## NMSE vs. Iterations Chart ### Overview The image is a line chart showing the Normalized Mean Squared Error (NMSE) on a logarithmic scale versus the number of iterations. The chart displays a red line that represents the NMSE, which decreases in steps as the number of iterations increases. Several text annotations are placed near the line, indicating different mathematical expressions. ### Components/Axes * **X-axis:** Iterations, ranging from 0 to 2500, with tick marks at intervals of 500. * **Y-axis:** NMSE (log scale), ranging from 10^0 to 10^-28. Tick marks are present at each order of magnitude. * **Data Series:** A single red line representing the NMSE. * **Annotations:** Several text boxes containing mathematical expressions, positioned near the red line. These boxes have either a light green or light pink background. ### Detailed Analysis * **Red Line (NMSE):** The red line starts at approximately 10^0 and decreases in a step-wise fashion. * From 0 to approximately 100 iterations, the NMSE drops from 10^0 to approximately 10^-6. * From 100 to approximately 300 iterations, the NMSE drops from 10^-6 to approximately 10^-12. * From 300 to approximately 1500 iterations, the NMSE remains constant at approximately 10^-12. * At approximately 1500 iterations, the NMSE drops sharply to approximately 10^-28, where it remains constant until 2500 iterations. * **Annotations:** * Near the top-left (around 0 iterations, 10^0 NMSE): "p1 * x + p2 * v + p3" (light pink background) * Around 100 iterations, 10^-6 NMSE: "+p3 * sin(x) + p6 * v^3 - p1 * v + p2 * x + p4 * cos(x) + p5 * v^2 + p7 * x^2 + p8 * v^4 + p9 * x * sin(v) + p10" (light pink background) * Around 300 iterations, 10^-12 NMSE: "+p3 * sin(x) + p5 * v^3 + p6 * x * v + p8 * x^3 p1 * x + p2 * v + p4 * cos(v) + p7 * sin(v)" (light green background) * Around 1500 iterations, 10^-12 NMSE: "p1 * x^3 + p2 * x * v + p3 * v^3 + p4 * sin(x) + p5 * x * cos(x)" (light green background) ### Key Observations * The NMSE decreases significantly in the first 1500 iterations, after which it stabilizes. * The annotations suggest that different mathematical expressions or models are being applied at different stages of the iterative process. * The step-wise decrease in NMSE indicates that the model is being refined or adjusted at specific iteration points. ### Interpretation The chart illustrates the convergence of an iterative process, likely an optimization or machine learning algorithm, as measured by the NMSE. The annotations provide insight into the potential changes in the model or algorithm being used during the iterations. The initial rapid decrease in NMSE suggests that the model quickly improves, while the later stabilization indicates that the model has reached a point of diminishing returns. The different mathematical expressions likely represent different terms or features being added or adjusted in the model to improve its performance. The sudden drop at 1500 iterations could indicate a significant change in the model or algorithm that leads to a substantial reduction in error.

## Chart: NMSE vs. Iterations for Model Complexity ### Overview This chart displays the Negative Mean Squared Error (NMSE) on a logarithmic scale as a function of the number of iterations. The chart illustrates how the NMSE decreases with increasing model complexity (number of terms in the polynomial) and iteration count. Multiple lines represent different polynomial models, showing their convergence behavior. ### Components/Axes * **X-axis:** Iterations (ranging from 0 to approximately 2500). * **Y-axis:** NMSE (log scale, ranging from 10^-30 to 10⁰). * **Lines:** Four distinct lines representing different polynomial models. * **Markers:** Circular