## Line Graph: Accuracy vs. Scale for Different Sample Sizes ### Overview The graph illustrates the relationship between "Scale" (x-axis) and "Accuracy (%)" (y-axis) for four distinct sample sizes (N = 8, 16, 64, 256). Each line represents a unique sample size, with accuracy peaking at specific scales before declining sharply. The largest sample size (N = 256) achieves the highest peak accuracy, while smaller samples exhibit lower peaks and more gradual declines. ### Components/Axes - **X-axis (Scale)**: Logarithmic scale ranging from 10⁻³ to 10³. - **Y-axis (Accuracy %)**: Linear scale from 0% to 50%. - **Legend**: Located in the top-left corner, mapping colors/symbols to sample sizes: - Red crosses (×): N = 256 - Yellow triangles (▲): N = 64 - Green circles (●): N = 16 - Blue squares (■): N = 8 ### Detailed Analysis 1. **N = 256 (Red ×)**: - Starts at 0% accuracy for scales < 10⁻². - Rises sharply to ~50% at scale 10⁻¹. - Peaks at 50% between 10⁻¹ and 10⁰, then declines steeply to 0% by 10¹. - Remains at 0% for scales ≥ 10². 2. **N = 64 (Yellow ▲)**: - Begins at ~5% accuracy at 10⁻³. - Increases to ~45% at 10⁻², peaking between 10⁻² and 10⁻¹. - Drops to ~20% at 10⁰, then declines to 0% by 10¹. 3. **N = 16 (Green ●)**: - Starts at ~5% at 10⁻³. - Rises to ~30% at 10⁻², peaking between 10⁻² and 10⁻¹. - Declines to ~15% at 10⁰, then drops to 0% by 10¹. 4. **N = 8 (Blue ■)**: - Begins at ~5% at 10⁻³. - Peaks at ~20% at 10⁻², then declines to ~10% at 10⁰. - Drops to 0% by 10¹. ### Key Observations - **Peak Accuracy**: Larger sample sizes (N = 256) achieve higher peak accuracy (~50%) compared to smaller sizes (N = 8 peaks at ~20%). - **Scale of Peak**: Larger N values peak at higher scales (e.g., N = 256 peaks at 10⁻¹, N = 8 at 10⁻²). - **Decline Pattern**: All lines exhibit a sharp drop after their peak, with larger N values experiencing steeper declines. - **Overlap**: Lines for N = 16 and N = 8 overlap slightly at scales < 10⁻². ### Interpretation The data suggests a trade-off between sample size and model performance: - **Larger Samples (N = 256)**: Achieve higher accuracy but overfit more severely, as evidenced by the abrupt decline post-peak. - **Smaller Samples (N = 8)**: Generalize better (slower decline) but with lower peak accuracy. - **Scale Sensitivity**: Accuracy is highly sensitive to scale, with optimal performance occurring at specific scales inversely related to sample size. - **Practical Implication**: Increasing sample size improves peak performance but may reduce robustness at larger scales, highlighting the need for regularization or adaptive scaling strategies.