## Chart Type: Box Plot of Normalized MSE ### Overview The image is a box plot comparing the normalized Mean Squared Error (MSE) on LSR-Transform for three different methods: PySR, KeplerAgent @1, and KeplerAgent @3. The y-axis represents the log base 10 of the normalized MSE, and the plot shows the distribution of the MSE for each method. A red dashed line is drawn at y = -2.0. ### Components/Axes * **Title:** Normalized MSE on LSR-Transform * **X-axis:** Method (categorical): PySR, KeplerAgent @1, KeplerAgent @3 * **Y-axis:** log10(Normalized MSE), with a scale from -15 to 1. * **Box Plots:** Each box plot represents the distribution of the log10(Normalized MSE) for a given method. The box extends from the first quartile (Q1) to the third quartile (Q3), with a line at the median. Whiskers extend from the box to show the range of the data. * **Horizontal Red Dashed Line:** Located at log10(Normalized MSE) = -2.0. * **Hashed Regions:** There are two hashed regions, one near the top of the plot and one near the bottom. The top region spans approximately from log10(Normalized MSE) = -0.5 to 0.5. The bottom region spans approximately from log10(Normalized MSE) = -13 to -12. * **Median and Mean Values:** The median and mean values are displayed below each box plot. ### Detailed Analysis **PySR (Blue Box Plot):** * The box extends from approximately -13 to -1.5. * Median = 4.47 x 10^-5 * Mean = 2.82 x 10^-1 **KeplerAgent @1 (Green Box Plot):** * The box extends from approximately -13 to -1.5. * Median = 1.40 x 10^-4 * Mean = 1.50 x 10^-1 **KeplerAgent @3 (Red Box Plot):** * The box extends from approximately -13 to -1.5. * Median = 1.94 x 10^-5 * Mean = 1.21 x 10^-1 ### Key Observations * The median MSE is lowest for KeplerAgent @3 (1.94 x 10^-5) and highest for KeplerAgent @1 (1.40 x 10^-4). * The mean MSE is lowest for KeplerAgent @3 (1.21 x 10^-1) and highest for PySR (2.82 x 10^-1). * The boxes for all three methods span a similar range on the y-axis. * The red dashed line at log10(Normalized MSE) = -2.0 falls within the boxes for all three methods. ### Interpretation The box plot compares the performance of three different methods (PySR, KeplerAgent @1, and KeplerAgent @3) in terms of normalized MSE on LSR-Transform. The results suggest that KeplerAgent @3 has the lowest median and mean MSE, indicating better performance compared to the other two methods. The wide range of the boxes suggests that there is significant variability in the MSE for all three methods. The red dashed line at log10(Normalized MSE) = -2.0 provides a reference point for comparing the performance of the methods. The hashed regions near the top and bottom of the plot may indicate a threshold or acceptable range for the MSE.

\n ## Box Plot: Normalized MSE on LSR-Transform ### Overview This image presents a box plot comparing the Normalized Mean Squared Error (MSE) across three different methods: PySR, KeplerAgent @1, and KeplerAgent @3. The y-axis represents the base-10 logarithm of the Normalized MSE, while the x-axis indicates the method used. A horizontal dashed red line is present, likely representing a benchmark or target MSE value. ### Components/Axes * **Title:** "Normalized MSE on LSR-Transform" (Top-center) * **X-axis Label:** "Method" (Bottom-center) * **Y-axis Label:** "log10(Normalized MSE)" (Left-center) * **Methods (Categories):** PySR, KeplerAgent @1, KeplerAgent @3 * **Benchmark Line:** A horizontal dashed red line at approximately -2.0 on the log10(Normalized MSE) scale. * **Box Plot Elements:** Each method has a box representing the interquartile range (IQR), with a line indicating the median. Whiskers extend from the boxes to show the range of the data, excluding outliers. ### Detailed Analysis The box plots show the distribution of Normalized MSE values for each method. * **PySR (Blue):** The PySR box plot is centered around a median value of approximately 4.47 x 10^-5. The box extends from roughly -3.0 to -1.0 on the log10(Normalized MSE) scale. The mean is reported as 2.82 x 10^-1. The whiskers extend to approximately -14 and 0. * **KeplerAgent @1 (Green):** The KeplerAgent @1 box plot has a median value of approximately 1.40 x 10^-4. The box extends from roughly -2.5 to -0.5 on the log10(Normalized MSE) scale. The mean is reported as 1.50 x 10^-1. The whiskers extend to approximately -14 and 0. * **KeplerAgent @3 (Red):** The KeplerAgent @3 box plot has a median value of approximately 1.94 x 10^-5. The box extends from roughly -2.5 to -0.5 on the log10(Normalized MSE) scale. The mean is reported as 1.21 x 10^-1. The whiskers extend to approximately -14 and 0. **Trends:** * PySR has the lowest median Normalized MSE. * KeplerAgent @1 has the highest median Normalized MSE. * KeplerAgent @3 has a median Normalized MSE between PySR and KeplerAgent @1. * All three methods have similar ranges, as indicated by the whiskers. ### Key Observations * PySR consistently performs better than KeplerAgent @1 and KeplerAgent @3, as indicated by its lower median and generally lower distribution of MSE values. * The benchmark line (dashed red line) appears to be above the median values for all three methods, suggesting that all methods achieve performance better than the benchmark. * The spread of data (as indicated by the box size and whisker length) is similar across all three methods, suggesting similar variability in performance. ### Interpretation The data suggests that PySR is the most effective method for the LSR-Transform task, as it consistently achieves the lowest Normalized MSE. KeplerAgent @3 performs better than KeplerAgent @1. The benchmark line indicates a performance threshold, and all methods surpass it. The similar spread of data across methods suggests that while PySR has a lower central tendency, the variability in performance is comparable. The fact that the whiskers extend to the same values for all methods suggests that the extreme values are similar, even if the central values differ. This could indicate that all methods are susceptible to the same types of errors or challenges in certain cases. The use of the log10 scale compresses the range of MSE values, making it easier to visualize differences between methods, especially when dealing with small values.

