Medical applications via minimal topological structure

Abstract

It is known that mathematical statics, mathematical modeling, and differential equations are used to give an in-depth understanding of many medical problems. On the edge of the information revolution, minimal structures show some qualitative properties issues that are difficult to deal with it, such as quality of education, nutrition, etc. The aim of this paper is to discuss two medical applications and show that a minimal structure space is suitable for analyzing several real-life problems. Then, the accuracy of the decision-making and attributes reduction of the medical information system are explained and obtained. Furthermore, we introduce a comparison between our approach and Pawlak’s approach to find accuracy for decision-making. Finally, the accuracy of decision-making via a variable precision model is improved.

Keywords

Minimal structure upper approximation lower approximation reduction of attributes accuracy

1 Introduction

For a long time, scientists believed that abstract topology is one of the pure mathematics branches and far away from real-life problems. This point of view is entirely different from the advent of inference rough set theory that used in data analysis, and so topology appears in many fields and its applications. Modern methods used a minimal structure space to analyze information systems. This space is a generalization of topological space and studies a significant role in many fields, as medical, engineering, education, chemistry models. The concept of a minimal structure space studied in [3, 16]. The notion of an information system consisting of the pair of elements (I, S) , where I is a finite set of objects, and S is the set of attributes. The rough set theory, a fast approach to attribute reduction in incomplete decision system with tolerance relation-based rough sets, and variable precision rough set model are introduced and discussed in [9 , 20] and the relationship between soft sets and rough sets is studied [2 , 4–7]. Moreover, it depends on two fundamental concepts, names, the lower and upper approximation of a set, no one of scientists expected the events of this theory in all fields, including data analysis and reduction. However, the mathematical statistics, mathematical model, and functional analysis can be used in the decision making of a medical problem. There are some deficiencies in analyzing some medical problems using these traditional mathematical tools. To overcome that, new approaches based on a minimal structure space are introduced. In the present paper, we aim to introduce a new method to analyze and reduction of a medical information system by minimal structure spaces. Also, we present a comparison between our method and Pawlak’s method [14] to find accuracy for decision-making, furthermore ways to improve it.

The rest of this paper is organized as follows. In Section 2, we review some basic concepts, such as minimal structure (briefly, $M$ -structure), $M$ -closure, $M$ -interior, the core of rough set, and variable precision model. In Section 3, minimal structure in heart failure are explained. The minimal structure in dengue fever are discussed in Section 4. A comparison between the Pawlak’s method [14] and our approach is introduced in Section 5. Finally, conclusion is given in Section 6.

2 Preliminaries

2.1 $M$ -Structure, $M$ -closure, and $M$ -interior

Definition 1 (cf. [16, 17]). Given a set $U,$ a subset $M \subseteq 2^{U}$ (the set of all subsets of $U$ ) containing ${\emptyset, U}$ is called a minimal structure (briefly, $M$ -structure) on $U$ , and the pair $(U, M)$ is called a minimal structure space (briefly, $M$ -space). The members of $M$ are called open sets, and the members of $2^{U} ∖ M$ are called closed sets. The set of all $M$ -structures on $U$ will be denoted by $M_{U}$ (notice that $(M_{U}, \subseteq)$ is a complete lattice).

Definition 2 (cf. [16, 17]). Let $U$ be a non-empty set, and $M$ an $M$ -structure on $U$ . For a subset $B$ of $U$ , the $M$ -closure of $B$ (denoted by $M$ - $C (B))$ and the $M$ -interior of $B$ (denoted by $M$ - $I (B))$ are defined by

$M$ - $C (B) = ⋂ {F : B \subseteq F, U ∖ F \in M}$ ,

$M$ - $I (B) = ⋃ {G : G \subseteq B, G \in M}$ .

Theorem 3 (cf. [10]). Let $(U, M)$ be a minimal structure space. For $B, D \subseteq U$ , the followings hold.

$M$ - $I (U) = U$ and $M$ - $C (\emptyset) = \emptyset .$

$M$ - $I (B) \subseteq B$ and $B \subseteq M$ - $C (B) .$

(i) If $B \in M$ , then $M$ - $I (B) = B .$

(ii) If $U ∖ B \in M$ , then $M$ - $C (B) = B$ .

If $B \subseteq D$ , then $M$ - $I (B) \subseteq M$ - $I (D)$ and $M$ - $C (B) \subseteq M$ - $C (D)$ .

$M$ - $I (M$ - $I (B)) = M$ - $I (B)$ and $M$ - $C (M$ - $C (B)) = M$ - $C (B)$ .

$M$ - $I (U ∖ B) = U ∖ M$ - $C (B)$ and $M$ - $C (U ∖ B)$ $= U ∖ M$ - $I (B)$ .

2.2 The core of rough set theory

Definition 4 (cf. [8, 14]). Let $U$ be a non-empty finite set of objects, $R$ be an equivalence relation on $U$ , and the pair $(U, R)$ is called the approximation space. Let $B \subseteq U$ , the core of rough set theory are:

The upper approximation of $B$ concerning $R$ , is the set of all objects, which can be possibly classified as $B$ concerning $R$ , and it is denoted by $R^{+} (B)$ . That is, $R^{+} (B) = ⋃_{b \in U} {R (b) : R (b) \cap B \neq \emptyset},$ where $R (b)$ denotes the equivalence class determined by b.

