## Combined Chart: Error vs. #particles and WCET vs. #particles ### Overview The image presents two charts side-by-side. The left chart is a violin plot showing the distribution of x-error and y-error as a function of the number of particles. The right chart is a bar plot showing the Worst-Case Execution Time (WCET) for 'pos' and 'braking' as a function of the number of particles. Both charts share the same x-axis: '#particles'. Both charts use a log scale for both axes. ### Components/Axes **Left Chart (Error vs. #particles):** * **Y-axis:** "Error (m)" - Logarithmic scale with markers at 10^-2, 10⁰, and 10². * **X-axis:** "#particles" - Logarithmic scale with markers at 10², 10³, 10⁴, and 10⁵. * **Legend (Top-Right):** * "x-error" - Light Blue * "y-error" - Light Orange **Right Chart (WCET vs. #particles):** * **Y-axis:** "WCET (s)" - Logarithmic scale with markers at 10^-1 and 10⁰. * **X-axis:** "#particles" - Logarithmic scale with markers at 10², 10³, 10⁴, and 10⁵. * **Legend (Top-Left):** * "pos" - Dark Blue * "braking" - Orange ### Detailed Analysis **Left Chart (Error vs. #particles):** * **x-error (Light Blue):** * At 10² particles: The distribution is centered around 10¹ m, with a wide spread. * At 10³ particles: The distribution is centered around 10^-1 m, with a smaller spread. * At 10⁴ particles: The distribution is centered around 10^-1 m, with a smaller spread. * At 10⁵ particles: The distribution is centered around 10^-1 m, with a smaller spread. * Trend: The x-error decreases significantly as the number of particles increases from 10² to 10³, then plateaus. * **y-error (Light Orange):** * At 10² particles: The distribution is centered around 10^-2 m, with a wide spread. * At 10³ particles: The distribution is centered around 10^-2 m, with a smaller spread. * At 10⁴ particles: The distribution is centered around 10^-1 m, with a smaller spread. * At 10⁵ particles: The distribution is centered around 10^-1 m, with a smaller spread. * Trend: The y-error remains relatively constant as the number of particles increases, with a slight increase between 10³ and 10⁴. **Right Chart (WCET vs. #particles):** * **pos (Dark Blue):** * At 10² particles: WCET is approximately 0.02 s. * At 10³ particles: WCET is approximately 0.2 s. * At 10⁴ particles: WCET is approximately 0.6 s. * At 10⁵ particles: WCET is approximately 2 s. * Trend: The WCET for 'pos' increases significantly as the number of particles increases. * **braking (Orange):** * At 10² particles: WCET is approximately 0.08 s. * At 10³ particles: WCET is approximately 0.15 s. * At 10⁴ particles: WCET is approximately 0.2 s. * At 10⁵ particles: WCET is approximately 0.4 s. * Trend: The WCET for 'braking' increases as the number of particles increases, but not as dramatically as 'pos'. ### Key Observations * Increasing the number of particles significantly reduces the x-error, especially between 10² and 10³ particles. * The y-error is less sensitive to the number of particles. * The WCET for both 'pos' and 'braking' increases with the number of particles, but 'pos' is more affected. * At 10^5 particles, the WCET for 'pos' is significantly higher than 'braking'. ### Interpretation The data suggests that increasing the number of particles improves the accuracy of the 'x' position estimate, with diminishing returns beyond 10³ particles. The 'y' position estimate is less affected by the number of particles. However, increasing the number of particles increases the computational cost, as reflected in the WCET. The 'pos' operation is more computationally expensive than 'braking' and its cost increases more rapidly with the number of particles. This information is useful for optimizing the particle filter algorithm, balancing accuracy and computational cost.

