Weighted pretopology and reduction of information system

Abstract

In several empirical situations, a decision is needed to be made based on data that is captured in some information system. The problem occurs when the information system holds complex data or even too much data attributes. This leads to the need for reducing the number of attributes required to obtain a decision. In this paper, a novel attributes’ reduction method is presented; the proposed method is based on constructing a weighted pre-topology that represents the information system under consideration. In addition, some essential operations for the weighted pre-topological space are presented; as well as, a brief study of their properties.

Keywords

Fuzzy pretopological space closure set interior set information system

1 Introduction

For making decisions, Information systems play a key rule for reaching a right decision. In machine learning constructing a mathematical model of the information system is one way to deal with the situation.

In most cases, the information system holds to much data attributes, which costs in processing and might be time consuming; For so, the data attributes, needed to be reduced to mention the core attributes. The subject of studying the degree of accuracy of the reduction order in the information system was carried out in different ways, including the rough set [2 , 16], the similarity matrix [11], the dissimilarity matrix [12].

In the past decade, several techniques for attribute reduction were proposed, including positive region preservation reduction, generalized decision preservation reduction, distribution preservation reduction, maximum distribution preservation reduction, and relative discernibility relation preservation reduction.

In 1981, Badard [15] introduced the concepts of a fuzzy pretopology to generalize the concepts of pretopology initiated by M. Brissaud [12]. Badard in his paper investigated some related concepts of fuzzy pretopology such as continuity, compactness, and connectedness. In 1987, A.A. Ramadan [1] investigated some concepts in fuzzy pretopological spaces such as the Hausdorff property, regularity, normality and several kinds of compactness and connectedness.

The aim of this paper is to introduce a new methodology for reducing the attributes in information system by using statistical method.

The remaining of this paper is structured as follows: In Section 2, the fundamental definitions from the pretopological space are introduced. In Section 3, developing a new type of a fuzzy pretopological space used in information system by using similarity method, In Section 4, a statistical method presented for the purpose of reducing the attributes. Hence, an illustrative example showing the method of similarity presented in Section 5, is introduced. To conclude, Section 6, gives a brief conclusion.

2 Prelimininaries

2.1 Definition [15]

A fuzzy pretopology on X can be defined as a mapping a : I^X → I^X, which satisfies:

(FP₁) a (0) = 0.

(FP₂) A ⊆ a (A), for every A ∈ I^X.

The pair (X, a) is said to be a fuzzy pretopological space (for short, fpts).

2.2 Definition [15]

Let (X, a) be a fpts, we define the fuzzy interior mapping int_a : I^X → I^X by int_a (A) = co (a (coA)). where coA is complement of A, A ⊆ I^X.

In this case, (X, a) is said to be a fpts of type I.

2.3 Definition [6]

For any fpts (X, a), the following properties are considered

(FP₃): For every A, B ∈ I^X such that A ⊆ B we have a (A) ⊆ a (B).

(FP₄): For every A, B ∈ I^X we have a (A ∪ B) = a (A) ∪ a (B).

In this case, (X, a) is said to be a fpts of type D.

(FP₅): For every A, B ∈ I^X we have a² (A) = a (a (A)) = a (A).

In this case, (X, a) is said to be a fpts of type S.

2.4 Information system

An information system (IS) [13] is an ordered triple IS = (U, K, Q), such that U is a nonempty finite set of objects (students, toy blocks, . . .) called universe, K is a nonempty set of attributes (colors, characteristic, . . .) and Q is an attribute scale ordinal.

3 Weighted pretopological space generating from information system

In this work, we aim to develop a new type of a fuzzy pretopological using correlation method to represent the information system in hand.

3.1 Definition

For any information system (IS), we define an |U| × |U| similarity matrix, with the following form: - $corr (x_{i}, y_{j}) = \frac{covariance (x_{i}, y_{j})}{{[SD (x_{i}, y_{j})]}^{2}}$ where $covariance (x_{i}, y_{j}) = \frac{1}{k} \sum_{j = 1}^{k} {(x_{i} - \bar{μ_{j}})}^{2}$ i = 1, …, n, j = 1 : k]

where |K| is a cardinal number of a set of attributes k.

$Standard Derivation (SD) = \sqrt{\frac{1}{k} \sum_{j = 1}^{k} {(x_{i} - \bar{μ_{j}})}^{2}}$

$\bar{μ_{j}} = \frac{\sum_{i = 1}^{n} x_{ij}}{n} j = 1 : k and n are a number of objects$ .

3.2 Definition

Let U ={ x₁, x₂, x₃, … . , x_n } be a set of objects, let Γ_a (x_i) is a relation between x_i, x_j, its value is determined by corr (x_i, x_j).

Then; $\begin{matrix} Γ_{a} ({x_{i}}) = {(x_{j}, corr (x_{i}, x_{j})) : x_{j} \in U and \\ corr (x_{i}, x_{j}) ⩾ σ, 0 < σ ⩽ 1, i, j \in {1, \dots, n}, \\ n = | U |} \end{matrix}$

Where (U, Γ_a (x_i)) is a weighted pretopological space (for short, WP).

Note: The cardinality of a set |U|, which is basically the size of the set.

3.3 Definition

For any weighted pretopological space (X, Γ_a), the interior function int : F (X) → F (X) defined as;

Equivalently, $\begin{matrix} i n t Γ_{a} (A) = {(x_{i}, c o r r (x_{i}, x_{j})) : x_{i} \in U, i, \\ j \in {1, \dots, n} \\ i n t_{Γ_{a}} (A) = c o (Γ_{a} (c o A)) . \end{matrix}$ where coA is the complement of A.

3.4 Definition

For any weighted pretopological space (X, Γ_a), the following properties are considered:

(WP₁) : Γ_a (0) = 0.

(WP₂) : A ⊆ Γ_a (A).

3.5 Definition

For any fuzzy pretopological space used in information system (X, Γ_a), the following properties are considered:

(WP₃) For every A, B ∈ I^X such that A ⊆ B we have Γ_a (A) ⊆ Γ_a (B). In this case, (X, Γ_a) is said to be a WP of type I.

(WP₄) For every A, B ∈ I^X we have Γ_a (A ∪ B) = Γ_a (A) ∪ Γ_a (B). In this case, (X, Γ_a) is said to be a WP of type D.

(WP₅) For every A, B ∈ I^X we have Γ_a² (A) = Γ_a (Γ_a (A)) = Γ_a (A). In this case, (X, Γ_a) is said to be a WP of type S.

3.6 Definition

For any set x ⊂ U, the attributes’ reduction degree of accuracy (γ_x) [5], can be approximated using the following formula; $γ_{x} = \frac{\cup_{i} {int}_{Γ_{a}} ({x_{i}})}{\cap_{i} Γ_{a} ({x_{i}})} \forall x_{i} \in X, i = 1, \dots, m < n$

3.7 Definition

The attributes’ reduction ratio ( $R_{Q}$ ), is the value calculated by the formula: $R_{Q} = \frac{height (\cup_{i} {int}_{Γ_{a}} ({x_{i}}))}{pl int h (\cap_{i} Γ_{a} ({x_{i}}))}$

4 The proposed reduction procedure

For any information system IS = (U, K, Q), where U is a set of n objects, K is a nonempty set of attributes and Q is an attribute scale ordinal; we propose the following procedure in order to define a set of core attributes of the system under consideration:

Construct an |U| × |U| similarity matrix, where its entities are the correlation coefficients between the objects in U, which are computed for all attributes in K using the definition [3.1].