markers are placed at specific points along each line, likely indicating key stages or evaluations. * **Legend:** Located in the top-left corner, labeling each line with its corresponding polynomial expression. ### Detailed Analysis The chart shows four lines, each representing a polynomial model with increasing complexity. * **Line 1 (Gray):** p₁⋅x + p₂⋅v + p₃. This line starts at approximately NMSE = 6.0 x 10^-7 at iteration 0. It decreases steadily until approximately iteration 1500, where it drops sharply to approximately 2.0 x 10^-25. The trend is initially a gentle slope downwards, followed by a near-vertical drop. * **Line 2 (Orange):** +p₃⋅sin(x) + p₆⋅v³ - p₁⋅v + p₂⋅x + p₄⋅cos(x) + p₅⋅v² + p₇⋅x² + p₈⋅v⁴ + p₉⋅x⋅sin(v) + p₁₀. This line begins at approximately NMSE = 2.0 x 10^-8 at iteration 0. It decreases slowly until approximately iteration 1500, where it drops sharply to approximately 2.0 x 10^-25. The trend is a very gradual decline, followed by a steep drop. * **Line 3 (Green):** +p₃⋅sin(x) + p₆⋅v³ + p₂⋅x + p₄⋅cos(v) + p₇⋅sin(v). This line starts at approximately NMSE = 2.0 x 10^-16 at iteration 0. It remains relatively flat until approximately iteration 1500, where it drops to approximately 2.0 x 10^-25. The trend is almost horizontal, then a sharp decline. * **Line 4 (Red):** p₁⋅x³ + p₂⋅x⋅v + p₃⋅v³ + p₄⋅sin(x) + p₅⋅x⋅cos(x). This line begins at approximately NMSE = 2.0 x 10^-25 at iteration 0. It remains flat until approximately iteration 1500, where it drops to approximately 2.0 x 10^-25. The trend is flat. The markers on the gray line are located at approximately: * (0, 6.0 x 10^-7) * (500, 5.0 x 10^-10) * (1000, 2.0 x 10^-12) * (1500, 2.0 x 10^-25) The markers on the orange line are located at approximately: * (0, 2.0 x 10^-8) * (500, 1.0 x 10^-10) * (1000, 2.0 x 10^-12) * (1500, 2.0 x 10^-25) The markers on the green line are located at approximately: * (0, 2.0 x 10^-16) * (500, 2.0 x 10^-16) * (1000, 2.0 x 10^-16) * (1500, 2.0 x 10^-25) The markers on the red line are located at approximately: * (0, 2.0 x 10^-25) * (500, 2.0 x 10^-25) * (1000, 2.0 x 10^-25) * (1500, 2.0 x 10^-25) ### Key Observations * All models converge to a similar NMSE value (approximately 2.0 x 10^-25) after 1500 iterations. * The simplest model (gray line) shows the most significant initial improvement in NMSE. * The more complex models (orange, green, and red lines) start with lower NMSE values and exhibit slower initial improvement. * The sharp drop in NMSE around iteration 1500 suggests a significant change in the optimization process or a convergence point. * The red line remains constant throughout the iterations, indicating it has already reached its optimal performance. ### Interpretation The chart demonstrates the relationship between model complexity, iteration count, and prediction error (NMSE). Initially, increasing model complexity leads to lower error, but beyond a certain point, adding more terms does not significantly improve performance. The convergence around iteration 1500 suggests that the optimization algorithm has reached a stable state for all models. The fact that all models converge to the same NMSE value indicates that the data may not require a highly complex model to achieve good performance. The red line's constant NMSE suggests it may have already reached its maximum representational capacity or is overfitting. This chart is likely used to evaluate the effectiveness of different polynomial models in fitting a given dataset and to determine the optimal level of model complexity. The logarithmic scale on the Y-axis emphasizes the large range of NMSE values and highlights the substantial error reduction achieved by the models.

