AI Research & Development Assignment — Parametric Curve Estimation

📌 Problem Statement

Estimate the unknown parameters θ, M, X in the parametric equations:

[ x(t) = t \cos(\theta) - e^{M|t|}\sin(0.3t)\sin(\theta) + X ]

[ y(t) = 42 + t \sin(\theta) + e^{M|t|}\sin(0.3t)\cos(\theta) ]

Constraints:

• (0^\circ < \theta < 50^\circ)
• (-0.05 < M < 0.05)
• (0 < X < 100)
• (6 < t < 60)

Dataset: xy_data.csv contains observed ((x,y)) points sampled from the curve.

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🧠 Approach

1. Data Preparation

   • Loaded xy_data.csv containing observed ((x,y)) values.
   • Generated a uniform parameter (t) array over ([6,60]) with the same length as the dataset.

2. Model Definition

   • Implemented the parametric equations in Python.
   • Converted θ from degrees to radians during computation.

3. Loss Function

   • Defined an L1 loss (sum of absolute differences) between observed and predicted points:

[ L(\theta, M, X) = \sum_i \left(|x_i - \hat{x}(t_i)| + |y_i - \hat{y}(t_i)|\right) ]

• L1 was chosen for robustness against outliers.

4. Optimization

   • Used Differential Evolution for global search within bounds.
   • Refined solution with Powell’s method for local minimization.

5. Evaluation

   • Reported total L1 loss, median residual, and 95th percentile residual.
   • Visualized observed vs fitted curve.

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📊 Results

Estimated Parameters:

• θ ≈ 0.826000 degrees
• M ≈ -0.030000
• X ≈ 11.579300

Final Submission Equation:

[ \left(t \cos(0.826000) - e^{-0.030000|t|}\sin(0.3t)\sin(0.826000) + 11.579300,\ 42 + t \sin(0.826000) + e^{-0.030000|t|}\sin(0.3t)\cos(0.826000)\right) ]

You can also view this equation plotted on Desmos.

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📈 Visualization

Observed data (blue points) vs fitted curve (red line):

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✨ Enhancements Beyond Requirements

To make this assignment stand out:

• ✅ Added scatter plot and saved it for README embedding.
• ✅ Compared L1 vs L2 loss and explained robustness.
• ✅ Modularized code with docstrings and comments.
• ✅ Provided both .py and .ipynb versions for reproducibility.
• ✅ Linked to external profiles for professional presence.
• ✅ Explained challenges (missing t, optimization stability) and how they were solved.

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👨‍💻 Author

Anuj Mundu

• MERN Stack Developer | ML/DL Enthusiast
• Focused on deploying robust dashboards and modular ML apps
• LinkedIn | GitHub | LeetCode