Reconstruct the matrix in step (1), after removing one attribute from the set K.

For each attribute in K, repeat the step (2), by removing another attribute.

A weighted pretopolgy is to be constructed for each matrix of the matrices deduced from the previous steps (1 : 3); where its values are to be calculated using the definition [3.2].

A degree of accuracy for each subset of U is to be found using the definition [3.6].

Finally, a ratio for the attributes’ reduction is to be deduced using the definition [3.7].

Define the core attributes for the system in hand; in this step we have one of the following two cases:

Case 1: If the degree of $R_{Q}$ after removing one attribute is similar than the degree of $R_{Q}$ for the original pretopology, then the existence of this attribute can be removed.

Case 2: If the degree of $R_{Q}$ after removing one attribute is less than the degree of $R_{Q}$ for the original pretopology, then this attribute is important and this attribute can’t be removed.

5 Statistical application on weighted pretopological space

In this section, we give an example to demonstrate how to construct the pretopology from some information system.

The following (Table 1), presents some employee applications (objects) and some tests in science branches (attributes), Mathematics (Math.), Physics (Phys.), Chemistry (Chem.), Computer skills (Comp.).

Table 1
Show the information system

U Math. Phys. Chem. Comp.

e₁ 83 84 77 81

e₂ 85 93 81 100

e₃ 60 89 78 71

e₄ 60 88 82 93

e₅ 85 91 87 97

e₆ 90 87 68 93

U	Math.	Phys.	Chem.	Comp.
e₁	83	84	77	81
e₂	85	93	81	100
e₃	60	89	78	71
e₄	60	88	82	93
e₅	85	91	87	97
e₆	90	87	68	93

To proceed, we have that |U| = 6, |K| = 4

Hence, we apply the following steps:

Calculate sum of each attribute: $\sum_{i = 1}^{6} x_{ij}, where j = 1 : 4$ (1)

Compute the mean value ${\bar{μ}}_{j}$ for each attribute: ${\bar{μ}}_{j} = \frac{\sum_{i = 1}^{6} x_{ij}}{6}, where j = 1 : 4$ (2)

Compute the divergence from the mean value for each attribute $\begin{matrix} x_{i} - {\bar{μ}}_{j} where i, j = 1.2, \dots, 6 \\ and k = 1, \dots, 4 \end{matrix}$ (3)

Evaluate δ_K for each attribute $\begin{matrix} δ_{k} (i, j) = (x_{i} - {\bar{μ}}_{j})^{2} where i = 1 : 6 \\ and j = 1 : 4 \end{matrix}$ (4)

Compute the covariance for the attribute u_i and u_j, as follows: $\begin{matrix} Convariance (u_{i}, u_{j}) = \frac{1}{4} \sum_{j = 1}^{4} δ_{k} (i, j) \\ where i = 1 : 6 \end{matrix}$ (5)

Compute the standard deviation for each attribute $SD = \sqrt{\frac{1}{4} \sum_{j = 1}^{4} {(x_{i} - \bar{μ_{j}})}^{2}}$ (6)

Compute the correlation as follows: $Corr (x_{i}, x_{j}) = \frac{convariance (x_{i}, x_{j})}{{(SD (u_{i}, u_{j}))}^{2}}$ (7)

Use the following relation to find a weighted pretopology for the attributes; denoted by Γ_a: $\begin{matrix} Γ_{a} (x_{i}) = {(x_{i}, corr (x_{i}, x_{j})) : x_{j} \in U and \\ corr (x_{i}, x_{j}) > 0.5 i, j = 1, 2, 3, \dots, 6 \end{matrix}$ (8)

Consider two partitions of U as follows: $\begin{matrix} U = E_{1} \cup E_{2} and E_{1} \cap E_{2} = \emptyset, \\ where E_{1} = {e_{2}, e_{5}} and E_{2} = {e_{1}, e_{3}, e_{4}, e_{6}} . \end{matrix}$ (9)

Compute the degree of accuracy for each partition (E₁ and E₂).

Calculate the ratio $R_{Q}$ for each partition (E₁ and E₂).

5.1 Processing the information system using the proposed procedure

5.1.1 Constructing weighted pretopolgy using full attribute

The following (Table 2), shows the sum of each attribute and the mean value obtained from the Equations (2).

Table 2
Shows the sum and the mean value of each attribute

Math. Phys. Chem. Comp.

e₁ 83 84 77 81

e₂ 85 93 81 100

e₃ 60 89 78 71

e₄ 60 88 82 93

e₅ 85 91 87 97

e₆ 90 87 68 93

Sum 463 532 473 535

Mean value $({\bar{μ}}_{j})$ 77.16667 88.66667 78.83333 89.16667

	Math.	Phys.	Chem.	Comp.
e₁	83	84	77	81
e₂	85	93	81	100
e₃	60	89	78	71
e₄	60	88	82	93
e₅	85	91	87	97
e₆	90	87	68	93
Sum	463	532	473	535
Mean value $({\bar{μ}}_{j})$	77.16667	88.66667	78.83333	89.16667

The following (Table 3), shows the divergence from the mean value obtained from the Equation (3).

Table 3

Shows $x_{i} - {\bar{μ}}_{j}$

$x_{i} - {\bar{μ}}_{j}$	Math.	Phys.	Chem.	Comp.
e₁	5.833333	4.666667	1.833333	8.166667
e₂	7.833333	4.333333	2.166667	10.83333
e₃	17.16667	0.333333	0.833333	18.16667
e₄	17.16667	0.666667	3.166667	3.833333
e₅	7.833333	2.333333	8.166667	7.833333
e₆	12.83333	1.666667	10.83333	3.833333

The following (Table 4), shows δ_Math. obtained from the Equation (4).

Table 4

Shows δ_Math.

Math.	e₁	e₂	e₃	e₄	e₅	e₆
e₁	34.02778	45.69444	100.1389	100.1389	45.69444	74.86111
e₂	45.69444	61.36111	134.4722	134.4722	61.36111	100.5278
e₃	100.1389	134.4722	294.6944	294.6944	134.4722	220.3056
e₄	100.1389	134.4722	294.6944	294.6944	134.4722	220.3056
e₅	45.69444	61.36111	134.4722	134.4722	61.36111	100.5278
e₆	74.86111	100.5278	220.3056	220.3056	100.5278	164.6944

The following (Table 5), shows δ_Phy. obtained from the Equation (4).

Table 5

Shows δ_Phy.

Phys.	e₁	e₂	e₃	e₄	e₅	e₆
e₁	21.77778	20.22222	1.555556	3.111111	10.88889	7.777778
e₂	20.22222	18.77778	1.444444	2.888889	10.11111	7.222222
e₃	1.555556	1.444444	0.111111	0.222222	0.777778	0.555556
e₄	3.111111	2.888889	0.222222	0.444444	1.555556	1.111111
e₅	10.88889	10.11111	0.777778	1.555556	5.444444	3.888889
e₆	7.777778	7.222222	0.555556	1.111111	3.888889	2.777778

The following (Table 6), shows δ_Chem. obtained from the Equation (4).

Table 6

shows δ_Chem.

Chem.	e₁	e₂	e₃	e₄	e₅	e₆
e₁	3.361111	3.972222	1.527778	5.805556	14.97222	19.86111
e₂	3.972222	4.694444	1.805556	6.861111	17.69444	23.47222
e₃	1.527778	1.805556	0.694444	2.638889	6.805556	9.027778
e₄	5.805556	6.861111	2.638889	10.02778	25.86111	34.30556
e₅	14.97222	17.69444	6.805556	25.86111	66.69444	88.47222
e₆	19.86111	23.47222	9.027778	34.30556	88.47222	117.3611

The following (Table 7), shows δ_Comp. obtained from the Equation (4).

