## Line Charts: Error Analysis of Different Methods ### Overview The image contains four line charts comparing the Mean Squared Error (MSE) of four different methods (BU, Rec, MinT, and WeaKL-BU) across varying values of σ2. The charts depict the error on Y1, Y2, Y1+Y2, and the hierarchical error. ### Components/Axes * **X-axis (all charts):** σ2, ranging from 0.0 to 1.0 in increments of 0.2. * **Y-axis (all charts):** MSE (Mean Squared Error). The range varies for each chart. * **Legends (all charts, top-right):** * Blue: BU * Orange: Rec * Green: MinT * Red: WeaKL-BU ### Detailed Analysis **1. Error on Y1 (Top-Left Chart):** * Y-axis ranges from 1.0 to 1.3. * **BU (Blue):** Relatively flat line, fluctuating slightly around MSE = 1.28 with a slight downward trend towards σ2 = 1.0. * **Rec (Orange):** Relatively flat line, fluctuating slightly around MSE = 1.18 with a slight upward trend towards σ2 = 1.0. * **MinT (Green):** Relatively flat line, fluctuating slightly around MSE = 1.34 with a slight downward trend towards σ2 = 1.0. * **WeaKL-BU (Red):** Starts at MSE ≈ 0.98 at σ2 = 0.0 and increases to approximately MSE = 1.18 at σ2 = 1.0. **2. Error on Y2 (Top-Right Chart):** * Y-axis ranges from 1.0 to 2.5. * **BU (Blue):** Starts at MSE ≈ 1.25 at σ2 = 0.0 and increases to approximately MSE = 2.5 at σ2 = 1.0. * **Rec (Orange):** Starts at MSE ≈ 1.20 at σ2 = 0.0 and increases to approximately MSE = 2.25 at σ2 = 1.0. * **MinT (Green):** Starts at MSE ≈ 1.30 at σ2 = 0.0 and increases to approximately MSE = 2.35 at σ2 = 1.0. * **WeaKL-BU (Red):** Starts at MSE ≈ 1.10 at σ2 = 0.0 and increases to approximately MSE = 2.15 at σ2 = 1.0. **3. Error on Y1+Y2 (Bottom-Left Chart):** * Y-axis ranges from 0.0 to 2.0. * **BU (Blue):** Starts at MSE ≈ 0.20 at σ2 = 0.0 and increases to approximately MSE = 1.95 at σ2 = 1.0. * **Rec (Orange):** Starts at MSE ≈ 0.10 at σ2 = 0.0 and increases to approximately MSE = 1.60 at σ2 = 1.0. * **MinT (Green):** Starts at MSE ≈ 0.10 at σ2 = 0.0 and increases to approximately MSE = 1.55 at σ2 = 1.0. * **WeaKL-BU (Red):** Starts at MSE ≈ 0.00 at σ2 = 0.0 and increases to approximately MSE = 1.55 at σ2 = 1.0. **4. Hierarchical Error (Bottom-Right Chart):** * Y-axis ranges from 2.0 to 6.0. * **BU (Blue):** Starts at MSE ≈ 2.25 at σ2 = 0.0 and increases to approximately MSE = 5.80 at σ2 = 1.0. * **Rec (Orange):** Starts at MSE ≈ 2.00 at σ2 = 0.0 and increases to approximately MSE = 5.20 at σ2 = 1.0. * **MinT (Green):** Starts at MSE ≈ 2.25 at σ2 = 0.0 and increases to approximately MSE = 5.30 at σ2 = 1.0. * **WeaKL-BU (Red):** Starts at MSE ≈ 1.80 at σ2 = 0.0 and increases to approximately MSE = 5.10 at σ2 = 1.0. ### Key Observations * In the "Error on Y1" chart, the errors for BU, Rec, and MinT are relatively stable across different values of σ2, while WeaKL-BU shows a clear increasing trend. * In the "Error on Y2", "Error on Y1+Y2", and "Hierarchical error" charts, all methods show an increasing trend in MSE as σ2 increases. * WeaKL-BU generally has the lowest error at σ2 = 0.0 in the "Error on Y2", "Error on Y1+Y2", and "Hierarchical error" charts. ### Interpretation The charts illustrate how the performance of different forecasting methods (BU, Rec, MinT, and WeaKL-BU) changes with respect to the parameter σ2. The "Error on Y1" chart suggests that the error on Y1 is relatively stable for BU, Rec, and MinT, while WeaKL-BU's error increases with σ2. The other three charts ("Error on Y2", "Error on Y1+Y2", and "Hierarchical error") show that the error for all methods increases as σ2 increases, indicating that higher values of σ2 negatively impact the accuracy of the forecasts for Y2, the combined Y1+Y2, and the hierarchical structure. WeaKL-BU seems to perform better at lower values of σ2 in terms of initial error, but its performance degrades as σ2 increases, similar to the other methods. The choice of method and the value of σ2 should be carefully considered based on the specific forecasting goals and the relative importance of accuracy at different levels of the hierarchy.

