# Attribute Relevance Score: A Novel Measure for Identifying Attribute Importance
## 4 Simulation Results
### Experiment #2: Benchmark Dataset
#### 4.2.2 Results #2.2
This stability may imply that the ARS effectively captures a fundamental relationship between \(\mathbf{v}_{2}^{(j)}\) and \(\mathbf{y}\) that remains informative despite the noise corruption. The resilience of \(\mathbf{v}_{2}^{(j)}\)'s ARS score under noise indicates that the relationship retains predictive utility, though at a much lower magnitude compared to \(\mathbf{v}_{1}^{(j)}\). For \(\mathbf{v}_{3}^{(j)}\), both versions \(\mathbf{v}_{3}^{(1)}\) and \(\mathbf{v}_{3}^{(10)}\) register low mean ARS scores (0.019, std: 0.033 and 0.018, std: 0.039, respectively), which is consistent with the weaker dependency of \(\mathbf{x}_{3}\) on \(\mathbf{y}\), particularly in the presence of external variability and more dominant dependencies. In comparison, traditional methods such as Pearson and Spearman correlations displayed higher mean values for informative variables. For instance, the Pearson correlation  
Figure 3: Performance of various dependency measures on informative variables \(\mathbf{v}_{1}^{(j)}\), \(\mathbf{v}_{2}^{(j)}\), and \(\mathbf{v}_{3}^{(j)}\) across different noise levels \((j)\). This figure illustrates the mean and standard deviation of each measure, highlighting the stability of ARS compared to other metrics. for \(\mathbf{v}_{1}^{(1)}\) was 0.781 (std: 0.028), indicating strong linear relationships. This pattern persisted across other informative variables, with measures like GMIC and MIC showing similarly elevated means (e.g., GMIC for \(\mathbf{v}_{1}^{(1)}\) was 0.595, std: 0.065), apparently reflecting substantial associations. However, it is important to note that this situation also arises for noninformative variables, as we will show in the next results. **Performance on noninformative variables**: In the case of noninformative variables \(\mathbf{v}_{4}^{(j)}\), \(\mathbf{v}_{5}^{(j)}\), and \(\mathbf{v}_{6}^{(j)}\), ARS effectively identified the lack of relevance by consistently returning scores of zero across all instances, with only occasional nonzero values attributable to random chance (see Figure 4). The ARS scores for these variables were zero in the vast majority of cases, resulting in extremely low mean values and minimal standard deviations. For example, for \(\mathbf{v}_{4}^{(1)}\), ARS returned a mean score of 0.009 with a standard deviation of 0.024, indicating that most scores were zero except for rare outliers. This demonstrates ARS's robustness in correctly identifying noise and nonrelevant attributes, as the minimal nonzero scores observed can be attributed to expected statistical fluctuations in practical datasets. However, other dependency measures exhibited higher variability and occasionally elevated scores, even for noninformative attributes, suggesting spurious associations. Metrics such as GMIC and MIC presented mean values of 0.139 (std: 0.027) and 0.240 (std: 0.035), respectively, for \(\mathbf{v}_{4}^{(1)}\), suggesting a tendency to overestimate the relevance of noisy attributes. Similarly, traditional correlation measures, such as Spearman and Pearson also showed notable variability (e.g., mean Spearman for \(\mathbf{v}_{4}^{(1)}\) was 0.088, std: 0.068), indicating their limitations in distinguishing between informative patterns and noise. Overall, the attribute relevance score (ARS) outperformed traditional and information-based metrics by providing a more stable and reliable assessment of attribute relevance across both informative and noninformative variables. While measures like Pearson, Spearman, and MIC can capture strong relationships when present, their high values--especially in noninformative scenarios--reduce their effectiveness in complex, noisy datasets. The fact that ARS returned zero scores for all noninformative variables, except for occasional outliers, underscores its robustness and practical utility in feature selection processes.