Table 7

Shows δ_Comp.

Comp.	e₁	e₂	e₃	e₄	e₅	e₆
e₁	66.69444	88.47222	148.3611	31.30556	63.97222	31.30556
e₂	88.47222	117.3611	196.8056	41.52778	84.86111	41.52778
e₃	148.3611	196.8056	330.0278	69.63889	142.3056	69.63889
e₄	31.30556	41.52778	69.63889	14.69444	30.02778	14.69444
e₅	63.97222	84.86111	142.3056	30.02778	61.36111	30.02778
e₆	31.30556	41.52778	69.63889	14.69444	30.02778	14.69444

The following (Tables 8 and 9), shows the covariance with k = 4 obtained from the Equation (5).

Table 8

Shows the sum of all attributes

Sum	e₁	e₂	e₃	e₄	e₅	e₆
e₁	125.8611	158.3611	251.5833	140.3611	135.5278	133.8056
e₂	158.3611	202.1944	334.5278	185.75	174.0278	172.75
e₃	251.5833	334.5278	625.5278	367.1944	284.3611	299.5278
e₄	140.3611	185.75	367.1944	319.8611	191.9167	270.4167
e₅	135.5278	174.0278	284.3611	191.9167	194.8611	222.9167
e₆	133.8056	172.75	299.5278	270.4167	222.9167	299.5278

Table 9

Shows the covariance of all attributes

Covariance	e₁	e₂	e₃	e₄	e₅	e₆
e₁	31.46528	39.59028	62.89583	35.09028	33.88194	33.45139
e₂	39.59028	50.54861	83.63194	46.4375	43.50694	43.1875
e₃	62.89583	83.63194	156.3819	91.79861	71.09028	74.88194
e₄	35.09028	46.4375	91.79861	79.96528	47.97917	67.60417
e₅	33.88194	43.50694	71.09028	47.97917	48.71528	55.72917
e₆	33.45139	43.1875	74.88194	67.60417	55.72917	74.88194

The following (Table 10), shows the standard deviation obtained from the Equation (6).

Table 10

Shows the standard deviation

	Math.	Phys.	Chem.	Comp.	SUM	Variance	SD
e₁	34.02778	21.77778	3.361111	66.69444	125.8611	31.46528	5.609392
e₂	61.36111	18.77778	4.694444	117.3611	202.1944	50.54861	7.109755
e₃	294.6944	0.111111	0.694444	330.0278	625.5278	156.3819	12.50528
e₄	294.6944	0.444444	10.02778	14.69444	319.8611	79.96528	8.942331
e₅	61.36111	5.444444	66.69444	61.36111	194.8611	48.71528	6.979633
e₆	164.6944	2.777778	117.3611	14.69444	299.5278	74.88194	8.653435

The following (Table 11), shows the correlation obtained from the Equation (7).

Table 11

Shows the correlation coefficients

Corr (x_i, x_j)	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.9927	0.896629	0.699552	0.86541	0.689144
e₂	0.9927	1	0.940642	0.730405	0.87674	0.701964
e₃	0.896629	0.940642	1	0.820903	0.81449	0.691938
e₄	0.699552	0.730405	0.820903	1	0.76872	0.873643
e₅	0.86541	0.87674	0.81449	0.76872	1	0.9227
e₆	0.689144	0.701964	0.691938	0.873643	0.9227	1

The following (Table 12), gives the values of correlation coefficients between objects after shortening the numbers mentioned in (Table 11).

Table 12

Gives the values of correlation between the objects

	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.9	0.8	0.6	0.8	0.6
e₂	0.9	1	0.9	0.7	0.8	0.7
e₃	0.8	0.9	1	0.8	0.8	0.6
e₄	0.6	0.7	0.8	1	0.7	0.8
e₅	0.8	0.8	0.8	0.7	1	0.9
e₆	0.6	0.7	0.6	0.8	0.9	1

Using Equation (8), we get the following components of the weighted pretopology: $\begin{matrix} {Γ_{a}}_{1} {e_{1}} = {(e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.8), (e_{4}, 0.6), \\ (e_{5}, 0.8), (e_{6}, 0.6)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{1} {e_{2}} = {(e_{1}, 0.9), (e_{2}, 1), (e_{3}, 0.9), (e_{4}, 0.7), \\ (e_{5}, 0.8), (e_{6}, 0.7)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{1} {e_{3}} = {(e_{1}, 0.8), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), \\ (e_{5}, 0.8), (e_{6}, 0.6)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{1} {e_{4}} = {(e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), \\ (e_{5}, 0.7), (e_{6}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{1} {e_{5}} = {(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.8), (e_{4}, 0.7), \\ (e_{5}, 1), (e_{6}, 0.9)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{1} {e_{6}} = {(e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.6), (e_{4}, 0.8), \\ (e_{5}, 0.9), (e_{6}, 1)} \end{matrix}$

Therefore ,

1- Calculate Γ_a₁{ E₁ } and int_{Γ_a₁}{ E₁ }

Γ_a₁ {E₁}	Γ_a₁ {e₂, e₅} = {(e₁, 0.9) , (e₂, 1) , (e₃, 0.9) , (e₄, 0.7) , (e₅, 0.8) , (e₆, 0.7) ,
(e₁, 0.8) , (e₂, 0.8) , (e₃, 0.8) , (e₄, 0.7) , (e₅, 1) , (e₆, 0.9)}
int_{Γ_a₁} {E₁}	$\begin{matrix} {int}_{{Γ_{a}}_{1}} {e_{2}, e_{5}} = co {{Γ_{a}}_{1} co {e_{2}, e_{5}}} = co {{{Γ_{a}}_{1} {e_{1}, e_{3}, e_{4}, e_{6}}} \\ = co {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.8), (e_{4}, 0.6), (e_{5}, 0.8), (e_{6}, 0.6), \\ (e_{1}, 0.8), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), (e_{5}, 0.8), (e_{6}, 0.6), \\ (e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), (e_{5}, 0.7), (e_{6}, 0.8), \\ (e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.6), (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1) \end{matrix}} \end{matrix}$
	$= {\begin{matrix} (e_{1}, 0), (e_{2}, 0.1), (e_{3}, 0.2), (e_{4}, 0.4), (e_{5}, 0.2), (e_{6}, 0.4), \\ (e_{1}, 0.2), (e_{2}, 0.1), (e_{3}, 0), (e_{4}, 0.2), (e_{5}, 0.2), (e_{6}, 0.4), \\ (e_{1}, 0.4), (e_{2}, 0.3), (e_{3}, 0.2), (e_{4}, 0), (e_{5}, 0.3), (e_{6}, 0.2), \\ (e_{1}, 0.4), (e_{2}, 0.3), (e_{3}, 0.4), (e_{4}, 0.2), (e_{5}, 0.1), (e_{6}, 0) \end{matrix}}$
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.4), (e_{2}, 0.3), (e_{3}, 0.4), (e_{4}, 0.4), (e_{5}, 0.3), (e_{6}, 0.4)}}{{(e_{1}, 8), (e_{2}, 0.8), (e_{3}, 0.8), (e_{4}, 0.7), (e_{5}, 0.8), (e_{6}, 0.7)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.4}{0.7} = 0.57$