The lower approximation of $B$ concerning $R$ , is the set of all objects, which can be certainly classified as $B$ concerning $R$ , and it is denoted by $R^{-} (B)$ . That is, $R^{-} (B) = ⋃_{b \in U} {R (b) : R (b) \subseteq B} .$

The boundary region of $B$ concerning $R$ , denoted by ${br}_{R} (B)$ . That is, ${br}_{R} (B) = R^{+} (B) ∖ R^{-} (B),$ which is the set of all objects, which can be classified neither as $B$ nor as $U ∖ B$ . The set $B$ is said to be rough concerning to $R$ if $R^{-} (B) \neq R^{+} (B)$ . That is ${br}_{R} (B) \neq \emptyset$ .

Definition 5 (cf. [8, 14]). Let $(U, R, D)$ be a decision approximation space, where $(U, R)$ is an approximation space and $D$ is an equivalence relation on $U$ called decision. Hence, $U / D = {Y_{1}, Y_{2}, . . ., Y_{n}}$ is a partition on $U$ . The lower and the upper approximations of $U / D$ concerning of $R$ are defined by $R^{-} (U / D) = {R^{-} (Y_{1}), R^{-} (Y_{2}), . . ., R^{-} (Y_{n})}$ and $R^{+} (U / D) = {R^{+} (Y_{1}), R^{+} (Y_{2}), . . ., R^{+} (Y_{n})} .$

2.3 Variable precision model

It is well known that for all $B, D \subseteq U,$ where $U$ is a non-empty finite universe set and $B \subseteq D$ if all elements of $B$ included in $D$ .

[1, 23] discuss that $B \subseteq D$ with an error ratio δ according to the value of measure $V (B, D)$ . That is, $B \subseteq_{δ} D$ if and only if $V (B, D) \leq δ$ , where $V (B, D) = 1 - \frac{card (B \cap D)}{card (B)}$ if $card (B) > 0$ , $V (B, D) = 0$ if $card (B) = 0$ , where the admissible classification error must be 0 < δ ≤ 0.5.

3 Minimal structure in heart failure

In this section, we show the importance of a minimal structure space in medical science for application in decision-making problems. Therefore, we have chosen to apply it to the heart failure problem. Data set containing the result of five symptoms for twelve patients. The study was conducted at the cardiology department, Al-Azhar University (Hospital of Sayed Glal University-Egypt). The study included twelve patients presenting to this hospital with different presenting symptoms, detailed history, physical examination, full labs, resting ECG, conventional echo assessment wren done. Finally, the heart failure diagnosis is confirmed or excluded.

3.1 Experimental results and discussion

In this subsection, we discuss the experimental results by introducing a preparatory study conducted on five symptoms of heart disease, according to Dickstein et al. [15] for twelve patients. The study was conducted at the cardiology department, Al-Azhar University (Hospital of Sayed Glal University-Egypt). The number of training data taken was 25 records, and the remaining 12 records were presented to this hospital with similar presenting symptoms, detailed history, physical examination, full labs, resting ECG, conventional echo assessment wren done. The data in the information system for only twelve patients due to similar patients, as Table 1, discuss the heart failure problem, The columns represent the symptoms (which yes ‘y’, it means that the patient has symptoms and no ‘n’, it means that the patient has no symptoms) of diagnosis of heart failure [15] (condition attributes), where H₁ is the breathlessness, H₂ is the orthopnea, H₃ is the paroxysmal nocturnal dyspnea, H₄ reduced exercise tolerance, H₅ is the ankle swelling. Attribute D is the decision of heart failure. The rows in Table 1, P = {p₁, p₂, p₃, p₄, p₅, p₆, p₇, p₈, p₉, p₁₀, p₁₁, p₁₂} represents the patients.

Table 1
Original medical information system

P p ₁ p ₂ p ₃ p ₄ p ₅ p ₆ p ₇ p ₈ p ₉ p ₁₀ p ₁₁ p ₁₂

H ₁ y n y n y n y y y n y y

H ₂ y n y n n n y y n n n n

H ₃ y n y n n n y n y n y n

H ₄ y y y y y y y y y y y y

H ₅ n y y n y n y y n y n y

D y n y n n n y y y n y n

P	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈	p ₉	p ₁₀	p ₁₁	p ₁₂
H ₁	y	n	y	n	y	n	y	y	y	n	y	y
H ₂	y	n	y	n	n	n	y	y	n	n	n	n
H ₃	y	n	y	n	n	n	y	n	y	n	y	n
H ₄	y	y	y	y	y	y	y	y	y	y	y	y
H ₅	n	y	y	n	y	n	y	y	n	y	n	y
D	y	n	y	n	n	n	y	y	y	n	y	n

We start that application by translating the description attributes (conditions attributes) {H₁, H₂, H₃, H₄, H₅} into qualitative terms as Table 2 that express from similarities between symptoms patients where the degree of similarity μ (x, y) is defined by $μ (x, y) = \frac{\sum_{i = 1}^{n} (a_{i} (x) = a_{i} (y))}{n},$ where n is the number of condition attributes.

