {\widehat{p}}_{24}\succ {\widehat{p}}_{1}$  Although values of both methods are different but they both proved analogous with similar ranking of patients. 
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Rahman, A.U.; Saeed, M.; Mohammed, M.A.; Jaber, M.M.; GarciaZapirain, B. A Novel Fuzzy Parameterized Fuzzy Hypersoft Set and Riesz Summability Approach Based Decision Support System for Diagnosis of Heart Diseases. Diagnostics 2022, 12, 1546. https://doi.org/10.3390/diagnostics12071546
Rahman AU, Saeed M, Mohammed MA, Jaber MM, GarciaZapirain B. A Novel Fuzzy Parameterized Fuzzy Hypersoft Set and Riesz Summability Approach Based Decision Support System for Diagnosis of Heart Diseases. Diagnostics. 2022; 12(7):1546. https://doi.org/10.3390/diagnostics12071546
Chicago/Turabian StyleRahman, Atiqe Ur, Muhammad Saeed, Mazin Abed Mohammed, Mustafa Musa Jaber, and Begonya GarciaZapirain. 2022. "A Novel Fuzzy Parameterized Fuzzy Hypersoft Set and Riesz Summability Approach Based Decision Support System for Diagnosis of Heart Diseases" Diagnostics 12, no. 7: 1546. https://doi.org/10.3390/diagnostics12071546