Pawlak proposed Rough set theory (RST) in 1982 [1]. RST is considered as a powerful mathematical research technique in pattern recognition, machine learning, information discovery, and other fields. Because Pawlak’s RST requires rigid equivalence relations, it can only extract expertise in information systems with definite attributes [2]. Some researchers have extended Pawlak’s idea by incorporating fuzzy equivalence relations, neighborhood relations and dominance relations into Pawlak rough sets to form neighborhood rough sets [3,4], fuzzy rough sets [5–9], and dominance-based rough sets [10–12]. The generalized models of rough set are commonly applied in the reduction of attributes [13–15], feature selection [16–19], extraction of rules [20–23], theory of decisions [24–26], incremental learning [27–29], Collaborative Filtering [30], Variable Precision Rough Set Model, [31], topological rough application [32], reduction in multi-valued information system [33] and so on. A significant application of RST is in the reduction of attributes in databases (feature selection). Given a dataset with distinct attribute values, the reduction of attributes finds subsets having the same attributes as the original. Its principle can be regarded as the most powerful result of RST, differentiating it from other theories. The reduction of attributes for selecting subsets attributes picks detailed and compact attributes and excludes redundant and inconsistent attributes from the learning tasks. Several algorithms for attribute reduction exist based on classical rough sets, rough neighborhood sets, entropy, and mutual knowledge. This study introduces the reduction of attributes using classical RST. Additionally, we introduce the strength of rules and similarity matrix that are also used in reducing attributes. Finally, we discuss many examples to explain these methods. This article is organized as the following: Section 2 provides a quick overview of the fundamental notion of the theory of rough sets and describes the reduction of attributes following the indiscernibility relation and discernibility matrix. Section 3 presents the definition of the strength of rules and some measures for evaluating attributes and discusses their properties. In Section 4, we introduce the definition of similarity matrix for finding the reduct of a decision information table. Finally, Section 5 provides the conclusion.

Fig. 1 presents four methods of reduction of attributes in information systems, which will be discussed in the following sections.

The rules are now generated based on the reduct and core. The reduced set of relations that maintains the same inductive relation categorization is known as a reduct. If P is minimum and the indiscernibility relation provided by P and Q is the same, the set P of attributes is the reduct of another set Q. The core is described as follows:

Core = ∩ reduct

Definition 3. The strength of rules

Let IS=(U,C,D) be a decision table. Every x∈U determines a sequence c₁ (x), …,c_n (x), d₁(x)……d_m((x), where {c1,...,cn}=C and {d1,...,dn}=D . The sequence called the decision rule induced by x in U and denoted by

c1(x),...,cn(x)→d1(x),...,dm(x) or C→xD.

Then the strength of rules is

σ∝,β(Ci,Dj)=|Ci(x)∝∩Dj(x)β||Dj(x)β|,

where Ci(x)∝=∪{x∈U|Ci(x)=α} , Dj(x)β=∪{x∈U|Dj(x)=β} , α∈valuesofCi , β∈valuesofDj , i=1,2,3,…n is the index of condition attributes, and j =1,2,3,…,m is the index of decision attribute.

Example 3.

From Tab. 2, and using Definition 3, we can calculate the strength of rules for all attributes as the following steps:

The rules strength for attribute C may be found as follows:

(C = 5) → (D = Low); the strength of this particular rule = 23 = 66.667%

(C = 4) → (D = Low); the strength of this particular rule = 13 = 33.333%

(C = 5) → (D = Medium); the strength of this particular rule = 12 = 50%

(C = 2) → (D = Medium); the strength of this particular rule = 12 = 50%

(C = 4) → (D = High); the strength of this particular rule = 23 = 66.667%

(C = 2) → (D = High); the strength of this particular rule = 13 = 33.333%

The average of the strength of rules for attribute C = 50%

The rules strength for attribute I may be found as follows:

(I = Fair) → (D = Low); the strength of this particular rule = 23 = 66.667%

(I = Good) → (D = Low); the strength of this particular rule = 13 = 33.333%

(I = Fair) → (D = Medium); the strength of this particular rule = 22 = 100%

(I = Excellent) → (D = High); the strength of this particular rule = 13 = 33.333%

(I = Good) → (D = High); the strength of this particular rule = 23 = 66.667%

The average of the strength of rules for attribute I = 60 %

The rules strength for attribute N may be found as follows:

(N = Medium) → (D = Low); the strength of this particular rule = 33 = 100%

(N = Low) → (D = Medium); the strength of this particular rule = 12 = 50%

(N = Medium) → (D = Medium); the strength of this particular rule = 12 = 50%

(N = Low) → (D = High); the strength of this particular rule = 33 = 100%

The average of the strength of rules for attribute N = 75%

The rules strength for attribute V may be found as follows:

(V = Medium) → (D = Low); the strength of this particular rule = 33 = 100%

(V = Low) → (D = Medium); the strength of this particular rule = 22 = 100%

(V = Medium) → (D = High); the strength of this particular rule= 13 = 33.333%

(V = Low) → (D = High); the strength of this particular rule = 23 = 66.667%

The average of the strength of rules for attribute V = 75%

Note: The attributes with the highest percentage of the strength of rules are N and V, and attributes with fewer proportions can be deleted. The reduct of set { C, I, N, V } is { N, V } . Therefore, Tab. 2 can be reduced to Tab. 8.

