# Attribute Relevance Score: A Novel Measure for Identifying Attribute Importance
## 4 Simulation Results
### Experiment #1: Computer-Generated Data
For the **rotated normal distributions (H-N)**, the ARS effectively captures the directional dependencies across different orientations. Datasets (H) to (J) and (L) to (N) maintain high scores across all measures (1.000), including ARS values around 0.980, indicating the robustness of the ARS in identifying dependencies even when the data are rotated. Notably, dataset (K), which presents no linear or clear directional correlation, shows an ARS value of 0.000, while other measures still return nonzero values. This highlights the ARS's unique sensitivity to the absence of any meaningful relationship, effectively distinguishing it from traditional dependency measures. The results for the **other nonlinear and complex distributions (O-U)** highlight the limitations of traditional linear measures in capturing intricate dependencies. For instance, in dataset (O), which exhibits a quadratic dependency, the ARS score (0.477) is notably higher than those of other measures, such as Pearson (0.052) and Spearman (0.046), demonstrating ARS's ability to detect nonlinear patterns. Similarly, datasets (R) and (T), featuring parabolic and trigonometric relationships, yield ARS values (0.297 and 0.215, respectively) that surpass those of the linear measures, suggesting that ARS offers a more refined assessment of these complex dependencies. Notably, datasets (D) and (K), which do not exhibit any significant linear or nonlinear relationships, have ARS scores of 0.000, whereas other measures still return nonzero values. This distinction underscores the ARS's unique sensitivity to the absence of meaningful relationships, effectively setting it apart from traditional dependency measures. However, for dataset (U), which represents a clustered structure formed by four multivariate normal distributions centered at distinct locations, it may be arguable whether any of the measures presented, including ARS, effectively capture the inherent clustering pattern. In this case, all measures, including ARS, tend to produce the lowest values observed, indicating the potential limitations of these metrics in identifying complex clustering structures. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Set** & \(r_{xy}\) & \(\rho\) & \(\tau\) & \(\phi_{k}\) & MI & MIC & MAS & MEV & GMIC & ARS Proposed \\ \hline (A) & 1.000 & 1.000 & 1.000 & 1.000 & 5.984 & 1.000 & 0.000 & 1.000 & 1.000 & 0.996 \\ (B) & 0.805 & 0.783 & 0.587 & 0.859 & 0.575 & 0.519 & 0.030 & 0.519 & 0.461 & 0.429 \\ (C) & 0.378 & 0.378 & 0.258 & 0.373 & 0.081 & 0.200 & 0.017 & 0.200 & 0.148 & 0.049 \\ (D) & 0.009 & 0.016 & 0.010 & 0.157 & 0.007 & 0.126 & 0.007 & 0.126 & 0.054 & 0.000 \\ (E) & 0.394 & 0.383 & 0.261 & 0.407 & 0.054 & 0.215 & 0.015 & 0.215 & 0.158 & 0.089 \\ (F) & 0.809 & 0.800 & 0.606 & 0.817 & 0.508 & 0.512 & 0.015 & 0.512 & 0.465 & 0.394 \\ (G) & 1.000 & 1.000 & 1.000 & 1.000 & 5.984 & 1.000 & 0.000 & 1.000 & 1.000 & 0.996 \\ (H) & 1.000 & 1.000 & 1.000 & 1.000 & 4.373 & 1.000 & 0.000 & 1.000 & 1.000 & 0.980 \\ (I) & 1.000 & 1.000 & 1.000 & 1.000 & 4.373 & 1.000 & 0.000 & 1.000 & 1.000 & 0.980 \\ (J) & 1.000 & 1.000 & 1.000 & 1.000 & 4.373 & 1.000 & 0.000 & 1.000 & 1.00 Overall, the results indicate that ARS is a robust metric for capturing both linear and nonlinear dependencies across diverse types of relationships. In comparison to traditional measures such as Pearson, Spearman, and Kendall--which are effective at identifying monotonic dependencies but limited with more complex nonlinear relationships--the ARS consistently outperformed these classical methods, particularly on datasets (O) to (U). By using ARS alongside conventional metrics, a more comprehensive framework is established for evaluating bivariate dependencies, fulfilling the objectives of this study and providing valuable insights across a broad range of real-world scenarios.