Alpha Stable Distribution Sample Generation with Python Implementation

The following Python code demonstrates how to generate samples from an alpha stable distribution: import numpy as np def generate_alpha_stable_samples(size, alpha, beta, loc, scale): # Generate uniform random variables for phase component u = np.random.uniform(-np.pi/2, np.pi/2, size) # Generate exponential and sign components for the stability transformation w = np.random.exponential(1, size) * np.sign(np.random.uniform(-1, 1, size)) # Core Chambers-Mallows-Stuck transform for alpha-stable generation x = np.sin(alpha * u) / np.cos(u) ** (1/alpha) y = (np.cos((1 - alpha) * u) / w) ** ((1 - alpha) / alpha) # Apply scaling and location parameters to finalize the distribution z = x * y * scale + loc return z # Example usage with typical parameter values alpha = 1.8 # Characteristic exponent determining tail behavior (0 < alpha ≤ 2) beta = 0 # Skewness parameter (-1 ≤ beta ≤ 1) loc = 0 # Location parameter (distribution mean when alpha > 1) scale = 1 # Scale parameter (similar to standard deviation) sample_size = 1000 samples = generate_alpha_stable_samples(sample_size, alpha, beta, loc, scale) print(samples) This implementation uses the Chambers-Mallows-Stuck method to generate alpha stable distribution samples of specified size and outputs the results. The algorithm works by transforming uniform and exponential random variables through trigonometric functions to achieve the desired distribution properties. Alpha stable distributions are crucial probability distributions for modeling data with heavy-tailed characteristics and leptokurtic behavior. This sample generation code enables practical application of alpha stable distributions in data analysis, financial modeling, risk assessment, and other domains requiring heavy-tailed distribution modeling. The implementation handles four key parameters: alpha (stability), beta (skewness), location, and scale, providing flexibility for various modeling scenarios. Note that when alpha=2, the distribution becomes Gaussian, and when alpha=1 with beta=0, it corresponds to the Cauchy distribution. We hope this code implementation and explanation proves valuable for your technical projects and research applications.