## Convergence Plot: Model Complexity vs. Normalized Mean Square Error (NMSE) ### Overview The image is a technical plot illustrating the convergence of a model's error (NMSE) over training iterations. It demonstrates how the error decreases in a stepwise fashion as increasingly complex mathematical terms are added to the model. The plot uses a logarithmic scale for the error axis to visualize the dramatic, multi-order-of-magnitude improvements. ### Components/Axes * **Chart Type:** Step-wise convergence plot (line chart). * **Y-Axis:** * **Label:** `NMSE (log scale)` * **Scale:** Logarithmic, ranging from `10^0` (1) at the top to `10^-28` at the bottom. * **Major Ticks:** `10^0`, `10^-4`, `10^-8`, `10^-12`, `10^-16`, `10^-20`, `10^-24`, `10^-28`. * **X-Axis:** * **Label:** `Iterations` * **Scale:** Linear, from `0` to `2500`. * **Major Ticks:** `0`, `500`, `1000`, `1500`, `2000`, `2500`. * **Data Series:** A single, solid red line representing the NMSE over iterations. * **Annotations:** Four text boxes with mathematical expressions, connected by gray arrows to specific points on the red line. The text boxes use colored backgrounds (light green and light pink) to highlight different terms. ### Detailed Analysis The red line shows a step-function decrease in NMSE. The error remains constant for long periods (plateaus) and then drops sharply at specific iteration points. Each drop is associated with the introduction of a new, more complex set of model terms, as indicated by the annotations. **1. Initial State (Iteration ~0):** * **Position:** Top-left corner of the plot. * **NMSE Value:** Approximately `10^0` (or 1). * **Annotation (Light Pink Background):** `p₁·x + p₂·v + p₃` * **Trend Description:** The line starts at a high error value. **2. First Major Drop (Iteration ~0-10):** * **Position:** The line drops vertically from `10^0` to approximately `10^-6`. * **NMSE Value:** Plateaus at ~`10^-6`. * **Annotation (Mixed Green/Pink Background):** * Green Terms: `+p₃·sin(x)`, `+p₆·v³`, `+p₅·v²`, `+p₇·x²`, `+p₈·v⁴`, `+p₉·x·sin(v)` * Pink Terms: `-p₁·v`, `+p₂·x`, `+p₄·cos(x)`, `+p₁₀` * **Trend Description:** A dramatic, near-vertical drop in error, followed by a long plateau. **3. Second Major Drop (Iteration ~1500):** * **Position:** The line drops vertically from ~`10^-6` to ~`10^-12`. * **NMSE Value:** Plateaus at ~`10^-12`. * **Annotation (Mixed Green/Pink Background):** * Green Terms: `+p₃·sin(x)`, `+p₅·v³`, `+p₆·x·v`, `+p₈·x³` * Pink Terms: `p₁·x`, `+p₂·v`, `+p₄·cos(v)`, `+p₇·sin(v)` * **Trend Description:** Another sharp, vertical drop in error, followed by a second long plateau. **4. Final Drop (Iteration ~1550):** * **Position:** The line drops vertically from ~`10^-12` to a value below `10^-28`. * **NMSE Value:** Plateaus at a very low value, appearing to be at or below `10^-30` (the axis ends at `10^-28`). * **Annotation (Light Green Background):** `p₁·x³ + p₂·x·v + p₃·v³ + p₄·sin(x) + p₅·x·cos(x)` * **Trend Description:** The final and most significant drop, leading to an extremely low, stable error. ### Key Observations 1. **Stepwise Convergence:** The model does not improve continuously. Error reduction occurs in discrete, massive jumps, each coinciding with a change in the model's functional form (the added terms). 2. **Magnitude of Improvement:** The total improvement in NMSE is extraordinary, spanning over 30 orders of magnitude (from `10^0` to `<10^-30`). 3. **Plateau Length:** The plateaus between drops are of similar length (~1500 iterations), suggesting a consistent training period is needed before the next model complexity level yields improvement. 4. **Annotation Color Coding:** The green highlighting in the annotations likely signifies the *newly added* terms at that stage, while pink may indicate terms carried over or modified from a previous stage. This is an inferred pattern based on visual grouping. ### Interpretation This plot is a powerful visualization of **model selection and symbolic regression**. It demonstrates the process of discovering an accurate mathematical model for a system. * **What it Suggests:** The data shows that a simple linear model (`p₁·x + p₂·v + p₃`) is a poor fit (high NMSE). Introducing non-linear and trigonometric terms (`sin`, `cos`, polynomials) in stages leads to exponentially better fits. The final model, a combination of cubic terms and trigonometric functions, achieves near-perfect accuracy (extremely low NMSE). * **Relationship Between Elements:** The x-axis (Iterations) represents the computational effort or search process. The y-axis (NMSE) is the objective measure of model quality. The annotations are the "discoveries" made during the search. The stepwise drops indicate moments where the algorithm found a significantly better model structure. * **Notable Anomalies/Trends:** The most striking trend is the **log-linear relationship between model complexity and error reduction**. Each major increase in the complexity of the term set (from linear, to including squares/cubes/trig, to a refined final set) results in a multiplicative (order-of-magnitude) decrease in error. This is characteristic of successful symbolic regression, where the correct functional form is more important than incremental parameter tuning. * **Underlying Message:** The plot argues that for this particular problem, the underlying physical or mathematical law is complex but discoverable. It highlights the efficiency of a search algorithm that can navigate a vast space of possible equations to find a parsimonious and highly accurate solution. The final model is not the most complex one shown (the second annotation has more terms), but it is the most effective, suggesting the algorithm also performs model simplification or finds a more fundamental representation.