Table 2

Similarities between symptoms of twelve of patients

	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈	p ₉	p ₁₀	p ₁₁	p ₁₂
p ₁	1	$\frac{1}{5}$	$\frac{4}{5}$	$\frac{2}{5}$	$\frac{2}{5}$	$\frac{2}{5}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	$\frac{1}{5}$	$\frac{4}{5}$	$\frac{3}{5}$
p ₂	$\frac{1}{5}$	1	$\frac{2}{5}$	$\frac{4}{5}$	$\frac{4}{5}$	$\frac{4}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	1	$\frac{2}{5}$	$\frac{3}{5}$
p ₃	$\frac{4}{5}$	$\frac{2}{5}$	1	$\frac{1}{5}$	$\frac{3}{5}$	$\frac{1}{5}$	1	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{2}{5}$
p ₄	$\frac{2}{5}$	$\frac{4}{5}$	$\frac{1}{5}$	1	$\frac{3}{5}$	1	0	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{4}{5}$
p ₅	$\frac{2}{5}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	1	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{4}{5}$
p ₆	$\frac{2}{5}$	$\frac{4}{5}$	$\frac{1}{5}$	1	$\frac{3}{5}$	1	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{4}{5}$
p ₇	$\frac{4}{5}$	$\frac{2}{5}$	1	0	$\frac{3}{5}$	$\frac{1}{5}$	1	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{2}{5}$
p ₈	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	$\frac{2}{5}$	$\frac{4}{5}$	$\frac{2}{5}$	$\frac{4}{5}$	1	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{3}{5}$
p ₉	$\frac{4}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	1	$\frac{2}{5}$	1	$\frac{4}{5}$
p ₁₀	$\frac{1}{5}$	1	$\frac{2}{5}$	$\frac{4}{5}$	$\frac{4}{5}$	$\frac{4}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	1	$\frac{2}{5}$	$\frac{3}{5}$
p ₁₁	$\frac{4}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	1	$\frac{2}{5}$	1	$\frac{4}{5}$
p ₁₂	$\frac{3}{5}$	$\frac{3}{5}$	$\frac{2}{5}$	$\frac{4}{5}$	$\frac{4}{5}$	$\frac{4}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	1

Now, we construct a minimal structure space via the relation that is related to the nature of the studied problem. Note that we define the relationship in each issue according to the expert’s requirements. In this case, $a R b \Leftrightarrow μ (a, b) > 0.8$ , where μ (a, b) is the sum of the similar symptoms between a and b divided on the number of symptoms.

From the Table 2, we have $R (p_{1}) = {p_{1}}$ , $R (p_{2}) = R (p_{10}) = {p_{2}, p_{10}}$ , $R (p_{3}) = R (p_{7}) = {p_{3}, p_{7}}$ , $R (p_{4}) = R (p_{6}) = {p_{4}, p_{6}}$ , $R (p_{5}) = {p_{5}}$ , $R (p_{8}) = {p_{8}}$ , $R (p_{9}) = R (p_{11}) = {p_{9}, p_{11}}$ , and $R (p_{12}) = {p_{12}}$ . The minimal structure on P is

$M = {P, \emptyset, {p_{1}}, {p_{2}, p_{10}}, {p_{3}, p_{7}}, {p_{4}, p_{6}}$ ,

{p₅} , {p₈} , {p₉, p₁₁} , {p₁₂} }, and thus

$2^{P} ∖ M = {P, \emptyset, P ∖ {p_{1}}, P ∖ {p_{2}, p_{10}},$

P ∖ {p₃, p₇} , P ∖ {p₄, p₆} , P ∖ {p₅} ,

P ∖ {p₈} , P ∖ {p₉, p₁₁} , P ∖ {p₁₂} }.

Through the rough set theory, we can calculate the accuracy of decision-making for two groups of patients:

$B_{1} = {p_{1}, p_{3}, p_{7}, p_{8}, p_{9}, p_{11}}$ , the set of patients having the disease.

$B_{2} = {p_{2}, p_{4}, p_{5}, p_{6}, p_{10}, p_{12}}$ , the set of patients not having heart failure.

Therefore, the lower approximation, the upper approximation, and the accuracy of $B_{1}$ are calculated as

$R^{-} (B_{1}) = {p_{1}, p_{3}, p_{7}, p_{8}, p_{9}, p_{11}},$ $R^{+} (B_{1}) = {p_{1}, p_{3}, p_{7}, p_{8}, p_{9}, p_{11}},$

and $γ (B_{1}) = 1,$ respectively.

Also, the lower approximation, the upper approximation, and the accuracy of $B_{2}$ are calculated as

$R^{-} (B_{2}) = {p_{2}, p_{4}, p_{5}, p_{6}, p_{10}, p_{12}},$ $R^{+} (B_{2}) = {p_{2}, p_{4}, p_{5}, p_{6}, p_{10}, p_{12}},$ and $γ (B_{2}) = 1,$ respectively.

Remark 6. The accuracy can be inferred in another way, that is, by creating a dissimilarity table instead of similarity. Also, it can be improved using a variable precision model in dissimilarity. By comparing our method (a minimal structure) with Pawlak’s method [14] to find the accuracy of decision-making, we are founding the same result when μ > 0.8.

3.2 Reduction of attributes

Rough set theory and a minimal structure using to analyze the information systems and construct a minimal subset of attributes give the same quality of accuracy.

From the Table 1, we eliminate attribute H₁ and construct a minimal structure according to problem requirement to calculate the accuracy of approximation as Table 3, and convert it to Table 4.