\n ## Charts: Error Analysis with Noise Level ### Overview The image presents four separate line charts, each depicting the Mean Squared Error (MSE) as a function of the noise level (σ₂). The charts analyze the error performance of four different methods (BU, Rec, MinT, and WeakKL-BU) across various error metrics: Error on Y1, Error on Y2, Error on Y1+Y2, and Hierarchical error. ### Components/Axes Each chart shares the following components: * **X-axis:** Labeled "σ₂", representing the noise level, ranging from 0.0 to 1.0 with increments of 0.2. * **Y-axis:** Labeled "MSE", representing the Mean Squared Error, with varying scales depending on the chart. * **Legend:** Located in the top-right corner of each chart, identifying the four methods: * BU (Blue) * Rec (Orange) * MinT (Green) * WeakKL-BU (Red) The charts are titled as follows: * Top-left: "Error on Y1" * Top-right: "Error on Y2" * Bottom-left: "Error on Y1+Y2" * Bottom-right: "Hierarchical error" ### Detailed Analysis **1. Error on Y1:** * The blue line (BU) shows a relatively flat trend, fluctuating around a MSE of approximately 1.3, with some minor oscillations. * The orange line (Rec) starts at approximately 1.2 and gradually increases to around 1.35 as σ₂ increases to 1.0. * The green line (MinT) starts at approximately 1.1 and increases to around 1.25 at σ₂ = 1.0. * The red line (WeakKL-BU) starts at approximately 1.05 and increases to around 1.2 at σ₂ = 1.0. **2. Error on Y2:** * The blue line (BU) exhibits a strong upward trend, starting at approximately 1.0 and increasing to around 2.4 at σ₂ = 1.0. * The orange line (Rec) shows a similar upward trend, starting at approximately 1.05 and increasing to around 2.3 at σ₂ = 1.0. * The green line (MinT) also increases, starting at approximately 1.1 and reaching around 2.2 at σ₂ = 1.0. * The red line (WeakKL-BU) starts at approximately 1.0 and increases to around 1.8 at σ₂ = 1.0. **3. Error on Y1+Y2:** * The blue line (BU) shows a strong upward trend, starting at approximately 0.1 and increasing to around 1.6 at σ₂ = 1.0. * The orange line (Rec) starts at approximately 0.1 and increases to around 0.8 at σ₂ = 1.0. * The green line (MinT) starts at approximately 0.05 and increases to around 0.5 at σ₂ = 1.0. * The red line (WeakKL-BU) starts at approximately 0.05 and increases to around 0.4 at σ₂ = 1.0. **4. Hierarchical error:** * The blue line (BU) exhibits a strong upward trend, starting at approximately 2.2 and increasing to around 5.5 at σ₂ = 1.0. * The orange line (Rec) shows a similar upward trend, starting at approximately 2.3 and increasing to around 5.2 at σ₂ = 1.0. * The green line (MinT) also increases, starting at approximately 2.4 and reaching around 5.0 at σ₂ = 1.0. * The red line (WeakKL-BU) starts at approximately 2.2 and increases to around 4.5 at σ₂ = 1.0. ### Key Observations * The "BU" method consistently exhibits the highest MSE values across all error metrics, especially as the noise level (σ₂) increases. * The "WeakKL-BU" method generally performs the best, with the lowest MSE values across all error metrics. * The error increases with increasing noise level (σ₂) for all methods and all error metrics. * The "Error on Y1" chart shows the smallest range of MSE values, while the "Hierarchical error" chart shows the largest. * The "MinT" and "Rec" methods show