Skip to content

README.md

Latest commit

 

History

History
389 lines (272 loc) · 16.3 KB

README.md

File metadata and controls

389 lines (272 loc) · 16.3 KB

🧭Random Walk Simulations in 1D, 2D and 3D

License: MIT Python 3.8+ Jupyter Notebook Matplotlib Plotly

A comprehensive collection of random walk simulations in 1D, 2D, and 3D with interactive visualizations, animations, and statistical analysis.

1D Random Walk 2D Random Walk 3D Random Walk

📚 Table of Contents

📘 Introduction

A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers. This notebook provides a hands-on approach to understanding random walks by:

  • Simulating paths in 1D, 2D, and 3D.
  • Analyzing statistical properties like mean displacement and standard deviation.
  • Visualizing trajectories and distributions using matplotlib and plotly.
  • Creating interactive animations to observe the walk's evolution over time.
  • Exploring the effects of bias on the walker's path.

✨ Features

  • Multi-Dimensional Simulations: Explore random walks in 1D, 2D, and 3D space.
  • Statistical Analysis: Calculate and visualize mean position, standard deviation, and final position distributions over multiple trials.
  • Biased Walks: Simulate and analyze random walks where steps in certain directions are more probable.
  • Interactive Visualizations: Utilize plotly and ipywidgets for dynamic plots.
  • Animations: Generate .mp4 or .gif animations of random walks using matplotlib.animation.
  • Quadrant Analysis (2D): Track the walker's time in each quadrant.
  • Step Size Comparison (2D): Analyze the effect of different step sizes.
  • Simultaneous Walks (3D): Visualize multiple walkers starting from different positions.

📁 Folder Structure

.
├── 🎞️ ANIMATION/
│   ├── 🎞️ 1D/         # Animations for 1D walks (.mp4, .gif)
│   ├── 🎞️ 2D/         # Animations for 2D walks (.mp4, .gif)
│   └── 🎞️ 3D/         # Animations for 3D walks (.mp4, .gif)
├── 🖼️ IMAGES/
│   ├── 🖼️ 1D/         # Static plots for 1D walks (.png, .jpg)
│   ├── 🖼️ 2D/         # Static plots for 2D walks (.png, .jpg)
│   └── 🖼️ 3D/         # Static plots for 3D walks (.png, .jpg)
├── 📊 STATISTICS/     # Optional: CSV or other files with statistical results
├── 🧪 RandomWalk.ipynb  # The main Jupyter Notebook file
├── 📦 requirements.txt # List of required Python packages
└── 📄 README.md       # This file

🛠️ Setup and Installation

You can follow these steps to set up your environment and run the notebook.

✅ Prerequisites

  • Python 3.7 or higher
  • pip (Python package installer)
  • git (for cloning the repository)

🧬 Cloning the Repository

git clone https://github.com/PuspenduPH/Random-Walk-Simulations
cd Random-Walk-Simulations

🧰 Setting Up a Virtual Environment (Recommended)

Using a virtual environment prevents dependency conflicts.

  1. Create a virtual environment (e.g., named randomwalk_env): 
    python -m venv randomwalk_env
  2. Activate the environment:
    • Windows: 
      randomwalk_env\Scripts\activate
    • Linux/macOS: 
      source randomwalk_env/bin/activate
  3. (Optional) Upgrade pip: 
    python -m pip install --upgrade pip

📦 Installing Dependencies

Install all required Python libraries using the requirements.txt file:

pip install -r requirements.txt

Alternatively, install them manually:

pip install numpy scipy scikit-learn matplotlib plotly ipywidgets pandas tqdm ipympl

Note: FFmpeg is necessary to save animations in .mp4 format and must be installed manually (see Important Notes below).

🚀 Usage

▶️ Running the Notebook

  1. Ensure your virtual environment is activated.
  2. Navigate to the repository directory in your terminal.
  3. Launch Jupyter Notebook or JupyterLab: 
    jupyter notebook
# OR
jupyter lab
  4. Open the random_walk_notebook.ipynb (or your actual notebook file name) in the Jupyter interface.
  5. Run the cells sequentially or explore specific sections.

⚠️ Important Notes

This notebook contains interactive visualizations, animations, and data manipulation tools. Follow these instructions to avoid common errors and ensure smooth execution:

  • Animation in Jupyter:

    • Install the ipympl package with: 
      pip install ipympl
    • Use the following magic commands at the top of your notebook: 
      %matplotlib ipympl  # For displaying animations
%matplotlib widget  # For interactive widgets
    • Tip: Use only one magic command in a cell, depending on your requirements. If you need any help installing ipympl yo can follow this link

  • Saving Animations as .mp4:

    • Install FFmpeg:
      • Windows:
        • Download FFmpeg from ffmpeg.org.
        • Extract and add the bin folder to your system's PATH.
      • macOS/Linux: 
        brew install ffmpeg     # For macOS with Homebrew
sudo apt install ffmpeg # For Ubuntu/Debian

  • General Tips:

    • Use FuncAnimation from matplotlib.animation for creating animations.
    • Utilize PillowWriter if you prefer saving animations as .gif.
    • Use tqdm for progress tracking in loops or simulations.
    • Suppress unnecessary warnings: 
      import warnings
warnings.filterwarnings("ignore")
    • Use Python’s traceback module for inline error logging during debugging.