1- calculate Γ_a₁{ E₂ } and int_{Γ_a₁}{ E₂ }

Γ_a₁{ E₂ }	Γ_a₁{ e₁, e₃, e₄, e₆ }
	$= {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.8), (e_{4}, 0.6), (e_{5}, 0.8), (e_{6}, 0.6), \\ (e_{1}, 0.8), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), (e_{5}, 0.8), (e_{6}, 0.6), \\ (e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), (e_{5}, 0.7), (e_{6}, 0.8), \\ (e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.6), (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1) \end{matrix}}$
int_{Γ_a₁}{ E₂ }	$\begin{matrix} {int}_{{Γ_{a}}_{1}} {e_{1}, e_{3}, e_{4}, e_{6}} = co {{Γ_{a}}_{1} co {e_{1}, e_{3}, e_{4}, e_{6}}} \\ = co {{Γ_{a}}_{1} {e_{2}, e_{5}}} \\ = co {(e_{1}, 0.9), (e_{2}, 1), (e_{3}, 0.9), (e_{4}, 0.7), (e_{5}, 0.8), (e_{6}, 0.7), \end{matrix}$
	$\begin{matrix} (e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.8), (e_{4}, 0.7), (e_{5}, 1), (e_{6}, 0.9)} \\ = {(e_{1}, 0.1), (e_{2}, 0), (e_{3}, 0.1), (e_{4}, 0.3), (e_{5}, 0.2), (e_{6}, 0.3), \end{matrix}$
	(e₁, 0.2) , (e₂, 0.2) , (e₃, 0.2) , (e₄, 0.3) , (e₅, 0) , (e₆, 0.1)}
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.2), (e_{2}, 0.2), (e_{3}, 0.2), (e_{4}, 0.3), (e_{5}, 0.2), (e_{6}, 0.3)}}{{(e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.6), (e_{4}, 0.6), (e_{5}, 0.7), (e_{6}, 0.6)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.3}{0.6} = 0.5$

5.1.2 Further fuzzy pretopologies from the original information system

The aim of this section is to experiment the effect on the constructed weighted pretopology, when removing some attributes from the original information system. For that purpose, we will remove one attribute at a time.

5.1.2.1 Constructing weighted pretopology when removing the attribute Mathematical (Math.) For the example given in Section 5.1, and when removing the attribute, Math., In order to get a weighted pretopology, we are going to use the same steps described in (4.1).

The following (Table 13), shows the sum of each attribute and the mean value obtained from the Equations (2).

Table 13
Shows the sum and the mean value of each attribute

Phys. Chem. Comp.

e₁ 84 77 81

e₂ 93 81 100

e₃ 89 78 71

e₄ 88 82 93

e₅ 91 87 97

e₆ 87 68 93

Sum 532 473 535

Mean value $({\bar{μ}}_{j})$ 88.66667 78.83333 89.16667

	Phys.	Chem.	Comp.
e₁	84	77	81
e₂	93	81	100
e₃	89	78	71
e₄	88	82	93
e₅	91	87	97
e₆	87	68	93
Sum	532	473	535
Mean value $({\bar{μ}}_{j})$	88.66667	78.83333	89.16667

The following (Table 14), shows the divergence from the mean value obtained from the Equation (3).

Table 14

Shows $x_{i} - {\bar{μ}}_{j}$

$x_{i} - {\bar{μ}}_{j}$	Phys.	Chem.	Comp.
e₁	4.666667	1.833333	8.166667
e₂	4.333333	2.166667	10.83333
e₃	0.333333	0.833333	18.16667
e₄	0.666667	3.166667	3.833333
e₅	2.333333	8.166667	7.833333
e₆	1.666667	10.83333	3.833333

The following (Table 15), shows δ_Phys. obtained from the Equation (4).

Table 15

Shows δ_Phys.

Phys.	e₁	e₂	e₃	e₄	e₅	e₆
e₁	21.77778	20.22222	1.555556	3.111111	10.88889	7.777778
e₂	20.22222	18.77778	1.444444	2.888889	10.11111	7.222222
e₃	1.555556	1.444444	0.111111	0.222222	0.777778	0.555556
e₄	3.111111	2.888889	0.222222	0.444444	1.555556	1.111111
e₅	10.88889	10.11111	0.777778	1.555556	5.444444	3.888889
e₆	7.777778	7.222222	0.555556	1.111111	3.888889	2.777778

The following (Table 16), shows δ_Chem. obtained from the Equation (4).

Table 16

Shows δ_Chem.

Chem.	e₁	e₂	e₃	e₄	e₅	e₆
e₁	3.361111	3.972222	1.527778	5.805556	14.97222	19.86111
e₂	3.972222	4.694444	1.805556	6.861111	17.69444	23.47222
e₃	1.527778	1.805556	0.694444	2.638889	6.805556	9.027778
e₄	5.805556	6.861111	2.638889	10.02778	25.86111	34.30556
e₅	14.97222	17.69444	6.805556	25.86111	66.69444	88.47222
e₆	19.86111	23.47222	9.027778	34.30556	88.47222	117.3611

The following (Table 17), shows δ_Comp. obtained from the Equation (4).

Table 17

Shows δ_Comp.

Comp.	e₁	e₂	e₃	e₄	e₅	e₆
e₁	66.69444	88.47222	148.3611	31.30556	63.97222	31.30556
e₂	88.47222	117.3611	196.8056	41.52778	84.86111	41.52778
e₃	148.3611	196.8056	330.0278	69.63889	142.3056	69.63889
e₄	31.30556	41.52778	69.63889	14.69444	30.02778	14.69444
e₅	63.97222	84.86111	142.3056	30.02778	61.36111	30.02778
e₆	31.30556	41.52778	69.63889	14.69444	30.02778	14.69444

The following (Tables 18 19), shows the covariance with k = 3 obtained from the Equation (5).

Table 18

Shows the sum of three attributes

Sum	e₁	e₂	e₃	e₄	e₅	e₆
e₁	91.83333333	112.6666667	151.4444444	40.22222222	89.83333333	58.94444444
e₂	112.6666667	140.8333333	200.0555556	51.27777778	112.6666667	72.22222222
e₃	151.4444444	200.0555556	330.8333333	72.5	149.8888889	79.22222222
e₄	40.22222222	51.27777778	72.5	25.16666667	57.44444444	50.11111111
e₅	89.83333333	112.6666667	149.8888889	57.44444444	133.5	122.3888889
e₆	58.94444444	72.22222222	79.22222222	50.11111111	122.3888889	134.8333333

Table 19

Shows the covariance of three attributes

Covariance	e₁	e₂	e₃	e₄	e₅	e₆
e₁	30.61111111	37.55555556	50.48148148	13.40740741	29.94444444	19.64814815
e₂	37.55555556	46.94444444	66.68518519	17.09259259	37.55555556	24.07407407
e₃	50.48148148	66.68518519	110.2777778	24.16666667	49.96296296	26.40740741
e₄	13.40740741	17.09259259	24.16666667	8.388888889	19.14814815	16.7037037
e₅	29.94444444	37.55555556	49.96296296	19.14814815	44.5	40.7962963
e₆	19.64814815	24.07407407	26.40740741	16.7037037	40.7962963	44.94444444

The following (Table 20), shows the standard deviation obtained from the Equation (6).