similar performance across all charts. ### Interpretation The charts demonstrate the sensitivity of different methods (BU, Rec, MinT, WeakKL-BU) to noise when estimating errors in Y1, Y2, their combination (Y1+Y2), and a hierarchical error structure. The consistent superior performance of "WeakKL-BU" suggests that this method is more robust to noise than the others. The increasing MSE values with increasing noise level (σ₂) are expected, indicating that the accuracy of all methods degrades as the noise increases. The differences in the scales of the Y-axes across the charts suggest that the magnitude of the errors varies significantly depending on the error metric. The "Hierarchical error" is substantially larger than the errors in Y1 or Y2 individually, indicating that the hierarchical structure introduces additional complexity and potential for error. The relatively flat trend of the "BU" method in the "Error on Y1" chart might indicate that this method is less sensitive to noise when estimating errors in Y1 specifically, but it consistently performs poorly in other scenarios. This suggests that the "BU" method might be suitable only for specific applications where noise is minimal and the error metric is limited to Y1. The other methods show a more consistent increase in error with noise, suggesting a more predictable behavior.

## Line Charts: Comparative Error Analysis of Four Methods ### Overview The image displays a 2x2 grid of four line charts. Each chart plots the Mean Squared Error (MSE) on the y-axis against a parameter σ₂ (sigma squared) on the x-axis, ranging from 0.0 to 1.0. The charts compare the performance of four distinct methods: BU, Rec, MinT, and WeaKL-BU. The titles indicate the error is measured on different targets or combinations: Y1, Y2, Y1+Y2, and a "Hierarchical error." ### Components/Axes * **Chart Titles (Top of each subplot):** * Top-Left: "Error on Y1" * Top-Right: "Error on Y2" * Bottom-Left: "Error on Y1+Y2" * Bottom-Right: "Hierarchical error" * **X-Axis (All charts):** Label: "σ₂". Scale: Linear, from 0.0 to 1.0 with major ticks at 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. * **Y-Axis (All charts):** Label: "MSE". Scale varies per chart. * **Legend (Present in all charts, positioned top-left or center-left):** * Blue line: "BU" * Orange line: "Rec" * Green line: "MinT" * Red line: "WeaKL-BU" ### Detailed Analysis **1. Chart: Error on Y1 (Top-Left)** * **Y-Axis Scale:** 1.0 to 1.3. * **Trends & Approximate Values:** * **BU (Blue):** Relatively flat, fluctuating between ~1.32 and ~1.35 across the entire σ₂ range. * **MinT (Green):** Also relatively flat, closely following the BU line, fluctuating between ~1.31 and ~1.34. * **Rec (Orange):** Shows a steady, near-linear increase. Starts at ~1.18 (σ₂=0.0) and ends at ~1.27 (σ₂=1.0). * **WeaKL-BU (Red):** Starts lowest at ~1.00 (σ₂=0.0). Increases with a noticeable upward slope, crossing the Rec line around σ₂=0.7, and ends at ~1.21 (σ₂=1.0). It exhibits more volatility than the other lines. **2. Chart: Error on Y2 (Top-Right)** * **Y-Axis Scale:** 1.0 to 2.5. * **Trends & Approximate Values:** * All four methods show a clear, accelerating upward trend (convex curve). * **BU (Blue):** Consistently the highest error. Starts at ~1.35 (σ₂=0.0) and rises to ~2.55 (σ₂=1.0). * **MinT (Green):** Starts at ~1.18 (σ₂=0.0), ends at ~2.50 (σ₂=1.0). Generally the second-highest. * **Rec (Orange):** Very closely follows MinT, starting at ~1.15 and ending at ~2.48. * **WeaKL-BU (Red):** Consistently the lowest error. Starts at ~1.00 (σ₂=0.0) and rises to ~2.35 (σ₂=1.0). **3. Chart: Error on Y1+Y2 (Bottom-Left)** * **Y-Axis Scale:** 0.0 to 2.0. * **Trends & Approximate Values:** * **BU (Blue):** Shows a steady, near-linear increase from ~0.70 (σ₂=0.0) to ~2.00 (σ₂=1.0). It is the highest error method for σ₂ < ~0.9. * **Rec, MinT, WeaKL-BU:** All start very low (near 0.0 at σ₂=0.0) and follow a similar, accelerating upward curve. They converge closely at higher σ₂ values. * At σ₂=1.0, the approximate final values are: Rec ~1.60, MinT ~1.58, WeaKL-BU ~1.55. **4. Chart: Hierarchical error (Bottom-Right)** * **Y-Axis Scale:** 2 to 6. * **Trends & Approximate Values:** * All methods show a strong, accelerating upward trend. * **BU (Blue):** Highest error throughout. Starts at ~3.4 (σ₂=0.0) and ends at ~5.9 (σ₂=1.0). * **MinT (Green):** Second highest. Starts at ~2.7, ends at ~5.5. * **Rec (Orange):** Third. Starts at ~2.5, ends at ~5.4. * **WeaKL-BU (Red):** Lowest error throughout. Starts at ~2.0, ends at ~5.2. ### Key Observations 1. **Consistent Hierarchy:** In three of the four charts (Y2, Y1+Y2, Hierarchical), the performance hierarchy is consistent: WeaKL-BU (best/lowest MSE) < Rec ≈ MinT < BU (worst/highest MSE). The "Error on Y1" chart is an exception where BU and MinT are similar and highest. 2. **Impact of σ₂:** For all methods and all error metrics, MSE increases as σ₂ increases. The relationship is often non-linear (convex), especially for Y2 and Hierarchical error. 3. **Convergence at High σ₂:** In the "Error on Y1+Y2" chart, the three non-BU methods converge to very similar MSE values as σ₂ approaches 1.0. 4. **Scale of Error:** The "Hierarchical error" yields the largest MSE values (up to ~6), suggesting it is a more complex or aggregated error metric compared to errors on individual components Y1 and Y2. ### Interpretation The data demonstrates a comparative analysis of four statistical or machine learning methods (BU, Rec, MinT, WeaKL-BU) under varying levels of noise or variance (σ₂). The key finding is the **superior performance of the WeaKL-BU method**, which consistently achieves the lowest Mean Squared Error across most scenarios, particularly for the Y2 component and the aggregated hierarchical error. The charts suggest that the BU method is the most sensitive to increasing σ₂, showing the steepest error growth in the combined and hierarchical metrics. The Rec and MinT methods perform similarly to each other, occupying a middle ground. The distinct behavior in the "Error on Y1" chart—where BU and MinT are flat and high—implies that for the Y1 component specifically, these methods have a high but stable error floor unaffected by σ₂, whereas Rec and WeaKL-BU's errors are lower but more sensitive to this parameter. The "Hierarchical error" chart is particularly telling, as it likely represents the overall system or total error. The clear, separated, and increasing curves here provide strong evidence that the choice of method has a significant and growing impact on overall performance as the underlying parameter σ₂ increases. WeaKL-BU's advantage is most pronounced in this holistic view.

