Quaternion Generator and Analysis

This project generates indeterministic quaternion complex numbers in the form:

[ Q = ah + bi + cj + dk ]

where ( h, i, j, k ) are quaternion basis vectors, and the components ( a, b, c, d ) are drawn from independent Gaussian distributions. The generator is implemented as a Linux /dev/quaternion device driver, and a suite of Python scripts provides statistical analysis and visualizations.

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Features

1. Quaternion Generator (/dev interface)

• Generates quaternions with independent Gaussian-distributed components.
• Components range between (-16) and (+16), scaled from fixed-point arithmetic.
• Output is streamed via /dev/quaternion.

2. Statistical Analysis

Python scripts analyze the generated quaternions:

• Multivariate Gaussian test: Verifies the data's Gaussian distribution.
• Mahalanobis distance analysis: Compares Mahalanobis distances to the theoretical chi-squared distribution.
• Variance and correlation analysis: Measures variances and correlations between components.

3. Visualizations

• 3D Scatter Plot: Visualizes quaternion components in a 3D space.
• Pairwise Scatter Plot: Pairwise relationships and distributions of components.
• Mahalanobis Histogram: Compares Mahalanobis distances with the chi-squared distribution.

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Installation

Prerequisites

1. Linux Kernel Development Tools

   sudo apt install build-essential linux-headers-$(uname -r)

2. Python Environment
   • Install Python 3.8+
   • Create a virtual environment:

     python3 -m venv myenv
     source myenv/bin/activate

   • Install required Python libraries:

     pip install numpy matplotlib seaborn

Building the Kernel Module

1. Clone the repository:

   git clone https://github.com/Qubitdyne/dev_quaternion.git
   cd quaternion-generator

2. Build and load the kernel module:

   make
   sudo insmod quaternion.ko

3. Verify the device:

   ls /dev/quaternion

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Usage

1. Generate Quaternion Data

Output quaternions to a file:

sudo cat /dev/quaternion | head -n 10000000 > quaternions.txt

2. Analyze Data

Run the provided Python scripts for analysis and visualization:

Multivariate Gaussian Test

python3 multivariate_gtest.py

Example output:

Mean: [0.03224541 0.02656311 0.02722455 0.02464791]
Covariance Matrix:
 [[ 8.53811937e+01 -1.12989828e-02 -7.35170146e-03  3.54985011e-02]
 [-1.12989828e-02  8.53231369e+01 -1.99751844e-02  5.74578123e-03]
 [-7.35170146e-03 -1.99751844e-02  8.53506152e+01 -3.93849926e-02]
 [ 3.54985011e-02  5.74578123e-03 -3.93849926e-02  8.53545864e+01]]

Mahalanobis Analysis

python3 mahalanobis_analysis.py

Example output:

Mean Mahalanobis Distance: 1.943155791389275
Standard Deviation of Mahalanobis Distances: 0.47343972202782014
KS Statistic: 0.5698722823418253
P-Value: 0.0

Variance Analysis

python3 variance_analysis.py

Example output:

Variances of Components (h, i, j, k): [85.38118516 85.3231284  85.35060664 85.35457789]
Correlation Matrix:
 [[ 1.00000000e+00 -1.32380747e-04 -8.61198815e-05  4.15829670e-04]
 [-1.32380747e-04  1.00000000e+00 -2.34074477e-04  6.73290131e-05]
 [-8.61198815e-05 -2.34074477e-04  1.00000000e+00 -4.61438694e-04]
 [ 4.15829670e-04  6.73290131e-05 -4.61438694e-04  1.00000000e+00]]

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Visualizations

3D Scatter Plot

Pairwise Scatter Plot

Mahalanobis Histogram

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License

This project is licensed under the MIT License. See LICENSE for details.

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Contributing

Contributions are welcome! Please fork the repository and submit a pull request.

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Acknowledgments

This project was inspired by exploring the relationship between randomness, Gaussian distributions, and quaternion mathematics in computational systems.