Table 3
(H - H₁)

P p ₁ p ₂ p ₃ p ₄ p ₅ p ₆ p ₇ p ₈ p ₉ p ₁₀ p ₁₁ p ₁₂

H ₂ Y N Y N N N Y Y N N N N

H ₃ Y N Y N N N Y N Y N Y N

H ₄ Y Y Y Y Y Y Y Y Y Y Y Y

H ₅ N Y Y N Y N Y Y N Y N N

D Y N Y N N N Y Y Y N Y N

P	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈	p ₉	p ₁₀	p ₁₁	p ₁₂
H ₂	Y	N	Y	N	N	N	Y	Y	N	N	N	N
H ₃	Y	N	Y	N	N	N	Y	N	Y	N	Y	N
H ₄	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y
H ₅	N	Y	Y	N	Y	N	Y	Y	N	Y	N	N
D	Y	N	Y	N	N	N	Y	Y	Y	N	Y	N

Table 4

Similarities between symptoms of twelve of patients

	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈	p ₉	p ₁₀	p ₁₁	p ₁₂
p ₁	1	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{2}{4}$
p ₂	$\frac{1}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	1	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$
p ₃	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{1}{4}$	1	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{1}{4}$
p ₄	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{3}{4}$	1	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	1
p ₅	$\frac{1}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	1	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$
p ₆	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{3}{4}$	1	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	1
p ₇	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{1}{4}$	1	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{1}{4}$
p ₈	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	1	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	$\frac{2}{4}$
p ₉	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{1}{4}$	1	$\frac{2}{4}$	1	$\frac{3}{4}$
p ₁₀	$\frac{1}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	1	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$
p ₁₁	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{1}{4}$	1	$\frac{2}{4}$	1	$\frac{3}{4}$
p ₁₂	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{3}{4}$	1	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	1

Then, the minimal structure on P is

$M = {P, \emptyset, {p_{1}}, {p_{2}, p_{5}, p_{10}}, {p_{3}, p_{7}},$

{p₄, p₆, p₁₂} , {p₈} , {p₉, p₁₁} }, and thus

$2^{P} ∖ M = {P, \emptyset, P ∖ {p_{1}}, P ∖ {p_{2}, p_{5}, p_{10}},$

P ∖ {p₃, p₇} , P ∖ {p₄, p₆, p₁₂},

P ∖ {p₈} , P ∖ {p₉, p₁₁} }. Therefore,

The lower approximation of $B_{1}$ is $R^{-} (B_{1}) =$ {p₁, p₃, p₇, p₈, p₉, p₁₁}, the upper approximation of $B_{1}$ is $R^{+} (B_{1}) = {p_{1}, p_{3}, p_{7},$ p₈, p₉, p₁₁} , and the accuracy of $B_{1}$ is $γ (B_{1}) = 1$ .

The lower approximation of $B_{2}$ is $R^{-} (B_{2}) =$ {p₂, p₄, p₅, p₆, p₁₀, p₁₂}, the upper approximation of $B_{2}$ is $R^{+} (B_{2}) = {p_{2}, p_{4}, p_{5}, p_{6},$ p₁₀, p₁₂}, and the accuracy of $B_{2}$ is $γ (B_{2}) = 1$ .

That is the same accuracy of original information system, the attribute H₁ is dispensable, and {H₂, H₃, H₄, H₅} is reduce for the information system.

Similarly, there are two reduces as {H₁, H₂, H₃, H₅} when we eliminate attribute H₄ and {H₁, H₂, H₃, H₄} when we eliminate attribute H₅.

Further, from the Table 1, we eliminate attribute H₂ and construct a minimal structure according to problem requirement to calculate the accuracy of approximation as Table 5, and convert it to Table 6.

Table 5

(H - H₂)

P	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈	p ₉	p ₁₀	p ₁₁	p ₁₂
H ₁	Y	N	Y	N	Y	N	Y	Y	Y	N	Y	Y
H ₃	Y	N	Y	N	N	N	Y	N	Y	N	Y	N
H ₄	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y
H ₅	N	Y	Y	N	Y	N	Y	Y	N	Y	N	N
D	Y	N	Y	N	N	N	Y	Y	Y	N	Y	N

Table 6

Similarities between symptoms of twelve of patients

	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈	p ₉	p ₁₀	p ₁₁	p ₁₂
p ₁	1	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{1}{4}$	1	$\frac{3}{4}$
p ₂	$\frac{1}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{1}{4}$	$\frac{2}{4}$
p ₃	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$
p ₄	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{2}{4}$	1	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$
p ₅	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$
p ₆	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{2}{4}$	1	$\frac{1}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$
p ₇	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$
p ₈	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$
p ₉	1	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{1}{4}$	1	$\frac{3}{4}$
p ₁₀	$\frac{1}{4}$	1	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{1}{4}$	1	$\frac{1}{4}$	$\frac{2}{4}$
p ₁₁	1	$\frac{1}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	1	$\frac{1}{4}$	1	$\frac{3}{4}$
p ₁₂	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	$\frac{3}{4}$	$\frac{2}{4}$	$\frac{3}{4}$	1

Then, the minimal structure on P is

$M = {P, \emptyset, {p_{1}, p_{9}, p_{11}}, {p_{2}, p_{10}}, {p_{3}, p_{7}},$

{p₄, p₆} , {p₅, p₈} , {p₁₂} },

and

$2^{P} ∖ M = {P, \emptyset, P ∖ {p_{1}, p_{9}, p_{11}}, P ∖ {p_{2}, p_{10}},$

P ∖ {p₃, p₇} , P ∖ {p₄, p₆} ,

P ∖ {p₅, p₈} , P ∖ {p₁₂} }, Therefore,

The lower approximation of $B_{1}$ is $R^{-} (B_{1}) =$ {p₁, p₃, p₇, p₉, p₁₁}, the upper approximation of $B_{1}$ is $R^{+} (B_{1}) = {p_{1}, p_{3}, p_{5}, p_{7},$ p₈, p₉, p₁₁}, and the accuracy of $B_{1}$ is $γ (B_{1}) = \frac{5}{7}$ .