## Line Graph: NMSE vs. Iterations ### Overview The image is a line graph depicting the relationship between the Normalized Mean Squared Error (NMSE) and the number of iterations. The y-axis is logarithmic (ranging from 10⁻²⁸ to 10⁰), and the x-axis represents iterations (0 to 2500). Multiple data series are plotted, each associated with distinct mathematical expressions involving variables like `p1`, `p2`, `x`, `v`, and trigonometric functions (e.g., `sin(x)`, `cos(x)`). The graph shows exponential decay in NMSE as iterations increase, with distinct trends for different series. --- ### Components/Axes - **Y-axis**: "NMSE (log scale)" with values from 10⁻²⁸ to 10⁰ (logarithmic scale). - **X-axis**: "Iterations" with values from 0 to 2500. - **Legend**: Located on the right side of the graph, with color-coded labels for each data series. Colors include red, gray, and green, each corresponding to specific mathematical expressions. - **Data Series**: - **Red Line**: `p1·x + p2·v + p3` (initial NMSE ~10⁰, drops to ~10⁻⁴). - **Gray Lines**: - `+p3·sin(x) + p6·v³ - p1·v + p2·x + p4·cos(x)` (NMSE ~10⁻⁴ to ~10⁻⁸). - `+p5·v² + p7·x² + p8·v⁴ + p9·x·sin(v) + p10` (NMSE ~10⁻⁸ to ~10⁻¹²). - `+p3·sin(x) + p5·v³ + p6·x·v + p8·x³ + p1·x` (NMSE ~10⁻¹² to ~10⁻¹⁶). - `+p2·v + p4·cos(v) + p7·sin(v)` (NMSE ~10⁻¹⁶ to ~10⁻²⁰). - **Green Lines**: - `+p1·x³ + p2·x·v + p3·v³ + p4·sin(x) + p5·x·cos(x)` (NMSE ~10⁻²⁰ to ~10⁻²⁸). --- ### Detailed Analysis 1. **Red Line (`p1·x + p2·v + p3`)**: - Starts at 10⁰ (iteration 0) and drops sharply to 10⁻⁴ by iteration 500. - Remains flat at 10⁻⁴ until iteration 1500, then continues to decline slightly. 2. **Gray Lines**: - **First Gray Line**: - Starts at 10⁻⁴ (iteration 500) and drops to 10⁻⁸ by iteration 1000. - Remains flat at 10⁻⁸ until iteration 1500. - **Second Gray Line**: - Starts at 10⁻⁸ (iteration 1000) and drops to 10⁻¹² by iteration 1500. - **Third Gray Line**: - Starts at 10⁻¹² (iteration 1500) and drops to 10⁻¹⁶ by iteration 2000. - **Fourth Gray Line**: - Starts at 10⁻¹⁶ (iteration 2000) and drops to 10⁻²⁰ by iteration 2500. 3. **Green Lines**: - **First Green Line**: - Starts at 10⁻²⁰ (iteration 2000) and drops to 10⁻²⁸ by iteration 2500. --- ### Key Observations - **Exponential Decay**: All data series show a logarithmic decrease in NMSE as iterations increase, indicating rapid convergence. - **Complexity vs. Performance**: Series with higher-order terms (e.g., `x³`, `v⁴`, `sin(x)`) achieve lower NMSE values, suggesting that more complex models improve accuracy faster. - **Color Consistency**: The legend matches the colors of the data series (red, gray, green) accurately. - **Flat Plateaus**: Some series exhibit flat regions (e.g., red line at 10⁻⁴), indicating periods of minimal improvement. --- ### Interpretation The graph demonstrates that increasing the number of iterations significantly reduces NMSE, reflecting improved model performance. The use of complex mathematical expressions (e.g., trigonometric functions, higher powers) correlates with faster convergence, as these terms likely capture more nuanced patterns in the data. The red line (simplest model) shows the slowest improvement, while green lines (most complex) achieve the lowest NMSE. This suggests that model complexity and iteration count are critical factors in optimizing accuracy. The flat plateaus may indicate saturation points where further iterations yield diminishing returns.