## Line Graphs: Error Analysis Across Different Metrics ### Overview The image contains four line graphs arranged in a 2x2 grid, each depicting the Mean Squared Error (MSE) of different methods as a function of the parameter σ₂ (ranging from 0.0 to 1.0). The graphs are labeled as follows: 1. **Top-left**: Error on Y1 2. **Top-right**: Error on Y2 3. **Bottom-left**: Error on Y1+Y2 4. **Bottom-right**: Hierarchical error Each graph includes a legend with four methods: BU (blue), Rec (orange), MinT (green), and WeaKL-BU (red). The y-axis is labeled "MSE" with approximate values ranging from 0.0 to 2.5 (top-left and top-right) and 0.0 to 6.0 (bottom-left and bottom-right). --- ### Components/Axes - **X-axis**: σ₂ (0.0 to 1.0) in all subplots. - **Y-axis**: MSE (Mean Squared Error) with varying ranges: - Top-left (Y1): ~1.0 to 1.3 - Top-right (Y2): ~1.0 to 2.5 - Bottom-left (Y1+Y2): ~0.0 to 2.0 - Bottom-right (Hierarchical error): ~2.0 to 6.0 - **Legends**: Positioned in the top-right corner of each subplot, with colors: - BU: Blue - Rec: Orange - MinT: Green - WeaKL-BU: Red --- ### Detailed Analysis #### Top-left: Error on Y1 - **BU (blue)**: Flat line at ~1.3 MSE. - **Rec (orange)**: Slightly increasing trend, starting at ~1.2 and rising to ~1.3. - **MinT (green)**: Decreasing trend, starting at ~1.3 and dropping to ~1.1. - **WeaKL-BU (red)**: Increasing trend, starting at ~1.0 and rising to ~1.2. #### Top-right: Error on Y2 - **BU (blue)**: Steeply increasing from ~1.2 to ~2.5. - **Rec (orange)**: Gradual increase from ~1.2 to ~2.2. - **MinT (green)**: Moderate increase from ~1.1 to ~2.0. - **WeaKL-BU (red)**: Steep increase from ~1.0 to ~2.2. #### Bottom-left: Error on Y1+Y2 - **BU (blue)**: Gradual increase from ~0.5 to ~1.5. - **Rec (orange)**: Slight increase from ~0.2 to ~0.5. - **MinT (green)**: Moderate increase from ~0.1 to ~0.4. - **WeaKL-BU (red)**: Steep increase from ~0.0 to ~0.5. #### Bottom-right: Hierarchical error - **BU (blue)**: Steep increase from ~3.0 to ~6.0. - **Rec (orange)**: Moderate increase from ~2.5 to ~5.0. - **MinT (green)**: Gradual increase from ~2.0 to ~4.5. - **WeaKL-BU (red)**: Steep increase from ~2.0 to ~5.5. --- ### Key Observations 1. **Consistency of Trends**: - In all subplots, **WeaKL-BU (red)** and **MinT (green)** generally show lower MSE values compared to **BU (blue)** and **Rec (orange)**. - **BU (blue)** consistently exhibits the highest MSE across all metrics, especially in the hierarchical error subplot. 2. **σ₂ Dependency**: - As σ₂ increases, MSE values rise for all methods, but the rate of increase varies. For example: - In the hierarchical error subplot, **BU (blue)** increases sharply, while **WeaKL-BU (red)** and **MinT (green)** show more gradual rises. 3. **Performance Differences**: - **WeaKL-BU (red)** and **MinT (green)** outperform **BU (blue)** and **Rec (orange)** in most cases, particularly in the Y1+Y2 and hierarchical error subplots. --- ### Interpretation The data suggests that **WeaKL-BU** and **MinT** are more robust to variations in σ₂ compared to **BU** and **Rec**, as evidenced by their lower MSE values across all metrics. The hierarchical error subplot highlights the sensitivity of **BU** to σ₂, with its MSE increasing dramatically as σ₂ approaches 1.0. This implies that **WeaKL-BU** and **MinT** may be better suited for scenarios with higher uncertainty (larger σ₂). The consistent underperformance of **BU** across all metrics raises questions about its stability or adaptability in dynamic environments.