The lower approximation of $B_{2}$ is $R^{-} (B_{2}) =$ {p₂, p₄, p₆, p₁₀, p₁₂}, the upper approximation of $B_{2}$ is $R^{+} (B_{2}) = {p_{2}, p_{4}, p_{5}, p_{6},$ p₈, p₁₀, p₁₂}, and the accuracy of $B_{2}$ is $γ (B_{2}) = \frac{5}{7}$ .

Therefore, H₂ is not dispensable attribute.

Similarly, we eliminate H₃ attribute, thus the minimal structure on P is

$M = {P, \emptyset, {p_{1}}, {p_{2}, p_{10}}, {p_{3}, p_{7}, p_{8}},$

{p₄, p₆} , {p₅} , {p₉, p₁₁, p₁₂} }, and

$2^{P} ∖ M = {P, \emptyset, P ∖ {p_{1}}, P ∖ {p_{2}, p_{10}},$

P ∖ {p₃, p₇, p₈} , P ∖ {p₄, p₆} ,

P ∖ {p₅} , P ∖ {p₉, p₁₁, p₁₂} }, Therefore,

The lower approximation of $B_{1}$ is $R^{-} (B_{1}) =$ {p₁, p₃, p₇, p₈}, the upper approximation of $B_{1}$ is $R^{+} (B_{1}) = {p_{1}, p_{3}, p_{7}, p_{8}, p_{9},$ p₁₁, p₁₂}, and the accuracy of $B_{1}$ is $γ (B_{1}) = \frac{4}{7}$ .

The lower approximation of $B_{2}$ is $R^{-} (B_{2}) =$ {p₂, p₄, p₅, p₆, p₁₀}, the upper approximation of $B_{2}$ is $R^{+} (B_{2}) = {p_{2}, p_{4}, p_{5}, p_{6}, p_{9},$ p₁₀, p₁₁, p₁₂}, and the accuracy of $B_{2}$ is $γ (B_{2}) = \frac{5}{8}$ .

Then H₃ is not dispensable attribute.

Remark 7. One of the basic concepts of rough set theory is the concept of core that is the common part of reduction. So, we can conclude the core of attributes is {H₂, H₃}.

4 Minimal structure in Dengue fever

In this section, we consider a problem of Dengue fever. This disease is transmitted to humans via virus-carrying Dengue mosquitoes [21]. Symptoms of Dengue fever start from three to four days of infection. Recovery usually takes two to seven days [17]. The disease is common in more than 120 countries around the world, mainly Asia and South America [19]. It causes about 60 million symptomatic infections worldwide and 13600 status deaths. So, we concerned with this problem and tried to analyze via a minimal structure space, reduction of condition attributes, and accuracy of decision attributes are obtained. The data discuss the Dengue fever problem. The columns of the following Table 7 represents the attributes (symptoms of Dengue fever), where T is a high fever (very high (vh), high (h), normal (l)), headache with vomiting H, muscle and joint pains J, and a characteristic skin rash S [21]. Attribute D is the decision of problem and the rows of attributes P = {p₁, p₂, p₃, p₄, p₅, p₆, p₇, p₈} are the patients.

Table 7
Original Dengue fever information system

P J H S T Dengue fever

p ₁ y y y h y

p ₂ y n n h n

p ₃ y n n h y

p ₄ n n n vh n

p ₅ n y y h n

p ₆ y y n vh y

p ₇ y y n l n

p ₈ y y n vh y

P	J	H	S	T	Dengue fever
p ₁	y	y	y	h	y
p ₂	y	n	n	h	n
p ₃	y	n	n	h	y
p ₄	n	n	n	vh	n
p ₅	n	y	y	h	n
p ₆	y	y	n	vh	y
p ₇	y	y	n	l	n
p ₈	y	y	n	vh	y

4.1 Experimental results and discussion

We start the application by transforming variables description into qualitative values as shown in Table 8.

Table 8
Similarities between symptoms of eight of patients

p ₁ p ₂ p ₃ p ₄ p ₅ p ₆ p ₇ p ₈

p ₁ 1 $\frac{1}{2}$ $\frac{1}{2}$ 0 $\frac{3}{4}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$

p ₂ $\frac{1}{2}$ 1 1 $\frac{1}{2}$ $\frac{1}{4}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$

p ₃ $\frac{1}{2}$ 1 1 $\frac{1}{2}$ $\frac{1}{4}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$

p ₄ 0 $\frac{1}{2}$ $\frac{1}{2}$ 1 $\frac{1}{4}$ $\frac{1}{2}$ $\frac{1}{4}$ $\frac{1}{2}$

p ₅ $\frac{3}{4}$ $\frac{1}{4}$ $\frac{1}{4}$ $\frac{1}{4}$ 1 $\frac{1}{4}$ $\frac{1}{4}$ $\frac{1}{4}$

p ₆ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{4}$ 1 $\frac{3}{4}$ 1

p ₇ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{4}$ $\frac{1}{4}$ $\frac{3}{4}$ 1 $\frac{3}{4}$

p ₈ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{4}$ 1 $\frac{3}{4}$ 1

	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈
p ₁	1	$\frac{1}{2}$	$\frac{1}{2}$	0	$\frac{3}{4}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
p ₂	$\frac{1}{2}$	1	1	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
p ₃	$\frac{1}{2}$	1	1	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
p ₄	0	$\frac{1}{2}$	$\frac{1}{2}$	1	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{2}$
p ₅	$\frac{3}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	1	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$
p ₆	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{4}$	1	$\frac{3}{4}$	1
p ₇	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{3}{4}$	1	$\frac{3}{4}$
p ₈	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{4}$	1	$\frac{3}{4}$	1