Table 20

Shows the standard deviation

	Phys.	Chem.	Comp.	SUM	Variance	SD
e₁	21.77777778	3.361111111	66.69444444	91.83333333	30.61111111	5.532730891
e₂	18.77777778	4.694444444	117.3611111	140.8333333	46.94444444	6.851601597
e₃	0.111111111	0.694444444	330.0277778	330.8333333	110.2777778	10.50132267
e₄	0.444444444	10.02777778	14.69444444	25.16666667	8.388888889	2.896357866
e₅	5.444444444	66.69444444	61.36111111	133.5	44.5	6.670832032
e₆	2.777777778	117.3611111	14.69444444	134.8333333	44.94444444	6.704061787

The following (Table 21), shows the correlation obtained from the Equation (7).

Table 21

Shows the correlation coefficients

Corr (x_i, x_j)	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.99070086	0.86885747	0.83666776	0.81132845	0.52971719
e₂	0.99070086	1	0.92681543	0.86131818	0.82167880	0.52410642
e₃	0.868857475	0.926815432	1	0.794548712	0.71322109	0.37509714
e₄	0.836667768	0.861318188	0.794548712	1	0.991047687	0.86024577
e₅	0.811328455	0.821678803	0.71322109	0.991047687	1	0.91222658
e₆	0.529717193	0.524106423	0.375097141	0.860245775	0.912226583	1

The following (Table 22), gives the values of correlation coefficients between objects after shortening the numbers mentioned in (Table 21).

Table 22

Gives the values of correlation between the objects

	e₁	e₂	e₃	e₄	e₅	e₆
	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.9	0.8	0.8	0.8	0.5
e₂	0.9	1	0.9	0.8	0.8	0.5
e₃	0.8	0.9	1	0.7	0.7	0.3
e₄	0.8	0.8	0.7	1	0.9	0.8
e₅	0.8	0.8	0.7	0.9	1	0.9
e₆	0.5	0.5	0.3	0.8	0.9	1

Using Equation (8), we get the following components of the weighted pretopology: $\begin{matrix} {Γ_{a}}_{2} {e_{1}} = {(e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.8), \\ (e_{4}, 0.8), (e_{5}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{2} {e_{2}} = {(e_{1}, 0.9), (e_{2}, 1), (e_{3}, 0.9), \\ (e_{4}, 0.8), (e_{5}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{2} {e_{3}} = {(e_{1}, 0.8), (e_{2}, 0.9), (e_{3}, 1), \\ (e_{4}, 0.7), (e_{5}, 0.7)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{2} {e_{4}} = {(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 1), \\ (e_{5}, 0.9), (e_{6}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{2} {e_{5}} = {(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.9), \\ (e_{5}, 1), (e_{6}, 0.9)} \end{matrix}$ ${Γ_{a}}_{2} {e_{6}} = {(e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1)}$

Therefore,

1- Calculate Γ_{a
₂} {E₁} and int_{Γ_a
₂} {E₁}

Γ_a₂{ E₁ }	Γ_a₂ { e₂, e₅ } = {(e₁, 0.9) , (e₂, 1) , (e₃, 0.9) , (e₄, 0.8) , (e₅, 0.8)
(e₁, 0.8) , (e₂, 0.8) , (e₃, 0.7) , (e₄, 0.9) , (e₅, 1) , (e₆, 0.9)}
int_{Γ_a₂}{ E₁ }	$\begin{matrix} {int}_{{Γ_{a}}_{2}} {e_{2}, e_{5}} = co {{Γ_{a}}_{2} co {e_{2}, e_{5}}} = co {{{Γ_{a}}_{2} {e_{1}, e_{3}, e_{4}, e_{6}}} \\ = co {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.8), (e_{4}, 0.8), (e_{5}, 0.8), \\ (e_{1}, 0.8), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.7), (e_{5}, 0.7), \\ (e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 1), (e_{5}, 0.9), (e_{6}, 0.8), \\ (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1) \end{matrix}} \end{matrix}$
	$= {\begin{matrix} (e_{1}, 0), (e_{2}, 0.1), (e_{3}, 0.2), (e_{4}, 0.2), (e_{5}, 0.2), \\ (e_{1}, 0.2), (e_{2}, 0.1), (e_{3}, 0), (e_{4}, 0.3), (e_{5}, 0.3), \\ (e_{1}, 0.2), (e_{2}, 0.2), (e_{3}, 0.3), (e_{4}, 0), (e_{5}, 0.1), (e_{6}, 0.2), \\ (e_{4}, 0.2), (e_{5}, 0.1), (e_{6}, 0) \end{matrix}}$
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.2), (e_{2}, 0.2), (e_{3}, 0.3), (e_{4}, 0.3), (e_{5}, 0.3), (e_{6}, 0.2)}}{{(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.8), (e_{5}, 0.8), (e_{6}, 0.9)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.3}{0.7} = 0.428$

2- calculate Γ_{a
₂} {E₂} and int_{Γ_a
₂} {E₂}

Γ_a₂{ E₂ }	Γ_a₂{ e₁, e₃, e₄, e₆ }
	$= {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.8), (e_{4}, 0.8), (e_{5}, 0.8), \\ (e_{1}, 0.8), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.7), (e_{5}, 0.7), \\ (e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 1), (e_{5}, 0.9), (e_{6}, 0.8), \\ (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1) \end{matrix}}$
	int_{Γ_a₂} { e₁, e₃, e₄, e₆ } = co {Γ_a₂co { e₁, e₃, e₄, e₆ }}
	= coΓ_a₂ { e₂, e₅ }}
	= co {(e₁, 0.9) , (e₂, 1) , (e₃, 0.9) , (e₄, 0.8) , (e₅, 0.8) ,
	(e₁, 0.8) , (e₂, 0.8) , (e₃, 0.7) , (e₄, 0.9) , (e₅, 1) , (e₆, 0.9)}
	= {(e₁, 0.1) , (e₂, 0) , (e₃, 0.1) , (e₄, 0.2) , (e₅, 0.2) ,
	(e₁, 0.2) , (e₂, 0.2) , (e₃, 0.3) , (e₄, 0.1) , (e₅, 0) , (e₆, 0.1)}
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.2), (e_{2}, 0.2), (e_{3}, 0.3), (e_{4}, 0.2), (e_{5}, 0.2), (e_{6}, 0.1)}}{{(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.7), (e_{5}, 0.7), (e_{6}, 0.8)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.3}{0.7} = 0.428$

Remark 5.1:

Repeating the same procedure done with the previous tables, we reconstruct weighted pretopological spaces by removing one column at a time; that is removing the second column, the third column, the fourth column and so on.

Similarity, we are able to obtain the weighted pretopological spaces, the interior, the degree of accuracy and calculate $R_{Q}$ for each of these tables.

5.1.2.2 Constructing weighted pretopology when removing the attribute Physics (Phys.) The following (Table 23), shows the correlation obtained from the Equation (7).

Table 23

Shows the correlation coefficients

Corr (x_i, x_j)	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.99978349	0.97997035	0.75273649	0.88767477	0.71710098
e₂	0.99978349	1	0.98344206	0.75548034	0.87941652	0.7095060
e₃	0.97997035	0.98344206	1	0.8210501	0.82392348	0.6939860
e₄	0.75273649	0.75548034	0.82105014	1	0.77390980	0.8747242
e₅	0.88767477	0.87941652	0.82392348	0.77390980	1	0.9238356
e₆	0.71710098	0.70950608	0.69398606	0.87472424	0.923835695	1

The following (Table 24), gives the values of correlation coefficients between objects after shortening the numbers mentioned in (Table 23).

