## 3 Proposed Method for Attribute Weighting
### The Description of Principal Component Analysis and the Entropy Method
#### 3.2.1 Principal Component Analysis  
Principal Component Analysis (PCA) is a well-known non-parametric technique in unsupervised learning [20]. This statistical technique has become a commonly used approach to address dimensionality reduction in large datasets. The primary purpose of PCA lies in transforming a multivariate dataset, characterized by n attributes, into a new dataset that preserves the essence, the same dimensionality, and the key information of the original set [21]. The PCA process involves a linear transformation of the data, specifically an orthogonal transformation of variable vectors, which may be correlated, into a set of linearly uncorrelated variables [22]. During this transformation, eigenvectors, known as principal components, and their corresponding eigenvalues are obtained. It is crucial to note that the first principal component exhibits the maximum variability present in the original data, while successive components capture the remaining variability in descending order [23]. Eigenvalues are usually presented in descending order due to the inherent nature of the decomposition process; the main purpose is to capture as much of the total variability present in the original dataset as possible. The explanation of total variance is closely linked to the eigenvalues derived from the covariance matrix of the dataset or from the correlation matrix. The choice between the covariance matrix and the correlation matrix depends on the objectives and characteristics of the data. These eigenvectors form a one-dimensional orthogonal basis. As a result, principal components are expressed through a set of indices summarized using the linear combination of coefficients and values of variables present in the dataset [24; 25]. Given its ability to reduce complexity and preserve essential information, PCA has established itself as a crucial tool in complex data analysis and has found applications in a wide range of disciplines. The following outlines the PCA calculation process, which occurs in several stages. Step 1. Calculation of the Covariance Matrix  
Consider the dataset X with n variables and m rows. From the provided data, the covariance matrix is calculated. The covariance matrix C is a symmetric \(n\times n\) matrix that displays the covariance between every pair of variables [26; 27]. Refer to Equation (1). \[Cov\big{(}X_{i},Y_{j}\big{)}=\frac{\sum_{k=1}^{n}\big{(}X_{ik}-\overline{X}_{i} \big{)}\big{(}Y_{jk}-\overline{Y}_{j}\big{)}}{n-1}, \tag{1}\]  
where \(\overline{X}_{i},\overline{Y}_{j}\) are the means of \(X_{ij}Y_{j}\), respectively. Step 2. Decomposition of the Covariance Matrix. The next step involves decomposing the covariance matrix in terms of the associated eigenvalues and eigenvectors. Eigenvalues represent the amount of explained variance for each component, while eigenvectors define the direction in which there is greater variability in the data. The computation of eigenvalues and eigenvectors is performed using the characteristic equation in Equation (2) [28; 29]. \[Cv=\lambda v, \tag{2}\]  
where \(v\) is an eigenvector, and \(\lambda\) is the corresponding eigenvalue. The eigenvalues are determined by solving the characteristic polynomial with Equation (3). \[det(C-\lambda I)=0, \tag{3}\]  
where \(I\) is the identity matrix, and \(\lambda\) is a real value. The eigenvectors corresponding to each eigenvalue are determined by solving the system of Equation (4). \[(C-\lambda I)v=0, \tag{4}\]  
These eigenvectors, or principal components, form a matrix \(V\) with dimensions \(n\times n\). The eigenvector matrix consists of linearly independent vectors, each with a magnitude of one and a right angle between them. Each vector is associated with its eigenvalue. Step 3. Eigenvalue Sorting  
As the matrix C is a positive definite matrix, it enables the calculation of the total explained variance of the data based on its eigenvalues.