A minimal structure on P (where μ (p_i, p_j) >0.8) that is related this problem is $M = {P, \emptyset, {p_{1}},$ {p₂, p₃} , {p₄} , {p₅} , {p₆, p₈} , {p₇}}, and thus $2^{P} ∖ M = {P, \emptyset, P ∖ {p_{1}}, P ∖ {p_{2}, p_{3}}, P ∖ {p_{4}}, P ∖ {p_{5}},$ P ∖ {p₆, p₈} , P ∖ {p₇}}.

For $B_{1} = {p_{1}, p_{3}, p_{6}, p_{8}}$ , by rough set theory from Definition 2, we calculate the accuracy of $B_{1}$ and which is the set of patients having Dengue fever, through lower and upper approximations as

$R^{-} (B_{1}) = {p_{1}, p_{6}, p_{8}}, R^{+} (B_{1}) = {p_{1}, p_{2},$ p₃, p₆, p₈}, and $γ (B_{1}) = \frac{3}{5}$ .

For $B_{2} = {p_{2}, p_{4}, p_{5}, p_{7}}$ , by rough set theory from Definition 2, we calculate the accuracy of $B_{2}$ and which is the set of patients not having Dengue fever, through lower and upper approximations as

$R^{-} (B_{2}) = {p_{4}, p_{5}, p_{7}}$ , $R^{+} (B_{2}) = {p_{2},$ p₃, p₄, p₅, p₇}, and $γ (B_{2}) = \frac{3}{5}$ .

Remark 8. By comparing with our method (a minimal structure) and Pawlak’s method [14] to find the accuracy of decision-making, we are founding the same result when μ (p_i, p_j) >0.8.

4.2 Reduction of attributes

A minimal structure and rough set theory are using to analyze information system and construct a minimal subset of attributes give the same quality of the accuracy.

From the Table 7, we eliminate attribute J and construct a minimal structure according to problem requirement to calculate the accuracy of approximation as Table 9; we notice that, accuracy is changed. So, we can not remove attribute J.

Table 9
Attributes without J

p ₁ p ₂ p ₃ p ₄ p ₅ p ₆ p ₇ p ₈

p ₁ 1 $\frac{1}{3}$ $\frac{1}{3}$ 0 1 $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$

p ₂ $\frac{1}{3}$ 1 1 $\frac{2}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$

p ₃ $\frac{1}{3}$ 1 1 $\frac{2}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$

p ₄ 0 $\frac{2}{3}$ $\frac{2}{3}$ 1 0 $\frac{2}{3}$ $\frac{1}{3}$ $\frac{2}{3}$

p ₅ 1 $\frac{1}{3}$ $\frac{1}{3}$ 0 1 $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$

p ₆ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{2}{3}$ $\frac{1}{3}$ 1 $\frac{2}{3}$ 1

p ₇ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{2}{3}$ 1 $\frac{2}{3}$

p ₈ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{2}{3}$ $\frac{1}{3}$ 1 $\frac{2}{3}$ 1

	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈
p ₁	1	$\frac{1}{3}$	$\frac{1}{3}$	0	1	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
p ₂	$\frac{1}{3}$	1	1	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
p ₃	$\frac{1}{3}$	1	1	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
p ₄	0	$\frac{2}{3}$	$\frac{2}{3}$	1	0	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{2}{3}$
p ₅	1	$\frac{1}{3}$	$\frac{1}{3}$	0	1	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
p ₆	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	1	$\frac{2}{3}$	1
p ₇	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	1	$\frac{2}{3}$
p ₈	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	1	$\frac{2}{3}$	1

A minimal structure on P (where μ (p_i, p_j) >0.8) that is related this problem is $M = {P, \emptyset, {p_{1}, p_{5}},$ {p₂, p₃} , {p₄} , {p₆, p₈} , {p₇}}, and thus $2^{P} ∖ M = {P, \emptyset, P ∖ {p_{1}, p_{5}}, P ∖ {p_{2}, p_{3}}, P ∖ {p_{4}}, {p_{6}, p_{8}},$ P ∖ {p₇}}.

$R^{-} (B_{1}) = {p_{6}, p_{8}}$ , $R^{+} (B_{1}) = {p_{1}, p_{2}, p_{3},$ p₅, p₆, p₈}, and $γ (B_{1}) = \frac{1}{3}$ .

$R^{-} (B_{2}) = {p_{4}, p_{7}}$ , $R^{+} (B_{2}) = {p_{1}, p_{2}, p_{3},$ p₄, p₅, p₇}, and $γ (B_{2}) = \frac{1}{3}$ .

Therefore, attribute J is not dispensable.

At eliminate H attribute as shown in Table 10, the minimal structure on P that is related this problem is $M = {P, \emptyset, {p_{1}}, {p_{2}, p_{3}}, {p_{4}}, {p_{5}},$ {p₆, p₈} , {p₇}}, and thus $2^{P} ∖ M = {P, \emptyset, P ∖ {p_{1}},$ P ∖ {p₂, p₃} , P ∖ {p₄} , P ∖ {p₅} , P ∖ {p₆, p₈} , P ∖ {p₇}}.

