Docs: Add anisotropic GMM example

[henry2004y]

Adds an example of GMM fitting for anisotropic data to the AMReX documentation notebook.

This new section includes:

• Generation of synthetic anisotropic data.
• Fitting a GMM to the data.
• Extracting and verifying the physical parameters.
• Visualization of the results.

[gemini-code-assist]

Summary of Changes

Hello @henry2004y, I'm Gemini Code Assist¹! I'm currently reviewing this pull request and will post my feedback shortly. In the meantime, here's a summary to help you and other reviewers quickly get up to speed!

This pull request significantly enhances the AMReX documentation by introducing a practical example of applying Gaussian Mixture Models (GMM) to anisotropic data. The addition guides users through the entire process, from generating synthetic anisotropic datasets to fitting GMMs, extracting specific physical parameters like parallel and perpendicular temperatures, and visualizing the results. This new content aims to provide a clearer understanding and practical application of GMMs for complex, directionally dependent data distributions.

Highlights

• Anisotropic GMM Example Added: A new section has been integrated into the AMReX documentation notebook, providing a comprehensive example of Gaussian Mixture Model (GMM) fitting for anisotropic data.
• Synthetic Data Generation: The example includes the generation of synthetic anisotropic data, simulating distributions where properties vary across different directions, relevant for fields like plasma physics.
• GMM Fitting and Parameter Extraction: It demonstrates the process of fitting a GMM to the generated anisotropic data and accurately extracting physical parameters, such as parallel and perpendicular temperatures.
• Visualization of Results: The new section features clear visualizations of the anisotropic data, overlaid with GMM fit ellipses that represent the fitted components and their standard deviation contours.

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 68.31%. Comparing base (8e4373d) to head (ecdb77d).
⚠️ Report is 1 commits behind head on main.

Additional details and impacted files

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  Lines        4191     4191           
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  Misses       1328     1328           

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[gemini-code-assist]

Code Review

This pull request adds a helpful example for anisotropic GMM fitting to the documentation notebook. The new section is well-structured with a clear explanation, code for data generation and fitting, and visualization of the results.

My review includes a critical fix for a KeyError that would prevent the example code from running, and a suggestion to remove a redundant import for better code hygiene. After addressing these points, the example will be a great addition to the documentation.

[gemini-code-assist]

The keys used to access the temperature parameters are incorrect. The get_gmm_parameters function returns dictionaries with keys "T_parallel" and "T_perpendicular" (note the capital 'T'). Using "temp_parallel" and "temp_perp" will cause a KeyError when the code is executed.

I've corrected the keys and also used more descriptive variable names for the extracted temperatures to improve clarity.

    temp_par_ext = round(params[\"T_parallel\"], 5)
    temp_per_ext = round(params[\"T_perpendicular\"], 5)
    print(f\"Component {i+1}: Center = {center}, Temp Parallel = {temp_par_ext}, Temp Perp = {temp_per_ext}\")

[gemini-code-assist]

The Ellipse class is already imported in a previous cell of this notebook (at line 708). To avoid redundancy and keep the notebook clean, this import statement and the following blank line can be removed.