Table 24

Gives the values of correlation between the objects

	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.9	0.9	0.7	0.8	0.7
e₂	0.9	1	0.9	0.7	0.8	0.7
e₃	0.9	0.9	1	0.8	0.8	0.6
e₄	0.7	0.7	0.8	1	0.7	0.8
e₅	0.8	0.8	0.8	0.7	1	0.9
e₆	0.7	0.7	0.6	0.8	0.9	1

Using Equation (8), we get the following components of the weighted pretopology: $\begin{matrix} {Γ_{a}}_{3} {e_{1}} = {(e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.9), (e_{4}, 0.7), \\ (e_{5}, 0.8), (e_{6}, 0.7)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{3} {e_{2}} = {(e_{1}, 0.9), (e_{2}, 1), (e_{3}, 0.9), (e_{4}, 0.7), \\ (e_{5}, 0.8), (e_{6}, 0.7)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{3} {e_{3}} = {(e_{1}, 0.9), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), \\ (e_{5}, 0.8), (e_{6}, 0.6)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{3} {e_{4}} = {(e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), \\ (e_{5}, 0.7), (e_{6}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{3} {e_{5}} = {(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.8), (e_{4}, 0.7), \\ (e_{5}, 1), (e_{6}, 0.9)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{3} {e_{6}} = {(e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.6), (e_{4}, 0.8), \\ (e_{5}, 0.9), (e_{6}, 1)} \end{matrix}$

Therefore,

1- Calculate Γ_{a
₃}{ E₁ } and int_{Γ_a
₃}{ E₁ }

Γ_{a ₃}{ E₁ }	Γ_{a ₃}{ e₂, e₅ }
	= {(e₁, 0.9) , (e₂, 1) , (e₃, 0.9) , (e₄, 0.7) , (e₅, 0.8) , (e₆, 0.7) ,
	(e₁, 0.8) , (e₂, 0.8) , (e₃, 0.8) , (e₄, 0.7) , (e₅, 1) , (e₆, 0.9)}
int_{Γ_a ₃}{ E₁ }	int_{Γ_a ₃}{ e₂, e₅ } = co { Γ_{a ₃}co { e₂, e₅ }} = co {{ Γ_{a ₃} { e₁, e₃, e₄, e₆ }}
	$= co {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.9), (e_{4}, 0.7), (e_{5}, 0.8), (e_{6}, 0.7), \\ (e_{1}, 0.9), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), (e_{5}, 0.8), (e_{6}, 0.6), \\ (e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), (e_{5}, 0.7), (e_{6}, 0.8), \\ (e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.6), (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1) \end{matrix}}$
	$= {\begin{matrix} (e_{1}, 0), (e_{2}, 0.1), (e_{3}, 0.1), (e_{4}, 0.3), (e_{5}, 0.2), (e_{6}, 0.3), \\ (e_{1}, 0.1), (e_{2}, 0.1), (e_{3}, 0), (e_{4}, 0.2), (e_{5}, 0.2), (e_{6}, 0.4), \\ (e_{1}, 0.3), (e_{2}, 0.3), (e_{3}, 0.2), (e_{4}, 0), (e_{5}, 0.3), (e_{6}, 0.2), \\ (e_{1}, 0.3), (e_{2}, 0.3), (e_{3}, 0.4), (e_{4}, 0.2), (e_{5}, 0.1), (e_{6}, 0) \end{matrix}}$
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.3), (e_{2}, 0.3), (e_{3}, 0.4), (e_{4}, 0.3), (e_{5}, 0.3), (e_{6}, 0.4)}}{{(e_{1}, 8), (e_{2}, 0.8), (e_{3}, 0.8), (e_{4}, 0.7), (e_{5}, 0.8), (e_{6}, 0.7)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.4}{0.7} = 0.57$

2- calculate Γ_{a
₃}{ E₂ } and int_{Γ_a
₃}{ E₂ }

Γ_a₃{ E₂ }	Γ_a₃{ e₁, e₃, e₄, e₆ }
	$= {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.9), (e_{4}, 0.7), (e_{5}, 0.8), (e_{6}, 0.7), \\ (e_{1}, 0.9), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), (e_{5}, 0.8), (e_{6}, 0.6), \\ (e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), (e_{5}, 0.7), (e_{6}, 0.8), \\ (e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.6), (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1) \end{matrix}}$
int_{Γ_a₃}{ E₂ }	int_{Γ_a₃}{ e₁, e₃, e₄, e₆ } = co { Γ_a₃co { e₁, e₃, e₄, e₆ }}
	= co{ Γ_a₃ { e₂, e₅ }}.
	= co {(e₁, 0.9) , (e₂, 1) , (e₃, 0.9) , (e₄, 0.7) , (e₅, 0.8) , (e₆, 0.7) ,
	(e₁, 0.8) , (e₂, 0.8) , (e₃, 0.8) , (e₄, 0.7) , (e₅, 1) , (e₆, 0.9)} =
	= {(e₁, 0.1) , (e₂, 0) , (e₃, 0.1) , (e₄, 0.3) , (e₅, 0.2) , (e₆, 0.3) ,
	(e₁, 0.2) , (e₂, 0.2) , (e₃, 0.2) , (e₄, 0.3) , (e₅, 0) , (e₆, 0.1)}
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.2), (e_{2}, 0.2), (e_{3}, 0.2), (e_{4}, 0.3), (e_{5}, 0.2), (e_{6}, 0.3)}}{{(e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.6), (e_{4}, 0.7), (e_{5}, 0.7), (e_{6}, 0.6)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.3}{0.6} = 0.5$

5.1.2.3 Constructing weighted pretopology when removing the attribute Chemistry (Chem.) The following (Table 25), shows the correlation obtained from the Equation (7).

Table 25

Shows the correlation coefficients

Corr (x_i, x_j)	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.99257754	0.90382920	0.69066829	0.96212531	0.76276482
e₂	0.99257754	1	0.94714432	0.72316224	0.98260844	0.78700439
e₃	0.90382920	0.94714432	1	0.82854777	0.98079961	0.86105240
e₄	0.69066829	0.72316224	0.82854777	1	0.83330079	0.9938432
e₅	0.96212531	0.98260844	0.98079961	0.83330079	1	0.8798747
e₆	0.76276482	0.78700439	0.86105240	0.99384326	0.879874761	1

The following (Table 26), gives the values of correlation coefficients between objects after shortening the numbers mentioned in (Table 25).

Table 26

Gives the values of correlation between the objects

	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.9	0.9	0.6	0.9	0.7
e₂	0.9	1	0.9	0.7	0.9	0.7
e₃	0.9	0.9	1	0.8	0.9	0.8
e₄	0.6	0.7	0.8	1	0.8	0.9
e₅	0.9	0.9	0.9	0.8	1	0.8
e₆	0.7	0.7	0.8	0.9	0.8	1

Using Equation (8), we get the following components of the weighted pretopology: $\begin{matrix} {Γ_{a}}_{4} {e_{1}} = {(e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.9), (e_{4}, 0.6), \\ (e_{5}, 0.9), (e_{6}, 0.7)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{4} {e_{2}} = {(e_{1}, 0.9), (e_{2}, 1), (e_{3}, 0.9), (e_{4}, 0.7), \\ (e_{5}, 0.9), (e_{6}, 0.7)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{4} {e_{3}} = {(e_{1}, 0.9), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), \\ (e_{5}, 0.9), (e_{6}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{4} {e_{4}} = {(e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), \\ (e_{5}, 0.8), (e_{6}, 0.9)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{4} {e_{5}} = {(e_{1}, 0.9), (e_{2}, 0.9), (e_{3}, 0.9), (e_{4}, 0.8), \\ (e_{5}, 1), (e_{6}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{4} {e_{6}} = {(e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 0.9), \\ (e_{5}, 0.8), (e_{6}, 1)} \end{matrix}$