Reminder: Always restart the kernel after switching %matplotlib backends or installing new Jupyter-related packages to avoid rendering issues.

📓 Notebook Content Overview

The notebook is divided into sections exploring random walks in increasing dimensions:

1. One-Dimensional Random Walk

  • Simulation: Basic simulation of a single 1D random walk.

  • Analysis: Calculating displacement and distance from the origin.

  • Multiple Trials: Running simulations multiple times to observe average behaviour.

  • Biased Random Walk: Introducing probabilities for moving left or right.

Probability p=0.7 Probability p=0.3
1DRandomWalk_trials_p_0.7 1DRandomWalk_trials_p_0.3

  • Mean and Standard Deviation Analysis: Examining how mean position and standard deviation evolve.

  • Distribution of Walker's Position: Visualizing the probability distribution of the walker's position after N steps.

  • Animation: Animating the 1D random walk.

  • Animation with Position Distribution: Animating the walk alongside its evolving position distribution.

    rdwalk_1D_dist_s50_t40_p0.5.mp4

2. Two-Dimensional Random Walk

  • Simulation & Plotting: Simulating and plotting a 2D walk using matplotlib and plotly.

  • Contour Plot: Visualizing the density of visited locations.

  • Quadrant Analysis: Tracking the proportion of time spent in each quadrant.

  • Quadrant Analysis with Animation: Animating the walk and the evolving quadrant distribution.

  • Analysis with Contour Plot and Heatmap: Combined visualisation of path and visit frequency.

  • Distribution of Walker's Position: Analyzing the 2D distribution of positions over multiple trials.

  • Distribution of Final Position: Visualising where walkers end up after N steps.

  • Animation: Basic animation of the 2D random walk path.

  • Animation with Position Frequency: Animating the walk alongside a heatmap of visited positions.

  • Step Size Comparison: Analysing walks with different step lengths.

  • 2D Biased Random Walk: Implementing bias in movement directions (e.g., tendency to move North-East).

    2D Random Walk with Angular Bias 2D Random Walk with Target Bias

  • Comparison with/without Bias: Visual comparison of biased vs. unbiased walks.

  • Biased Walk with Different Probabilities: Exploring various bias configurations.

    rdwalk_2D_with_counts_s100_t20_p0.4_q0.1_r0.25_s0.25.mp4

3. Three-Dimensional Random Walk

  • Simulation (Single Trial): Simulating and visualizing a single walk in 3D space using matplotlib or plotly.

    random_walk_3d_s100.mp4

  • Simulation (Multiple Trials): Running and analyzing multiple 3D walks.

    random_walk_3d_multi_s50_t20_1.1.mp4

  • Multiple Trials with Different Initial Positions: Simulating walkers starting from various points.

    random_ip_walk_3d_multi_s50_t20.mp4

  • Distribution of Walker's Final Position: Visualizing the 3D distribution of endpoints.

  • Simultaneous Random Walk: Animating multiple walks concurrently.

  • Biased Random Walk: Introducing directional bias in 3D steps.

    • 3D Random Walk with Target Bias

      distance_biased_rdwalk_3d_s50_t15.mp4

    • 3D Random Walk with Angular Bias

      angular_biased_rdwalk_3d_s50_t15.mp4

  • Visualise Biased Walks with IpyWidgets: Using widgets to explore different bias parameters interactively.

  • Comparison with/without Bias: Comparing trajectories of biased and unbiased 3D walks.

    final_walk_comparison_3d_s50_t15.mp4

🎞️ Animation & Visual Assets

Animations and visual outputs are organized into separate directories by dimensionality:

  • ANIMATION/1D/ – Animations for 1D random walk.
  • ANIMATION/2D/ – Animations for 2D random walk.
  • ANIMATION/3D/ – Animations for 3D random walk.
  • IMAGES/1D/, IMAGES/2D/, IMAGES/3D/ – Supporting images and figures.

Please add, update, or replace assets as your experiments change.

📊 Statistics and Analysis

The STATISTICS folder is intended for:

  • Storing detailed statistical reports and datasets related to random walk experiments.
  • Tracking parameters such as mean, variance, skewness, and other performance metrics of the simulations.

This modular organization makes maintaining a historical record of simulation outcomes and analysis easier.

🧩 Interactive Features

The notebook includes several interactive components:

  • Adjustable parameters for walk length, bias, and step size
  • Real-time visualization updates
  • Animated simulations with playback controls
  • Parameter exploration with ipywidgets
  • Statistical analysis with immediate feedback

🔬 Advanced Analysis

Statistical investigations included in the notebook:

  • Mean square displacement analysis
  • First return times
  • Probability distributions
  • Central limit theorem demonstrations
  • Quadrant occupation statistics
  • Bias effects on displacement

🔗 References

🤝 Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

  1. Fork the repository.
  2. Create your feature branch (git checkout -b amazing-feature).
  3. Commit your changes (git commit -m 'Add some amazing feature').
  4. Push to the branch (git push origin amazing-feature).
  5. Open a Pull Request.

📄 License

Distributed under the MIT License. See LICENSE for more information.