An Innovative Method for Attribute Reduction: Weighted Attribute Concepts for Probabilistic Analysis of Decision Attributes

The concept of an indiscernibility relation, along with features of discernibility and indiscernibility matrices, was first introduced by Pawlak [1]. Yao [2] advanced this concept, constructing three regions - positive, negative, and boundary - to illustrate the advantages of the three-way decision rule in the rough set model and decision theory. The relationship between the error level of tolerance and incorrect decision cost was also elucidated.

Wu et al. [3] explored the fundamental notions of formal concept analysis, employing granular concepts. The object-oriented concept lattice's properties were examined, and an algorithm was proposed to probe the concept of lattice and its properties. Li et al. [4] established a multi-granular decision rule for the decomposition of the concept lattice, leading to the generation of a new algorithm for disjoint normal functions.

Ezhilarasi et al. [5] advanced the classification of rough set theory for uncertain datasets using the weighted attribute concept. This concept facilitated analysis of a decision rule's subset, and the results were accentuated by prediction accuracy for the datasets. Kang et al. [6] proposed employing an algebraic structure and lattice structure to investigate knowledge of decision dependency, and the potential applications of decision dependency were discussed in a technology that merges rough set theory and formal concept analysis.

The formal concept of reliability engineering problems was determined by Rocco et al. [7], who offered fundamental views on formal concepts pertaining to the field of reliability. Qian et al. [8] extended formal concepts to three-way decisions, and the characteristics of formal context were investigated using the isomorphism relationship between type I and type II dual interactable.

Huang and Bian [9] applied a formal concept analysis to online tour planning, while Yao and Chen [10] introduced the formal concept analysis for two definable sets and defined an approximation operator. They also examined the interpretation of lattice theory and set theory. Srirekha et al. [11] investigated the projection of a distributive lattice by defining the trivial ordered set and its properties. The lattice homomorphism was analyzed based on an equivalence relation, and its condition was verified.

Jiang et al. [12] analyzed the relationship between generalized weighted averages and its properties using hesitant fuzzy, while Xiao and He [13] discussed the properties of the formal concept lattice. They also calculated the weight for each attribute using the degree of importance in rough set theory and entropy information. Shi and Chen [14] proposed a new method on rough set theory and granular theory, characterizing the effectiveness of weight. Li et al. [15] experimented with a combination of fundamental geographic information data and proposed numerous applications for geo-ontologies merging. Bao et al. [16] generalized a new method on attribute weight for a significant individual degree of each conditional attribute.

Weighted attribute concepts have been extensively applied to attribute reduction. The positive region and degree of individual attributes have been investigated, along with their applications in diverse fields. Combining entropy information with these concepts, a formal concept lattice was defined. In this paper, an algebraic structure for the weighted attribute concept is defined, and attribute reduction by core and non-core elements is investigated. Decision attributes are utilized to estimate the probability of the core, demonstrating the effectiveness of weighted attribute concepts. Optimal reduction features were determined using weighted attribute concepts.

The remainder of this paper is organized as follows: Section 2 discusses the basic knowledge about the formal context that generates the granular. Section 3 provides the attribute reduction obtained from the indiscernibility matrix. In Section 4, the weighted attribute concept is introduced with the knowledge of core and non-core elements. Section 5 demonstrates the significance of the proposed work through a real-time application with the implementation of weighted attribute concepts.

Here, we examine the values V_a={0, 1}, from an information table that describes an indiscernibility relation using these values. Thus, the granular has been represented in a hasse diagram and an indiscernibility matrix is investigated to obtain the reduct.

3.1 Representation of indiscernibility table

Definition 3.1: Consider an information system (O, At, V, R) where the non-empty finite set of Object $G \subseteq O$, the attribute $M \subseteq A t$, V is the value of the cartesian product of objects and attributes, and $R \subseteq O \times A t$

$V_a=\left\{\begin{array}{l}1 ; \text { if } g R m \in V \\ 0 ; \text { if } g R m \notin V\end{array}\right.$

where $g \in G$ and $m \in M$.

Definition 3.2: Let us consider $\left(O, A t, V_a=\{0,1\}, f, R\right)$ to be the formal context, where O and At be the non-empty set of objects, and attributes respectively, V_a is the value, $f: O \times A t \rightarrow V_a$ is a function and R be the binary relation such that $R \subseteq O \times A t$.

Definition 3.3: Let (O, At, V_a, R) be a formal context then the value of the indiscernibility relation can be defined as:

$R_{(i, j)}=\left\{(1,1) \in O \times A t \mid \forall 1 \in V_a,(i, j) \in(O \times A t)\right\}$

where, throughout this paper, we represent (i, j) as {ij}.

Definition 3.4: Let (O, At, V_a, R) be a formal context, consider $G \subseteq O$ and $M \subseteq A t$ then the Cartesian product of object and attribute is defined as:

$R_{i j}=\left\{(i, j) \in G \times M \mid \forall m \in M, R_m(i)=R_m(j)\right\}$

Definition 3.5: Let (G, M, V_a={1,0}, R) be the formal context, where G is a set of object {g₁, g₂, .., g_n} and M is a set of attribute {m₁, m₂, ... m_k} then V_a is the value which defined as:

$V_{i j}=\left\{\begin{array}{l}1 ; \text { if } g_i \in m_j \\ 0 ; \text { if } g_i \notin m_j\end{array}\right.$

3.2 Indiscernibility matrix

Definition 3.6: The indiscernibility matrix is defined as $I_{i j}=\left\{v_{i j} \in G \times M \mid R_{i j}=R_{k l}\right\}$. If the matrix satisfies the symmetric condition I_ij=I_ji then take for consideration as I_ii=M, otherwise (i, j) is a pair of discernible values and I_ij≠∅.

