🌌 MixStable — Alpha-Stable Mixture Estimation Package

A comprehensive Python package for simulating, estimating, and visualizing alpha-stable mixture distributions.
This toolkit is designed for statisticians, data scientists, and researchers working with heavy-tailed or skewed data, where Gaussian models fail to capture real-world complexity.

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Key Features

Robust Estimation Methods

• Empirical Characteristic Function (ECF) estimators: kernel-based and weighted OLS
• Maximum Likelihood Estimation (MLE)
• Quantile-based and CDF-based approaches

Mixture Modeling with EM

• EM algorithm for two-component alpha-stable mixtures
• Flexible estimator choice inside the EM loop
• Optional integration with Gibbs sampling for Bayesian refinement

Visualization & Simulation

• Generate synthetic alpha-stable data
• Built-in visualization tools and an interactive Streamlit dashboard

R Integration (via rpy2)

• Uses R’s stabledist package to evaluate stable densities and CDFs
• Leverage mature R statistical methods directly from Python

Testing & Evaluation

• Built-in diagnostics and model-fit tools
• Evaluation metrics and example datasets

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Installation

Install directly from PyPI:

pip install Mixstable

Or clone the repository and install in editable mode:

git clone https://github.com/AdamNajib/MixStable.git
cd src
pip install -e .

Install the required R package (for rpy2 integration):

install.packages("stabledist")

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Running the Interactive Dashboard

To launch the Streamlit app for interactive parameter tuning and visualization:

streamlit run interface/app.py

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Usage Example

Estimate parameters from synthetic data:

from mixstable import generate_sample, em

# Generate alpha-stable samples
samples = generate_sample.generate_alpha_stable(alpha=1.7, beta=0, gamma=1, delta=0, size=1000)

# Fit a two-component mixture model using EM
result = em.run_em_algorithm(samples, num_components=2, max_iter=50)
print(result['params'])

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Requirements

• Python ≥ 3.8
• R (if using r_interface)
• stabledist R package

Python dependencies (auto-installed from requirements.txt):

• numpy, scipy, matplotlib, seaborn , statsmodels
• rpy2, pandas, tqdm , scikit-learn , streamlit

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Project Structure

Mixstable/
│
├── src/   
│   ├── Mixstable/                    # Core modules
│   │   └── em.py, gibbs.py, ecf.py, ...
│   ├── interface/               # Streamlit dashboard
│   └── tests/                   # Test scripts
│
├── README.md
├── setup.py
├── pyproject.toml
└── requirements.txt

Package Overview

Module	Description
em.py	Expectation-Maximization algorithm implementation
gibbs.py	Gibbs sampling routines for Bayesian inference
ecf.py	Empirical characteristic function estimators
generate_sample.py	Alpha-stable data generation tools
interface/	Streamlit dashboard and helper scripts
tests/	Unit and integration tests

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Example Output

Example EM fit visualization (from Streamlit app):

Iteration 50/50
Component 1: α = 1.72, β = 0.05, γ = 1.1, δ = 0.02
Component 2: α = 1.54, β = -0.10, γ = 0.8, δ = 2.4
Log-likelihood: -1856.42

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Theoretical Background

Alpha-stable distributions generalize the Gaussian family to model heavy-tailed and skewed data. They are defined by four parameters:

Parameter	Symbol	Meaning
Stability	α	Controls tail thickness (0 < α ≤ 2)
Skewness	β	Controls asymmetry (−1 ≤ β ≤ 1)
Scale	γ	Controls spread (> 0)
Location	δ	Controls center (real number)

This package provides both simulation tools and inference routines for mixtures of alpha-stable laws, extending beyond Gaussian mixtures.

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References

Hajjaji, O., Manou-Abi, S. M., & Slaoui, Y. (2024). Parameter estimation for stable distributions and their mixture. Journal of Applied Statistics, 52(8), 1594–1627. https://doi.org/10.1080/02664763.2024.2434627

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Authors

Solym Manou-Abi Email: [solym.manou.abi@univ-poitiers.fr]

Adam Najib Email: [najibadam145@gmail.com] GitHub: @AdamNajib

Yousri Slaoui Email: [Yousri.Slaoui@math.univ-poitiers.fr]

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