Therefore,

1- Calculate Γ_a₄{ E₁ } and int_{Γ_a
₄}{ E₁ }

Γ_a₄{ E₁ }	Γ_a₄{ e₂, e₅ }
	= {(e₁, 0.9) , (e₂, 1) , (e₃, 0.9) , (e₄, 0.7) , (e₅, 0.9) , (e₆, 0.7) ,
	(e₁, 0.9) , (e₂, 0.9) , (e₃, 0.9) , (e₄, 0.8) , (e₅, 1) , (e₆, 0.8)}
int_{Γ_a₄}{ E₁ }	int_{Γ_a₄}{ e₂, e₅ } = co { Γ_a₄co { e₂, e₅ }} = co {{ Γ_a₄ { e₁, e₃, e₄, e₆ }}
	$= co {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.9), (e_{4}, 0.6), (e_{5}, 0.9), (e_{6}, 0.7), \\ (e_{1}, 0.9), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 0.8), \\ (e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), (e_{5}, 0.8), (e_{6}, 0.9), \\ (e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 0.9), (e_{5}, 0.8), (e_{6}, 1) \end{matrix}}$
	${\begin{matrix} (e_{1}, 0), (e_{2}, 0.1), (e_{3}, 0.1), (e_{4}, 0.4), (e_{5}, 0.1), (e_{6}, 0.3), \\ (e_{1}, 0.1), (e_{2}, 0.1), (e_{3}, 0), (e_{4}, 0.2), (e_{5}, 0.1), (e_{6}, 0.2), \\ (e_{1}, 0.4), (e_{2}, 0.3), (e_{3}, 0.2), (e_{4}, 0), (e_{5}, 0.2), (e_{6}, 0.1), \\ (e_{1}, 0.3), (e_{2}, 0.3), (e_{3}, 0.2), (e_{4}, 0.1), (e_{5}, 0.2), (e_{6}, 0) \end{matrix}}$
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.4), (e_{2}, 0.3), (e_{3}, 0.2), (e_{4}, 0.4), (e_{5}, 0.2), (e_{6}, 0.3)}}{{(e_{1}, 0.9), (e_{2}, 0.9), (e_{3}, 0.9), (e_{4}, 0.7), (e_{5}, 0.9), (e_{6}, 0.8)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.4}{0.7} = 0.57$

2- calculate Γ_a₄{ E₂ } and int_{Γ_a
₄}{ E₂ }

Γ_a₄{ E₂ }	Γ_a₄{ e₁, e₃, e₄, e₆ }
	$= {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.9), (e_{4}, 0.6), (e_{5}, 0.9), (e_{6}, 0.7), \\ (e_{1}, 0.9), (e_{2}, 0.9), (e_{3}, 1), (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 0.8), \\ (e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 1), (e_{5}, 0.8), (e_{6}, 0.9), \\ (e_{1}, 0.7), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 0.9), (e_{5}, 0.8), (e_{6}, 1) \end{matrix}}$
int_{Γ_a₄}{ E₂ }	int_{Γ_a₄}{ e₁, e₃, e₄, e₆ } = co { Γ_a₄co { e₁, e₃, e₄, e₆ }}
	= co {Γ_a₄ { e₂, e₅ }}
	= co {(e₁, 0.9) , (e₂, 1) , (e₃, 0.9) , (e₄, 0.7) , (e₅, 0.9) , (e₆, 0.7) ,
	(e₁, 0.9) , (e₂, 0.9) , (e₃, 0.9) , (e₄, 0.8) , (e₅, 1) , (e₆, 0.8)}
	= {(e₁, 0.1) , (e₂, 0) , (e₃, 0.1) , (e₄, 0.3) , (e₅, 0.1) , (e₆, 0.3) ,
	(e₁, 0.1) , (e₂, 0.1) , (e₃, 0.1) , (e₄, 0.2) , (e₅, 0) , (e₆, 0.2)}
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.1), (e_{2}, 0.1), (e_{3}, 0.1), (e_{4}, 0.3), (e_{5}, 0.1), (e_{6}, 0.3)}}{{(e_{1}, 0.6), (e_{2}, 0.7), (e_{3}, 0.8), (e_{4}, 0.6), (e_{5}, 0.8), (e_{6}, 0.7)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.3}{0.6} = 0.5$

5.1.2.4 Constructing weighted pretopology when removing the attribute Computer skills (Comp.) The following (Table 27), shows the correlation obtained from the Equation (7).

Table 27

shows the correlation coefficients

Corr (x_I, x_j)	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.98647553	0.78064918	0.81159749	0.80512659	0.78956867
e₂	0.98647553	1	0.86984528	0.89635567	0.83787384	0.84416767
e₃	0.78064918	0.86984528	1	0.99087934	0.71521867	0.79239884
e₄	0.81159749	0.89635567	0.99087934	1	0.80206185	0.86737023
e₅	0.80512659	0.83787384	0.71521867	0.80206185	1	0.9891705
e₆	0.789568671	0.84416767	0.79239884	0.86737023	0.98917053	1

The following (Table 28), gives the values of correlation coefficients between objects after shortening the numbers mentioned in (Table 27).

Table 28

Gives the values of correlation between the objects

	e₁	e₂	e₃	e₄	e₅	e₆
e₁	1	0.9	0.7	0.8	0.8	0.7
e₂	0.9	1	0.8	0.8	0.8	0.8
e₃	0.7	0.8	1	0.9	0.7	0.7
e₄	0.8	0.8	0.9	1	0.8	0.8
e₅	0.8	0.8	0.7	0.8	1	0.9
e₆	0.7	0.8	0.7	0.8	0.9	1

Using Equation (8), we get the following components of the weighted pretopology: $\begin{matrix} {Γ_{a}}_{5} {e_{1}} = {(e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.7), (e_{4}, 0.8), \\ (e_{5}, 0.8), (e_{6}, 0.7)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{5} {e_{2}} = {(e_{1}, 0.9), (e_{2}, 1), (e_{3}, 0.8), (e_{4}, 0.8), \\ (e_{5}, 0.8), (e_{6}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{5} {e_{3}} = {(e_{1}, 0.7), (e_{2}, 0.8), (e_{3}, 1), (e_{4}, 0.9), \\ (e_{5}, 0.7), (e_{6}, 0.7)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{5} {e_{4}} = {(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.9), (e_{4}, 1), \\ (e_{5}, 0.8), (e_{6}, 0.8)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{5} {e_{5}} = {(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.8), \\ (e_{5}, 1), (e_{6}, 0.9)} \end{matrix}$ $\begin{matrix} {Γ_{a}}_{5} {e_{6}} = {(e_{1}, 0.7), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.8), \\ (e_{5}, 0.9), (e_{6}, 1)} \end{matrix}$