Reduce Tab. 8: to become as Tab. 9 by removing the repeated rows of attribute values.

We find the core of Tab. 9; to keep the table consistent. Two decision values L and H remain if we eliminate V = M. This implies that we cannot make a sole judgment based on attribute V, so the value of V cannot be removed. Similarly, two decision values M and H remain when we eliminate N = L, implying that we cannot make a unique judgment based on attribute N. Therefore, the value of N cannot be removed. Now Tab. 9 is like Tab. 10.

Tab. 10: Displays each instance’s core. Tab. 10 can be further reduced; by merging double rows.

By removing the same rows again, we obtain Tab. 11.

Tab. 11 provides us with rules of judgment. No further reduction is necessary. The decisions on reduction and core are as follows.

1) If N(Medium) and V(Medium) ⇒ D(Low)

2) If V(Low) ⇒ D(Medium)

3) If N(Low) ⇒ D(High)

The similarity matrix is considered a novel method reducing condition attributes. It is easy to use and produces more accurate results, depending on the deletion or dispensing of the attribute that has the least influence on decision making under specific conditions.

Definition 4. Let IS=(U,C,D) be a decision table; then the distance d(xi,xj) between two objects xi,xj∈U according to one attribute a∈A can be calculated using the following equation:

d(xi,xj)={0,xi(a)≠xj(a),a∈A1,xi(a)=xj(a),a∈A

The ratio of the similarity between two objects denoted by δ(xi,xj) is given by the following equation:

δ(xi,xj)=∑d(xi,xj)|A|

Example 4.

From Tab. 2, and Definition 4, we obtain the similarity matrix for all objects as in Tab. 12:

From Tab. 2; we can find the indiscernibility relation of the decision attribute as follows:

U/IND(D) = {{c1,c2,c3},{c4,c5},{c6,c7,c8}}

From Tab. 12, selecting the ratio of similarity between two objects when δ(xi,xj)=1 according to the following equation: f(xi)={xj:δ(xi,xj)=1} ,

we get;

f(c1)={ c 1} , f(c2)={ c 2} , f(c3)={ c 3} , f(c4)={ c 4} , f(c5)={ c 5} , f(c6)={ c 6} , f(c7)={ c 7} , f(c8)={ c 8} ,

→ LOW (U(A)/D) = { c 1 ,c 2 ,c 3 ,c 4 ,c 5 ,c 6 ,c 7 ,c 8}

Then the degree of dependency is

μ(U(A)/D) ) = 88=1

By eliminating attribute C from Tab. 2, we obtain the following similarity matrix in Tab. 13.

From Tab. 13, selecting the ratio of similarity between two objects in the same manner as above, we get;

f(c1)={ c 1 ,c 2},f(c2)={ c 1 ,c 2},f(c3)={ c3},f(c4)={c4} , f(c5)={ c 5},f(c6)={ c 6},f(c7)= {c7 ,c 8},f(c8)={c7} , {c8}

→ LOW (U(A-{C})/D = { c 1 ,c 2 ,c 3 ,c 4 ,c 5 ,c 6 ,c 7 ,c 8}

Then The degree of dependency is

μ(U(A−{C})/D) = 88=1

Also, by eliminating attributes {I}, {N}, {V, {C, N}, {I, V}, {C, I}, {C, V}, {I, N}, {N, V} from Tab. 2 respectively, and computing its similarity matrices, we get the following degree of dependencies:

μ(U(A)/D) = 88 = 88=1

μ(U(A−{C})/D) = 88 = 88=1

μ(U(A−{I})/D) = 88 = 88=1

μ(U(A−{N})/D) = 88 = 88=1

μ(U(A−{V})/D) = 88 = 88=1

μ(U(A−{C,N})/D) = 88 = 88=1

μ(U(A−{I,V})/D) = 88=1

μ(U(A−{I,C})/D) = 58=0.625

μ(U(A−{V,C})/D) = 58=0.625

μ(U(A−{I,N})/D) = 48=0.5

μ(U(A−{N,V})/D) = 68=0.75

From previous calculations, we have;

μ(U(A)/D) =1

μ(U(A−{N,C})/D) =1

μ(U(A−{I,V})/D) =1

Therefore, the reduct(A) = {{ C,N } , { I,V }} .

We emphasised in our research the need of decreasing the size of the dataset before beginning any research and how rough set theory provides an effective approach for determining the minimal dataset’s reduct. We also discussed how the rough set may be unable to discover the minimal reduct by itself since doing so may need computing all combinations of attributes, which is not achievable in huge datasets. We proposed two methods to find the reduct of a dataset. One of them is the strength of rules which calculate the strength of rules for all attributes, and the other is similarity matrix, which is considered a novel method reducing condition attributes, it is easy to use and produce more accurate results.