Table 10

Attributes without H

	p ₁	p ₂	p ₃	p ₄	p ₅	p ₆	p ₇	p ₈
p ₁	1	$\frac{2}{3}$	$\frac{2}{3}$	0	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$
p ₂	$\frac{2}{3}$	1	1	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$
p ₃	$\frac{2}{3}$	1	1	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$
p ₄	0	$\frac{1}{3}$	$\frac{1}{3}$	1	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{2}{3}$
p ₅	$\frac{2}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$	1	0	0	0
p ₆	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	0	1	$\frac{2}{3}$	1
p ₇	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	0	$\frac{2}{3}$	1	$\frac{2}{3}$
p ₈	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	$\frac{2}{3}$	0	1	1	1

We notice that, $γ (B_{1}) = \frac{3}{5}$ . Also, for the set not having Dengue fever $B_{2}$ , notice that, $γ (B_{2}) = \frac{3}{5}$ . That is the same accuracy of the original information system. Similarly, at eliminate S attribute. So, {S, J, T} is reduction. Similarly, at eliminate T attribute we notice dramatic change of accuracy, where, $γ (B_{1}) = \frac{1}{6}$ , and $γ (B_{2}) = \frac{2}{7}$ . So T is indispensable attribute, and there are two D-reduces only as {J, H, T}, and {S, J, T}. The common part of D-reduces is {J, T} that is the D-core of attributes.

5 Improving accuracy based on a variable precision rough set model

Due to the importance of decision-making in a medical cases, we can try to increase the accuracy of decision-making by a variable precision rough set model δ, which proposed in [1, 23]. If we consider the definition of $M$ -interior and $M$ -closure with δ-majority parallel to classical definitions, we notice the contradiction. Therefore, we introduce a novel notion of generalization of $M$ -closure and $M$ -interior as follows:

Definition 9. Let $U$ be a non-empty set, $M$ an $M$ -structure on $U$ , $B \subseteq U$ , and 0 < δ ≤ 0.5 . Then we define

$M^{δ}$ - $C (B) = ⋂ {F \subseteq U : V (F, B) < 1 - δ}$ ,

$M^{δ}$ - $I (B) = ⋃ {G \cap B, V (G, B) \leq δ, G \in M}$ .

Based on Definition 9, we present the following

Theorem 10. Let $(U, M)$ be a minimal structure space and 0 < δ ≤ 0.5 . For $B, D \subseteq U$ , the followings hold:

$M^{δ}$ - $I (U) = U$ and $M^{δ}$ - $C (\emptyset) = \emptyset .$

$M^{δ}$ - $I (B) \subseteq B$ and $B \subseteq M^{δ}$ - $C (B) .$

(i) If $B \in M$ , then $M^{δ}$ - $I (B) = B .$

(ii) If $U ∖ B \in M$ , then $M^{δ}$ - $C (B) = B$ .

If $B \subseteq D$ , then $M^{δ}$ - $I (B) \subseteq M^{δ}$ - $I (D)$ and $M^{δ}$ - $C (B) \subseteq M^{δ}$ - $C (D)$ .

$M^{δ}$ - $I (M^{δ}$ - $I (B)) = M^{δ}$ - $I (B)$ and $M^{δ}$ - $C (M^{δ}$ - $C (B)) = M^{δ}$ - $C (B)$ .

$M^{δ}$ - $I (U ∖ B) = U ∖ M^{δ}$ - $C (B)$ and $M^{δ}$ - $C (U ∖ B) = U ∖ M^{δ}$ - $I (B)$ .

Proof. (1) From Definition 9 and definition of value of measure $V (B, D)$ , $M^{δ}$ - $I (U) = ⋃ {G \cap U, C (G, U) \leq δ, G \in M_{U}}$ , $U, \emptyset \in M$ , and then $V (U, U) = 0$ . Therefore, $M^{δ}$ - $I (U) = U$ . Similarly, since $V (G, \emptyset) = 1$ , then $M^{δ}$ - $C (\emptyset) = \emptyset$ .

(2) It is clear from Definition 9.

(3) (i) If $B \in M$ and $G = B,$ then $V (B, B) = 0,$ that is, $M^{δ}$ - $I (B) = B .$

(ii) If $U ∖ B \in M$ , then $B \in (M^{δ})^{c}$ , and by Definition 9, that is, $M^{δ}$ - $C (B) = B$ .

(4) is similar to (2).

(5) and (6) directly from (3) and by Definition 9.

Example 11. From Table 9, if we put the variable δ = 0.5 in the original information system, then

$R_{0.5}^{-} (B_{1}) = {p_{1}, p_{3}, p_{6}, p_{8}}$ , $R_{0.5}^{+} (B_{1}) = {p_{1}, p_{2}, p_{3}, p_{6}, p_{8}}$ , and $γ_{0.5} (B_{1}) = \frac{4}{5}$ .

$R_{0.5}^{-} (B_{2}) = {p_{2}, p_{4}, p_{5}, p_{7}}$ , $R_{0.5}^{+} (B_{2}) = {p_{2}, p_{3}, p_{4}, p_{5}, p_{7}}$ , and $γ_{0.5} (B_{2}) = \frac{4}{5}$ .

From the Example 11 above we can see the accuracy is improving based on a variable precision rough set model. Further, in the case of eliminate H and S attributes, the accuracy of $γ_{0.5} (B_{1}) = \frac{4}{5}$ and the accuracy of $B_{2}$ is $γ_{0.5} (B_{2}) = \frac{4}{5} .$

Table 11 shows that is improving for our method compared to Pawlak’s method [14].