$I D S^{\text {Fun }}=\wedge\left\{\vee R_{i j} \mid \forall i, j \in O, R_{i j} \neq \emptyset\right\}$

Example 3.1: Consider an information table with the 5 object along with the attribute price, Room, Furniture, also with decision attributes Prestigious flat consisting of "Yes" and "No".

Table 1. Information table

Table 2. Indiscernibility table

From Tables 1 and 2, objects 1 and 3 are considered as inconsistent data. Hence, an indiscernibility matrix has been framed from Figure 1 for the existence of an indiscernibility function (from Figure 2) to obtain the reduct as:

$\begin{gathered}I D S^{Flat}=\{M\} \wedge\{a\} \wedge\{a \vee c \vee g\} \wedge\{a \vee g\} \wedge\{f\} \wedge \{d\}=\{a \wedge f\} \vee\{a \wedge d\} .\end{gathered}$.

1.png

Figure 1. Hasse diagram from Table 2

2.png

Figure 2. Indiscernibility matrix

Thus, the two reduct are $T_1=\{a \wedge f\}$ and $T_2=\{a \wedge d\}$ can be represented as hasse diagram (Figure 3).

3.png

Figure 3. Reduct

This section investigates statistical data that describes breast cancer. These data have been classified from the image of the mammograms which has been identified from the test report of the patient survey who undergone the process of a breast cancer test. The weighted attribute concept helps us to determine the optimal reduct and easy to analyze the knowledge of the information table.

Work Rule:

Step 1: Consider an information table with condition and decision attributes.

Step 2: Create an indiscernibility relation table from the information table.

Step 3: Generate a granular hasse diagram from the table.

Step 4: From the granular, define the indiscernibility matrix and find its reduct.

Step 5: Determine the core and non core element.

Step 6: Form the table using entropy value and core values with probability value of decision attributes.

Step 7: Identify the non core element weight and analyse the weighted attribute concepts for study.

Example 5.1: Table 5 shows a mammogram analysis of the effect of detecting breast cancer using machine learning technology. The set of the object obtained from the 5 patient details which is consist of mammogram image {1, 2, 3, 4, 5} and the set of attributes obtained from the extraction of the feature of breast cancer that are collected from the knowledge of mean, standard deviation, smoothness, third moment, uniformity and entropy. From these details, the attribute was defined as {Contrast, IDM, Dissimilarity}. The contract is the measure of the image inertia or the local variations present in images as "a" (9.1-10.9), "b" (11.1-12.9); Inverse Difference Moment (IDM)-Homogentiy is the large values of cases which describes the largest element of principal diagonal as "c" (3.1-4), "d" (4.1-5); Dissimilarity is a directional moment that measures the image contrast that increases the linearly not exponentially as "e" (2.4-2.5),"f" (2.6-2.7), "g" (2.8-2.9); The analyzation of imperfect data can be classified by a new mathematical approach on Rough Set Theory. The Decision data is the specific decision rule of the action, result, and outcome.

Table 5. Information table

Table 6. Indiscernibility table

From Table 6, the structure of the formal concept lattice was defined using the data obtained from Table 5. The hasse diagram and indiscernibility matrix constructed in Figures 4 and 5. Therefore the indiscernibility function can be obtain attribute reduction without the loss of information the reduct can be constructed based on join and meet irreducible context.

4.png

Figure 4. Hasse diagram from Table 6

5.png

Figure 5. Indiscernibility matrix

$\Rightarrow I D S^{B C}=\{M\} \wedge\{a\} \wedge\{a \vee d \vee e\} \wedge\{a \vee f\} \wedge\{c\} \wedge\{a \vee d\} \Rightarrow\{a \wedge c \wedge d\} \vee\{a \wedge c \wedge f\}$

From the above IDS^BC, the two-reduction decision are $T_1=\{\mathrm{a} \wedge \mathrm{c} \wedge \mathrm{d}\}$ and $T_2=\{\mathrm{a} \wedge \mathrm{c} \wedge \mathrm{f}\}$.

6.png

Figure 6. Reduct of breast cancer info

Finally, the reduction of the attribute was examined using the probabilistic value of the core. Hence the weighted attribute concept has been analysed by Hasse diagram (Figure 6). Thus, the optimal weight of each element in the core and noncore attributes are defined in the Tables 7 and 8.

Table 7. Weight for reduct T₁={a, c, d}

Table 8. Weight for reduct T₂={a, c, f}

From the above example, the optimistic reduction obtained using the weighted attribute concept that are analyzed from the non-core elements. Thus, we investigate the information table and determine the minimal attribute of breast cancer that classifies the mammogram images.

From the above systematic study, the weighted attribute concept was analyzed from the attribute reduction and consistency set. The indiscernibility matrix was formed from the granular ordered pair of objects and attributes. The reduct was examined through the probability of weighted attribute concept of core and non-core elements. The weight of each attribute was characterized by the probability values and defined the significance of the tuber affected in the mammogram images. Hence from the above example the attribute contract ranges from a={9.1-10.9} and IDM range from c={3.1-4} are the essential attribute. In addition, we analyze the attribute IDM that ranges from d={4.1-5} and dissimilarity that ranges from f={2.6-2.7} having a clear knowledge about the weight that emphasizes with the decision attributes. Thus, the weight of noncore element d is lesser than the f. Therefore, we come to the conclusion that reduct {a, c, f} gives an optimal result than {a, c, d}.