## [Chart Type]: Dual-Panel Performance Analysis (Violin Plot and Bar Chart) ### Overview The image displays a two-panel figure analyzing the performance of a particle-based system (likely a filter or simulation) as a function of the number of particles. The left panel shows error distributions, and the right panel shows Worst-Case Execution Time (WCET). Both panels share a common x-axis representing the number of particles on a logarithmic scale. ### Components/Axes **Common X-Axis (Both Panels):** * **Label:** `#particles` * **Scale:** Logarithmic * **Markers/Ticks:** `10²`, `10³`, `10⁴`, `10⁵` **Left Panel (Violin Plot):** * **Title/Type:** Violin Plot of Error Distributions * **Y-Axis Label:** `Error (m)` * **Y-Axis Scale:** Logarithmic, ranging from approximately `10⁻²` to `10²` meters. * **Legend:** Located at the top-center of the panel. * `x-error`: Represented by blue-filled violin shapes and blue error bars. * `y-error`: Represented by orange-filled violin shapes and orange error bars. * **Data Representation:** For each particle count (`10²`, `10³`, `10⁴`, `10⁵`), there are two overlapping violin plots showing the probability density of the error data. Vertical lines with horizontal caps (error bars) indicate the range or confidence intervals. **Right Panel (Bar Chart):** * **Title/Type:** Bar Chart of Worst-Case Execution Time (WCET) * **Y-Axis Label:** `WCET (s)` * **Y-Axis Scale:** Logarithmic, ranging from approximately `10⁻¹` to `10⁰` seconds. * **Legend:** Located at the top-left of the panel. * `pos`: Represented by blue bars. * `braking`: Represented by orange bars. * **Data Representation:** For each particle count, two bars are grouped together, showing the WCET for "pos" (likely position computation) and "braking" tasks. ### Detailed Analysis **Left Panel - Error Distributions:** * **Trend Verification:** Both `x-error` (blue) and `y-error` (orange) distributions show a clear trend: as the number of particles increases from `10²` to `10⁵`, the distributions become narrower and shift downward on the logarithmic scale. This indicates a reduction in both the magnitude and variance of the error. * **Data Points & Observations:** * **At 10² particles:** The distributions are very wide, spanning nearly the entire y-axis range. The `y-error` (orange) distribution appears slightly wider and extends to higher error values than the `x-error` (blue). The central tendency (median/mean) for both is around `10⁰` to `10¹` meters. * **At 10³ particles:** A significant reduction in error is visible. The distributions are centered closer to `10⁻¹` meters. The `y-error` distribution still appears slightly broader than the `x-error`. * **At 10⁴ particles:** Error continues to decrease. The distributions are now centered below `10⁻¹` meters. The shapes are more compact. * **At 10⁵ particles:** The error is lowest. The distributions are tightly clustered around `10⁻²` meters. The `x-error` (blue) distribution appears to have a slightly lower median than the `y-error` (orange) at this point. **Right Panel - WCET:** * **Trend Verification:** Both `pos` (blue) and `braking` (orange) WCET show an increasing trend with the number of particles. The increase for `pos` is dramatically steeper than for `braking`. * **Data Points & Approximate Values:** * **At 10² particles:** * `pos` (blue): WCET is very low, approximately `2 x 10⁻²` seconds (0.02 s). * `braking` (orange): WCET is approximately `1.5 x 10⁻¹` seconds (0.15 s). * **At 10³ particles:** * `pos` (blue): Increases to approximately `10⁻¹` seconds (0.1 s). * `braking` (orange): Increases slightly to approximately `2 x 10⁻¹` seconds (0.2 s). * **At 10⁴ particles:** * `pos` (blue): Increases significantly to approximately `2 x 10⁰` seconds (2 s). * `braking` (orange): Remains relatively stable, around `1.5 x 10⁻¹` seconds (0.15 s). * **At 10⁵ particles:** * `pos` (blue): Shows a massive increase, reaching approximately `4 x 10⁰` seconds (4 s). This is the tallest bar in the chart. * `braking` (orange): Increases to approximately `9 x 10⁻¹` seconds (0.9 s). ### Key Observations 1. **Accuracy-Computation Trade-off:** There is a clear inverse relationship between accuracy (error) and computational cost (WCET). Increasing particles from `10²` to `10⁵` reduces error by roughly two orders of magnitude but increases the WCET for the `pos` task by over two orders of magnitude. 2. **Differential Cost Growth:** The computational cost (`WCET`) of the `pos` task scales much more poorly with particle count than the `braking` task. At `10⁵` particles, `pos` WCET is over 4 times greater than `braking` WCET. 3. **Error Convergence:** The error distributions for both x and y dimensions converge to similar, low values at high particle counts (`10⁵`), suggesting the system's accuracy becomes isotropic with sufficient particles. 4. **Initial Instability:** At the lowest particle count (`10²`), the error distributions are extremely wide, indicating highly unreliable and variable performance. ### Interpretation This figure illustrates a classic engineering trade-off in particle-based estimation or simulation systems (e.g., Particle Filters). The data demonstrates that **increasing the number of particles directly improves estimation accuracy (reduces error) but at a significant and non-linear computational cost.** * **What the data suggests:** The system's precision is highly dependent on particle count. The dramatic narrowing of the violin plots shows that more particles lead to more consistent and accurate estimates. However, the bar chart reveals that this accuracy comes at a price, particularly for the `pos` (position) computation, which appears to have a computational complexity that grows super-linearly with the number of particles. * **How elements relate:** The two panels are two sides of the same coin. The left panel shows the *benefit* (lower error), and the right panel shows the *cost* (higher WCET) of scaling up the system. An engineer would use this data to find an optimal operating point—perhaps at `10⁴` particles, where error is already quite low (`~0.1 m`) and the `pos` WCET (`~2 s`) might still be acceptable for a real-time system, whereas at `10⁵` particles, the WCET (`~4 s`) may be prohibitive despite marginally better accuracy. * **Notable Anomalies/Insights:** The most striking insight is the disproportionate growth of the `pos` WCET. This suggests that the algorithm for position computation does not scale efficiently with particle count and could be a bottleneck. Optimizing this specific component would yield the greatest performance gains. The relatively flat WCET for `braking` suggests it is a less complex or fixed-cost operation. The investigation should focus on why `pos` computation is so expensive and whether approximations or algorithmic changes can mitigate this.