Table 11
Comparison between Pawlak’s method [14] and our improved method

$U$ Pawlak’s method [14] The current method in Definition 9

$R^{-}$ $R^{+}$ γ $R_{δ = 0.5}^{-}$ $R_{δ = 0.5}^{+}$ γ _δ=0.5

$B_{1} = {p_{1}, p_{3}, p_{6}, p_{8}}$ {p₁, p₆, p₈} {p₁, p₂, p₃, p₆, p₈} $\frac{3}{5}$ {p₁, p₃, p₆, p₈} {p₁, p₂, p₃, p₆, p₈} $\frac{4}{5}$

$B_{2} = {p_{2}, p_{4}, p_{5}, p_{7}}$ {p₄, p₅, p₇} {p₂, p₃, p₄, p₅, p₇} $\frac{3}{5}$ {p₂, p₄, p₅, p₇} {p₂, p₃, p₄, p₅, p₇} $\frac{4}{5}$

$U$	Pawlak’s method [14]	The current method in Definition 9
$B_{1} = {p_{1}, p_{3}, p_{6}, p_{8}}$	{p₁, p₆, p₈}	{p₁, p₂, p₃, p₆, p₈}	$\frac{3}{5}$	{p₁, p₃, p₆, p₈}	{p₁, p₂, p₃, p₆, p₈}	$\frac{4}{5}$
$B_{2} = {p_{2}, p_{4}, p_{5}, p_{7}}$	{p₄, p₅, p₇}	{p₂, p₃, p₄, p₅, p₇}	$\frac{3}{5}$	{p₂, p₄, p₅, p₇}	{p₂, p₃, p₄, p₅, p₇}	$\frac{4}{5}$

6 Conclusion

In this article, we use a minimal structure space to analyze the information system and specify the accuracy of decision-making in some medical applications. Reduction and core of decision attributes are obtained. Then, a comparison between accuracy to decision-making in the case of similarity and dissimilarity symptoms patients. Finally, we give an improving accuracy of decision-making based on a variable precision rough set model. The subsequent work will consider control strategies (related to Traditional Chinese Medicine) and mathematical models (related to fuzzy quantifiers and their integral semantics [22]) of incipient stage of COVID-19 (i.e., Corona Virus Disease) in the group of old people.

Footnotes

Acknowledgment

The authors are grateful to the referees for their valuable comments and suggestions. This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project 2020/01/16509.

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$U$	Pawlak’s method [14]			The current method in Definition 9
	$R^{-}$	$R^{+}$	γ	$R_{δ = 0.5}^{-}$	$R_{δ = 0.5}^{+}$	γ _δ=0.5
$B_{1} = {p_{1}, p_{3}, p_{6}, p_{8}}$	{p₁, p₆, p₈}	{p₁, p₂, p₃, p₆, p₈}	$\frac{3}{5}$	{p₁, p₃, p₆, p₈}	{p₁, p₂, p₃, p₆, p₈}	$\frac{4}{5}$
$B_{2} = {p_{2}, p_{4}, p_{5}, p_{7}}$	{p₄, p₅, p₇}	{p₂, p₃, p₄, p₅, p₇}	$\frac{3}{5}$	{p₂, p₄, p₅, p₇}	{p₂, p₃, p₄, p₅, p₇}	$\frac{4}{5}$

Medical applications via minimal topological structure

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 M -Structure, M -closure, and M -interior

2.2 The core of rough set theory

2.3 Variable precision model

3 Minimal structure in heart failure

3.1 Experimental results and discussion

Table 1 Original medical information system P p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 p 12 H 1 y n y n y n y y y n y y H 2 y n y n n n y y n n n n H 3 y n y n n n y n y n y n H 4 y y y y y y y y y y y y H 5 n y y n y n y y n y n y D y n y n n n y y y n y n

Table 3 (H - H1) P p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 p 12 H 2 Y N Y N N N Y Y N N N N H 3 Y N Y N N N Y N Y N Y N H 4 Y Y Y Y Y Y Y Y Y Y Y Y H 5 N Y Y N Y N Y Y N Y N N D Y N Y N N N Y Y Y N Y N

Table 7 Original Dengue fever information system P J H S T Dengue fever p 1 y y y h y p 2 y n n h n p 3 y n n h y p 4 n n n vh n p 5 n y y h n p 6 y y n vh y p 7 y y n l n p 8 y y n vh y

Footnotes

Acknowledgment

References

2.1 $M$ -Structure, $M$ -closure, and $M$ -interior

Table 3
(H - H₁)

P p ₁ p ₂ p ₃ p ₄ p ₅ p ₆ p ₇ p ₈ p ₉ p ₁₀ p ₁₁ p ₁₂

H ₂ Y N Y N N N Y Y N N N N

H ₃ Y N Y N N N Y N Y N Y N

H ₄ Y Y Y Y Y Y Y Y Y Y Y Y

H ₅ N Y Y N Y N Y Y N Y N N

D Y N Y N N N Y Y Y N Y N

Table 7
Original Dengue fever information system

P J H S T Dengue fever

p ₁ y y y h y

p ₂ y n n h n

p ₃ y n n h y

p ₄ n n n vh n

p ₅ n y y h n

p ₆ y y n vh y

p ₇ y y n l n

p ₈ y y n vh y