# Attribute Relevance Score: A Novel Measure for Identifying Attribute Importance
## 4 Simulation Results
### Experiment #2: Benchmark Dataset
#### 4.2.2 Results #2.2  
To increase the dataset's complexity, additional noninformative predictor variables \(\mathbf{x}_{4}\), \(\mathbf{x}_{5}\), and \(\mathbf{x}_{6}\) are generated uniformly within the range 0 to 1 (\(U(0,1)\)). These variables do not contribute directly to \(\mathbf{y}\), simulating the presence of noise and further challenging the model's ability to differentiate between relevant and irrelevant features. To further enrich the dataset, we simulates the presence of progressively weaker signal-to-noise ratios, allowing the method to be tested on its ability to identify relevant variables in the presence of noise. To achieve this, we generate noise versions of the predictor variables using the following equation:  
\[\mathbf{v}_{i}^{(j)}=\mathbf{x}_{i}+\left(0.01+\frac{0.5(j-1)}{10-1}\right)N(0 ;0.3), \tag{16}\]  
for \(j=1,\ldots,10\) the level of noise, where the correlation between \(\mathbf{x}_{i}\) and \(\mathbf{v}_{i}^{(j)}\) systematically decreases as \(j\) increases. \(i=1,\ldots,6\) corresponds to the index values of the original \(\mathbf{x}_{i}\). This formula generates multiple noisy versions of the base variables \(\mathbf{x}_{i}\). This experimental setup enables a robust comparison of our method against alternative techniques by specifically evaluating their capacity to distinguish between relevant and nonrelevant features, even in the presence of noise. By conducting 100 repetitions with 100 observations each, we aim to  
Figure 2: Performance of ARS with different base models on Equation (15) without the noise term \(\eta\) on the benchmark dataset. The informative variables (\(\mathbf{x}_{1}\), \(\mathbf{x}_{2}\), and \(\mathbf{x}_{3}\)) and noninformative variables (\(\mathbf{x}_{4}\), \(\mathbf{x}_{5}\), and \(\mathbf{x}_{6}\)) are evaluated across three base models: decision trees, linear regression, and k-nearest neighbors. thoroughly assess the robustness, accuracy, and adaptability of the methods when faced with scenarios that include a mixture of informative and noninformative variables. The evaluation of our proposed attribute relevance score (ARS) method, using decision trees as the base model, conducted alongside established dependency measures, was performed using the benchmark dataset described above. This dataset provides a challenging environment, containing both informative and noninformative variables with varying levels of noise, designed to test the robustness and discriminative ability of each method. **Performance on informative variables**: For the variable \(\mathbf{v}_{1}^{(j)}\), both versions \(\mathbf{v}_{1}^{(1)}\) and \(\mathbf{v}_{1}^{(10)}\) exhibit high mean ARS scores (0.435 and 0.264, respectively) compared to the other predictors, indicating strong relevance (see Figure 3). This aligns with the role of \(\mathbf{x}_{1}\) as the primary driver of variability in \(\mathbf{y}\) due to its exponential dependency. Furthermore, the low standard deviation of the ARS across both versions highlights the stability and robustness of this measure in consistently capturing the predictive power of \(\mathbf{v}_{1}^{(j)}\). In the case of \(\mathbf{v}_{2}^{(j)}\), represented by \(\mathbf{v}_{2}^{(1)}\) and \(\mathbf{v}_{2}^{(10)}\), the mean ARS scores indicate relevance (0.087, std: 0.063 and 0.069, std: 0.064, respectively); however, these scores are notably lower than those obtained for \(\mathbf{v}_{1}^{(j)}\) and other methods. This discrepancy is expected, considering the second objective of our study, which assesses the effectiveness of approximating \(\mathbf{y}\) from \(\mathbf{x}\) using a specific error measure and algorithm. ARS quantifies this practical capability, capturing not only the statistical dependence but also the magnitude of predictive power within the context of the selected model and performance metric. In this context, despite the added structured noise in \(\mathbf{v}_{2}^{(j)}\), the mean ARS remains relatively stable, suggesting that the relevance of \(\mathbf{v}_{2}^{(j)}\) is not significantly diminished by noise alone.