# Attribute Relevance Score: A Novel Measure for Identifying Attribute Importance
## 4 Simulation Results  
### Experiment #1: Computer-Generated Data  
This study utilizes a synthetically generated dataset to evaluate the performance of the proposed methods under various types of bivariate relationships, including both linear and nonlinear dependencies. The data were created using multiple transformations and multivariate normal distributions to simulate distinct scenarios that reflect complexities commonly observed in real-world datasets. The goal is to assess the robustness and effectiveness of the proposed attribute relevance score (ARS) (using decision trees as the base model), and other established dependency measures across diverse data configurations. * **Multivariate normal distributions**: Seven datasets, each containing 1000 samples, were generated using multivariate normal distributions with correlation coefficients ranging from \(-1.0\) to \(1.0\). These datasets represent different degrees of linear relationships between variables, labeled from (A) to (G) in Figure 1.
* **Rotated normal distributions**: To examine the effect of rotational transformations on normally distributed data, seven datasets of 1000 samples each were created by applying specific rotation angles (ranging from 0 to \(\pi/2\)) to a bivariate normal distribution with equal variance and high covariance. These are labeled from (H) to (N) in Figure 1. A special case, labeled (K) with unique horizontal distribution,was included to demonstrate a scenario with zero correlation. These transformations model linear relationships oriented in diverse directions. * **Other nonlinear and complex distributions**: This category includes seven distinct datasets representing various nonlinear, asymmetric, and complex dependencies. They are labeled from (O) to (U) in Figure 1. Key examples are the following:
* (O): Displays a quadratic dependency with added noise. * (P) and (Q): Represent sequential rotations applied to uniformly distributed data. * (R) and (S): Exhibit parabolic and partially symmetric patterns. * (T): Simulates a trigonometric relationship combining sinusoidal and cosinusoidal transformations. * (U): Comprises a clustered structure formed by four multivariate normal distributions centered at distinct locations, resembling spatially separated clusters. To ensure comparability across subsets, all data were scaled using MinMax scaling within the range of \(-5\) to \(5\), preserving the inherent structural relationships while normalizing their ranges, as illustrated in Figure 1.  
Results #1  
Table 1 presents the performance of various dependency measures applied to the synthetically generated datasets described in the previous section. Each dataset was designed to capture different types of bivariate relationships, including both linear and nonlinear dependencies, to assess the robustness and effectiveness of the proposed attribute relevance score (ARS) and other correlation measures. The results demonstrate that for datasets based on **multivariate normal distributions (A-G)**, the ARS and other linear measures such as Pearson and Spearman coefficients perform as expected when the relationship between variables is linear. In particular, datasets (A) and (G), which exhibit perfect linear correlation, yield maximum values (1.000) for most measures, including Pearson, Spearman, Kendall, and \(\textit{Ph}_{k}\), with corresponding high ARS scores of 0.996. Similarly, datasets with moderate to low correlation levels (e.g., (B) and (F)) show a consistent decline in these measures, with ARS scores reflecting these variations (0.429 and 0.394, respectively). Figure 1: Scatter plots illustrating various bivariate relationships generated. Each subplot represents a different synthetic dataset: (**A–G**) are datasets generated from multivariate normal distributions with varying correlations; (**H–N**) are datasets generated from rotated normal distributions, illustrating different linear relationships; (**O–U**) represent other complex, nonlinear patterns.