## Dual-Axis Error Analysis: Particle Count vs. Performance Metrics ### Overview The image presents two side-by-side plots comparing error metrics and computational costs across varying particle counts (10² to 10⁵). The left plot uses violin plots to show error distributions, while the right uses bar charts for WCET comparisons. ### Components/Axes **Left Plot (Violin Plots):** - **X-axis**: "#particles" (log scale: 10², 10³, 10⁴, 10⁵) - **Y-axis**: "Error (m)" (log scale: 10⁻² to 10²) - **Legend**: - Blue: "x-error" - Orange: "y-error" - **Violin Plot Features**: - Median lines (horizontal black bars) - Error bars (vertical blue/orange lines) - Distribution density (shaded areas) **Right Plot (Bar Chart):** - **X-axis**: "#particles" (log scale: 10², 10³, 10⁴, 10⁵) - **Y-axis**: "WCET (s)" (log scale: 10⁻¹ to 10¹) - **Legend**: - Blue: "pos" - Orange: "braking" - **Bar Features**: - Height represents WCET magnitude - Error bars (vertical lines at bar tops) ### Detailed Analysis **Left Plot Trends:** - **x-error** (blue): - 10² particles: Median ~10⁻¹ m, range 10⁻²–10¹ m - 10³ particles: Median ~10⁻² m, range 10⁻³–10⁻¹ m - 10⁴ particles: Median ~10⁻³ m, range 10⁻⁴–10⁻² m - 10⁵ particles: Median ~10⁻⁴ m, range 10⁻⁵–10⁻³ m - **y-error** (orange): - 10² particles: Median ~10⁻¹ m, range 10⁻²–10¹ m - 10³ particles: Median ~10⁻² m, range 10⁻³–10⁻¹ m - 10⁴ particles: Median ~10⁻³ m, range 10⁻⁴–10⁻² m - 10⁵ particles: Median ~10⁻⁴ m, range 10⁻⁵–10⁻³ m - **Key Pattern**: x-error consistently exceeds y-error by ~10× across all particle counts. **Right Plot Trends:** - **pos** (blue): - 10² particles: ~10⁻¹ s - 10³ particles: ~10⁻¹ s - 10⁴ particles: ~10⁰ s - 10⁵ particles: ~10¹ s - **braking** (orange): - 10² particles: ~10⁻¹ s - 10³ particles: ~10⁻¹ s - 10⁴ particles: ~10⁻¹ s - 10⁵ particles: ~10⁰ s - **Key Pattern**: pos WCET increases exponentially with particle count, while braking remains stable until 10⁵ particles. ### Key Observations 1. **Error Reduction**: Both x- and y-errors decrease by ~10× per decade of particle count. 2. **WCET Divergence**: pos WCET grows exponentially (10× per decade), while braking WCET plateaus until 10⁵ particles. 3. **Asymmetry**: x-error dominates y-error in magnitude across all particle counts. ### Interpretation The data suggests a trade-off between precision and computational cost: - **Precision**: Increasing particle count reduces positional errors (x/y) by ~10× per decade, but with diminishing returns at higher counts. - **Computational Cost**: pos operations scale poorly with particle count (exponential growth), while braking operations remain efficient until extreme particle counts (10⁵). - **Design Implications**: For applications requiring high precision (e.g., robotics), particle counts above 10⁴ may be impractical due to WCET constraints. The x-error/y-error asymmetry indicates potential optimization opportunities in coordinate-specific error modeling.