## Box Plot: Normalized MSE on LSR-Transform ### Overview The image is a box plot comparing the performance of three different methods on a task called "LSR-Transform." Performance is measured by the logarithm (base 10) of the Normalized Mean Squared Error (MSE). Lower values indicate better performance. The plot includes a horizontal red dashed reference line. ### Components/Axes * **Title:** "Normalized MSE on LSR-Transform" * **Y-Axis:** * **Label:** `log10(Normalized MSE)` * **Scale:** Logarithmic. Major tick marks are visible at -15, -14, -13, -2.0, -1.5, 0, and 1. The axis has a break (indicated by diagonal hatching) between approximately -13 and -2.0, compressing this range. * **X-Axis:** * **Label:** `Method` * **Categories (from left to right):** 1. `PySR` (Blue box) 2. `KeplerAgent @1` (Green box) 3. `KeplerAgent @3` (Red/Brown box) * **Reference Line:** A red dashed horizontal line is positioned at `y = -2.0`. * **Data Annotations:** Below each box, the median and mean values are provided in scientific notation. ### Detailed Analysis **1. PySR (Blue Box, Left):** * **Visual Trend:** The box spans from approximately -14.0 to -1.3 on the y-axis. The median line is near the bottom of the box. The upper whisker extends to near 0.8, and the lower whisker extends to near -14.5. * **Annotated Values:** * `median = 4.47×10⁻⁵` (This corresponds to `log10(4.47e-5) ≈ -4.35`) * `mean = 2.82×10⁻¹` (This corresponds to `log10(0.282) ≈ -0.55`) * **Interpretation:** The large discrepancy between the median (-4.35) and mean (-0.55) indicates a highly right-skewed distribution. Most runs have very low error (median), but a few runs with very high error pull the mean up significantly. **2. KeplerAgent @1 (Green Box, Center):** * **Visual Trend:** The box spans from approximately -14.0 to -1.4. The median line is slightly above the bottom of the box. The upper whisker extends to near 0.5, and the lower whisker extends to near -14.5. * **Annotated Values:** * `median = 1.40×10⁻⁴` (This corresponds to `log10(1.40e-4) ≈ -3.85`) * `mean = 1.50×10⁻¹` (This corresponds to `log10(0.150) ≈ -0.82`) * **Interpretation:** Similar to PySR, the distribution is right-skewed (mean > median). The median error is slightly higher than PySR's median. **3. KeplerAgent @3 (Red/Brown Box, Right):** * **Visual Trend:** The box spans from approximately -14.0 to -1.5. The median line is near the bottom of the box. The upper whisker extends to near 0.0, and the lower whisker extends to near -14.5. * **Annotated Values:** * `median = 1.94×10⁻⁵` (This corresponds to `log10(1.94e-5) ≈ -4.71`) * `mean = 1.21×10⁻¹` (This corresponds to `log10(0.121) ≈ -0.92`) * **Interpretation:** This method has the lowest median error of the three. It also shows right-skew, but its mean is the lowest among the three methods. ### Key Observations 1. **Performance Ranking (by Median):** KeplerAgent @3 (best, lowest median) > PySR > KeplerAgent @1 (worst, highest median). 2. **Performance Ranking (by Mean):** KeplerAgent @3 (best, lowest mean) > KeplerAgent @1 > PySR (worst, highest mean). The mean ranking differs from the median ranking due to the different skew magnitudes. 3. **Distribution Shape:** All three methods exhibit strongly right-skewed distributions of log10(Normalized MSE). This means that while the typical (median) performance is very good (errors around 10⁻⁴ to 10⁻⁵), there is a long tail of runs with much higher errors (up to ~10⁰ or 1). 4. **Spread:** The interquartile range (height of the boxes) and the whisker lengths are broadly similar across methods, indicating comparable variability in performance, aside from the skew. 5. **Reference Line:** The red dashed line at `log10(Normalized MSE) = -2.0` (Normalized MSE = 0.01) serves as a visual benchmark. The median of all three methods is well below this line, indicating that the central tendency of each method achieves an error less than 1% of the normalized scale. ### Interpretation This chart evaluates symbolic regression or program synthesis methods (PySR and variants of KeplerAgent) on the LSR-Transform benchmark. The key takeaway is that **KeplerAgent @3 achieves the best median performance**, suggesting it is