## 4 Application Proposed Method for Attribute Weighting on a Case Study
### Implementation of the Case Study  
The methodology used in the case study is depicted in Figure 1. To carry out the implementation of the proposed method and the multicriteria decision-making process, the R programming language was employed to perform the necessary calculations in all phases [40]. FactoMineR library was chosen for its capability to facilitate multivariate data analysis, particularly Principal Component Analysis [41]. In the case of the Entropy Method, operations were performed manually. Phase 1. Data collection:  
Data Collection: The data used in this study were obtained from the National Population Council (CONAPO) [42]. These data underwent a comprehensive cleaning process, which included the removal of redundant data, imputation of missing values, and standardization of the dataset. In total, the dataset comprises 1814 instances, each representing observations associated with different localities and characterized by nine specific attributes. Table 2 provides a detailed summary of each attribute. Table 3 presents an excerpt with the dataset values to facilitate a comprehensive and transparent understanding of the data used in this analysis. \begin{table}
\begin{tabular}{c c} \hline \hline
**Metric** & **Description** \\ \hline ID & Locality code \\ \(x_{1}\) & Percentage of the population aged 15 and over who are illiterate \\ \(x_{2}\) & Percentage of the population aged 15 and over without basic education \\ \(x_{3}\) & Percentage of households without access to drainage or toilet facilities \\ \(x_{4}\) & Percentage of households without access to electricity \\ \(x_{5}\) & Percentage of households without access to piped water \\ \(x_{6}\) & Percentage of households with dirt floors \\ \(x_{7}\) & Percentage of households with overcrowding \\ \(x_{8}\) & Percentage of households without a refrigerator \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of dataset attributes. \begin{table}
\begin{tabular}{c c} \hline \hline
**Metric** & **Description** \\ \hline ID & Locality code \\ \(x_{1}\) & Percentage of the population aged 15 and over who are illiterate \\ \(x_{2}\) & Percentage of the population aged 15 and over without basic education \\ \(x_{3}\) & Percentage of households without access to drainage or toilet facilities \\ \(x_{4}\) & Percentage of households without access to electricity \\ \(x_{5}\) & Percentage of households without access to piped water \\ \(x_{6}\) & Percentage of households with dirt floors \\ \(x_{7}\) & Percentage of households with overcrowding \\ \(x_{8}\) & Percentage of households without a refrigerator \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of Mexico City. Later, a data normalization technique was applied to ensure that the dataset attributes were on the same scale. The normalization technique used was z-score normalization. In this method, dataset values are transformed to have a distribution with a mean of 0 and a variance of 1. Normalization plays a crucial role due to its impact on the homogeneity and comparability of the attributes, essential factors for the subsequent attribute-weighting process. Phase 2. Attribute Weighting  
The attribute-weighting phase stands out as the most crucial stage in the multiciteria decision-making process. To assign weights to the attributes, the proposed method, based on Principal Component Analysis and the Entropy Method, was applied, as detailed in Section 3.1. The following outlines the steps followed during the application of the proposed method for attribute weighting:  
Step 1: Calculate covariance matrix  
The first step involves calculating the covariance matrix. The covariance matrix provides a detailed perspective on the relationship among the various attributes that make up the dataset. Table 4 displays the values obtained from the covariance matrix. Step 2: Calculate eigenvalue vector  
From the covariance matrix, the vector of eigenvalues associated with the matrix was obtained.