## Histograms: Reward Distribution Comparison (RAP-MCTS vs SC-MCTS*) ### Overview The image contains two side-by-side histograms comparing reward distributions. The left histogram represents the baseline RAP-MCTS system, while the right histogram shows the SC-MCTS* system. Both use frequency counts on the y-axis, with distinct x-axis ranges and color-coded distributions. ### Components/Axes **Left Histogram (RAP-MCTS):** - **X-axis**: Reward values ranging from -640 to -560 (intervals of ~20) - **Y-axis**: Frequency (0 to 2000) - **Legend**: - Spearman: 0.01 - Pearson: 0.01 - P-value: 0.2624 - **Color**: Uniform blue bars **Right Histogram (SC-MCTS*):** - **X-axis**: Δa values ranging from -4 to 4 (intervals of ~1) - **Y-axis**: Frequency (0 to 2500) - **Legend**: - Spearman: 0.32 - Pearson: 0.32 - P-value: <0.0001 - **Color Gradient**: Purple (0.0) to Yellow (0.6) indicating "Proportion of Positive Δa" ### Detailed Analysis **Left Histogram (RAP-MCTS):** - Peak frequency (~2000) occurs at reward value -600 - Distribution spreads symmetrically between -640 and -560 - Frequencies decrease gradually toward the edges - No visible color variation (uniform blue) **Right Histogram (SC-MCTS*):** - Peak frequency (~2500) occurs at Δa = 0 - Distribution narrows significantly compared to RAP-MCTS - Right tail (positive Δa) shows higher frequency density - Color gradient transitions from purple (left) to yellow (right), indicating increasing proportion of positive Δa values ### Key Observations 1. **Distribution Shape**: - RAP-MCTS shows a wider, more uniform distribution - SC-MCTS* exhibits a sharper peak at 0 with steeper declines on both sides 2. **Statistical Significance**: - SC-MCTS* has a P-value <0.0001 vs RAP-MCTS' 0.2624, indicating highly significant differences - Spearman/Pearson correlations (0.32 vs 0.01) suggest stronger linear relationships in SC-MCTS* 3. **Δa Proportions**: - Right histogram's color gradient shows ~60% of rewards have positive Δa values (yellow region) ### Interpretation The SC-MCTS* system demonstrates: - **Improved Performance**: Narrower, more concentrated reward distribution around 0 suggests better target achievement - **Statistical Robustness**: Lower P-value (<0.0001) confirms significant improvement over baseline - **Positive Bias**: Color gradient reveals majority of rewards show positive Δa values, indicating systematic improvement - **Correlation Strength**: Higher Spearman/Pearson values (0.32) suggest more consistent linear relationships in reward outcomes The baseline RAP-MCTS shows: - **Wider Variability**: Broader distribution implies less consistent performance - **Negative Bias**: All rewards cluster in negative territory (-640 to -560) - **Weaker Correlations**: Near-zero Spearman/Pearson values indicate minimal linear relationships The color-coded Δa proportions in SC-MCTS* provide granular insight into reward quality distribution, revealing that higher positive deviations are more frequent in the improved system.