the most reliable method for producing low-error solutions on this task. The pervasive right-skew across all methods is a critical finding. It indicates that while these algorithms often find excellent solutions, they are not perfectly robust; a subset of runs fails to converge well, resulting in high-error outliers. This could be due to random initialization, the stochastic nature of the search, or particular difficulty with certain sub-problems in the benchmark. The comparison between `KeplerAgent @1` and `@3` suggests that the `@3` variant (which likely involves more computational resources, search depth, or ensemble size) provides a meaningful improvement in both median and mean error over the `@1` version. The fact that PySR's mean is the highest, despite having a median better than KeplerAgent @1, highlights how severely its performance is impacted by its worst-case runs. In summary, for the LSR-Transform task, KeplerAgent @3 is the most accurate and reliable method on average, but all methods show a vulnerability to producing occasional high-error results. The red line at -2.0 provides a clear visual threshold that all medians comfortably surpass.

## Box Plot: Normalized MSE on LSR-Transform ### Overview The image is a comparative box plot visualizing the distribution of **log₁₀(Normalized MSE)** across three methods: **PySR**, **KeplerAgent @1**, and **KeplerAgent @3**. The y-axis represents the logarithmic scale of normalized mean squared error, while the x-axis categorizes the methods. A red dashed line at **-2.0** and a gray shaded region between **-1.0 and 0.0** serve as reference thresholds. --- ### Components/Axes - **X-Axis (Methods)**: - PySR (blue box) - KeplerAgent @1 (green box) - KeplerAgent @3 (red box) - **Y-Axis (log₁₀(Normalized MSE))**: - Range: **-15 to 1** (logarithmic scale) - Key markers: - Red dashed line at **-2.0** - Gray shaded region from **-1.0 to 0.0** - **Legend**: - Colors correspond to methods: - Blue = PySR - Green = KeplerAgent @1 - Red = KeplerAgent @3 --- ### Detailed Analysis 1. **PySR (Blue Box)**: - **Median**: 4.47×10⁻⁵ (log₁₀ = -4.35) - **Mean**: 2.82×10⁻¹ (log₁₀ = -0.55) - **Spread**: Whiskers extend from ~-14.5 to ~-0.5. - **Outliers**: None visible. 2. **KeplerAgent @1 (Green Box)**: - **Median**: 1.40×10⁻⁴ (log₁₀ = -3.85) - **Mean**: 1.50×10⁻¹ (log₁₀ = -0.82) - **Spread**: Whiskers extend from ~-14.5 to ~-0.8. - **Outliers**: None visible. 3. **KeplerAgent @3 (Red Box)**: - **Median**: 1.94×10⁻⁵ (log₁₀ = -4.71) - **Mean**: 1.21×10⁻¹ (log₁₀ = -0.92) - **Spread**: Whiskers extend from ~-14.5 to ~-0.9. - **Outliers**: None visible. --- ### Key Observations 1. **Performance Thresholds**: - All medians fall **below the red dashed line (-2.0)**, indicating all methods perform better than the threshold. - The gray shaded region (-1.0 to 0.0) represents a "target" range, but no data points lie within it. 2. **Method Comparison**: - **KeplerAgent @3** achieves the **lowest median and mean**, suggesting superior performance. - **PySR** has the **highest mean** (2.82×10⁻¹), indicating poorer average performance compared to KeplerAgent methods. - **KeplerAgent @1** and **@3** show similar spread but differ in central tendency. 3. **Distribution Patterns**: - All methods exhibit **left-skewed distributions** (longer tails on the lower end). - **KeplerAgent @3** has the narrowest interquartile range, indicating consistency. --- ### Interpretation The data demonstrates that **KeplerAgent @3** outperforms both PySR and KeplerAgent @1 in terms of normalized MSE, with the lowest median and mean values. The logarithmic scale emphasizes the disparity in performance, particularly for PySR, whose mean (2.82×10⁻¹) is significantly higher than KeplerAgent methods. The absence of data in the gray shaded region (-1.0 to 0.0) suggests that none of the methods achieve the "target" range, though all remain below the critical threshold (-2.0). The consistent left-skewed distributions imply that errors are predominantly concentrated at lower magnitudes, with rare extreme outliers. This analysis highlights the effectiveness of KeplerAgent variants in minimizing error, with @3 being the most robust choice.