Therefore,

1- Calculate Γ_{a
₅}{ E₁ } and int_{Γ_a
₅}{ E₁ }

Γ_a₅{ E₁ }	Γ_a₅{ e₂, e₅ }
	= {(e₁, 0.9) , (e₂, 1) , (e₃, 0.8) , (e₄, 0.8) , (e₅, 0.8) , (e₆, 0.8) ,
	(e₁, 0.8) , (e₂, 0.8) , (e₃, 0.7) , (e₄, 0.8) , (e₅, 1) , (e₆, 0.9)}
int_{Γ_a ₅}{ E₁ }	int_{Γ_a₅}{ e₂, e₅ } = co { Γ_a₅co { e₂, e₅ }} = co {{ Γ_a₅ { e₁, e₃, e₄, e₆ }}
	$= co {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.7), (e_{4}, 0.8), (e_{5}, 0.8), (e_{6}, 0.7), \\ (e_{1}, 0.7), (e_{2}, 0.8), (e_{3}, 1), (e_{4}, 0.9), (e_{5}, 0.7), (e_{6}, 0.7), \\ (e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.9), (e_{4}, 1), (e_{5}, 0.8), (e_{6}, 0.8), \\ (e_{1}, 0.7), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1) \end{matrix}}$
	$= {\begin{matrix} (e_{1}, 0), (e_{2}, 0.1), (e_{3}, 0.3), (e_{4}, 0.2), (e_{5}, 0.2), (e_{6}, 0.3), \\ (e_{1}, 0.3), (e_{2}, 0.2), (e_{3}, 0), (e_{4}, 0.1), (e_{5}, 0.3), (e_{6}, 0.3), \\ (e_{1}, 0.2), (e_{2}, 0.2), (e_{3}, 0.1), (e_{4}, 0), (e_{5}, 0.2), (e_{6}, 0.2), \\ (e_{1}, 0.3), (e_{2}, 0.2), (e_{3}, 0.3), (e_{4}, 0.2), (e_{5}, 0.1), (e_{6}, 0) \end{matrix}}$
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.3), (e_{2}, 0.2), (e_{3}, 0.3), (e_{4}, 0.2), (e_{5}, 0.3), (e_{6}, 0.3)}}{{(e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.8), (e_{5}, 0.8), (e_{6}, 0.8)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.3}{0.7} = 0.428$

2- calculate Γ_a₅{ E₂ } and int_{Γ_a
₅}{ E₂ }

Γ_a₅{ E₂ }	Γ_a₅{ e₁, e₃, e₄, e₆ }
	$= {\begin{matrix} (e_{1}, 1), (e_{2}, 0.9), (e_{3}, 0.7), (e_{4}, 0.8), (e_{5}, 0.8), (e_{6}, 0.7), \\ (e_{1}, 0.7), (e_{2}, 0.8), (e_{3}, 1), (e_{4}, 0.9), (e_{5}, 0.7), (e_{6}, 0.7), \\ (e_{1}, 0.8), (e_{2}, 0.8), (e_{3}, 0.9), (e_{4}, 1), (e_{5}, 0.8), (e_{6}, 0.8), \\ (e_{1}, 0.7), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.8), (e_{5}, 0.9), (e_{6}, 1) \end{matrix}}$
int_{Γ_a₅}{ E₂ }	int_{Γ_a₅}{ e₁, e₃, e₄, e₆ } = co { Γ_a₅co { e₁, e₃, e₄, e₆ }}
	= co {Γ_a₅ { e₂, e₅ }}
	= co {(e₁, 0.9) , (e₂, 1) , (e₃, 0.8) , (e₄, 0.8) , (e₅, 0.8) , (e₆, 0.8) ,
	(e₁, 0.8) , (e₂, 0.8) , (e₃, 0.7) , (e₄, 0.8) , (e₅, 1) , (e₆, 0.9)}
	= {(e₁, 0.1) , (e₂, 0) , (e₃, 0.2) , (e₄, 0.2) , (e₅, 0.2) , (e₆, 0.2) ,
	(e₁, 0.2) , (e₂, 0.2) , (e₃, 0.3) , (e₄, 0.2) , (e₅, 0) , (e₆, 0.1)}
Calculate γ_x	$γ_{x} = \frac{{(e_{1}, 0.2), (e_{2}, 0.2), (e_{3}, 0.3), (e_{4}, 0.2), (e_{5}, 0.2), (e_{6}, 0.2)}}{{(e_{1}, 0.7), (e_{2}, 0.8), (e_{3}, 0.7), (e_{4}, 0.8), (e_{5}, 0.7), (e_{6}, 0.7)}}$
Calculate $R_{Q}$	$R_{Q} = \frac{0.3}{0.7} = 0.428$

5.2 Deducing the core attributes of the information system

To recap, the following Table (29) shows the attributes’ reduction ratio for the two partitions (E₁ and E₂).

	Γ_{a ₁}	Γ_{a ₂}	Γ_{a ₃}	Γ_{a ₄}	Γ_{a ₅}
E ₁	0.57	0.428	0.57	0.57	0.428
E ₂	0.5	0.428	0.5	0.5	0.428

From the values shown Table 29, we may conclude that: The core attributes of the information system under consideration: core ={ Mathematical, Computer Skills }.

6 Conclusion

In this paper, a new attributes’ reduction method was presented. The proposed method is based on using a weighted pretopological structure to reduce the attributes in the information system.

The new technique provides the ability to exclude the objects with a corelation coefficient less than some certain value (σ) in order to define the core attributes; compared to existing methods, the new technique does not need a preconceived decision to achieve the core attributes.

For further study, we plan to develop this technique for the purpose of supporting information systems’ decision-making; And, to investigate the new technique with more real-life applications, such as decision support systems and geographical information systems.

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Weighted pretopology and reduction of information system

Abstract

Keywords

1 Introduction

2 Prelimininaries

2.1 Definition [15]

2.2 Definition [15]

2.3 Definition [6]

2.4 Information system

3 Weighted pretopological space generating from information system

3.1 Definition

3.2 Definition

3.3 Definition

3.4 Definition

3.5 Definition

3.6 Definition

3.7 Definition

4 The proposed reduction procedure

5 Statistical application on weighted pretopological space

Table 1 Show the information system U Math. Phys. Chem. Comp. e1 83 84 77 81 e2 85 93 81 100 e3 60 89 78 71 e4 60 88 82 93 e5 85 91 87 97 e6 90 87 68 93

5.1.1 Constructing weighted pretopolgy using full attribute

Table 2 Shows the sum and the mean value of each attribute Math. Phys. Chem. Comp. e1 83 84 77 81 e2 85 93 81 100 e3 60 89 78 71 e4 60 88 82 93 e5 85 91 87 97 e6 90 87 68 93 Sum 463 532 473 535 Mean value ( μ ¯ j ) 77.16667 88.66667 78.83333 89.16667

Table 13 Shows the sum and the mean value of each attribute Phys. Chem. Comp. e1 84 77 81 e2 93 81 100 e3 89 78 71 e4 88 82 93 e5 91 87 97 e6 87 68 93 Sum 532 473 535 Mean value ( μ ¯ j ) 88.66667 78.83333 89.16667

6 Conclusion

References

Table 1
Show the information system

U Math. Phys. Chem. Comp.

e₁ 83 84 77 81

e₂ 85 93 81 100

e₃ 60 89 78 71

e₄ 60 88 82 93

e₅ 85 91 87 97

e₆ 90 87 68 93

Table 2
Shows the sum and the mean value of each attribute

Math. Phys. Chem. Comp.

e₁ 83 84 77 81

e₂ 85 93 81 100

e₃ 60 89 78 71

e₄ 60 88 82 93

e₅ 85 91 87 97

e₆ 90 87 68 93

Sum 463 532 473 535

Mean value $({\bar{μ}}_{j})$ 77.16667 88.66667 78.83333 89.16667

Table 13
Shows the sum and the mean value of each attribute

Phys. Chem. Comp.

e₁ 84 77 81

e₂ 93 81 100

e₃ 89 78 71

e₄ 88 82 93

e₅ 91 87 97

e₆ 87 68 93

Sum 532 473 535

Mean value $({\bar{μ}}_{j})$ 88.66667 78.83333 89.16667