Generalized Box-Muller method for generating q-Gaussian random deviates

TL;DR

This paper introduces a generalized Box-Muller algorithm for efficiently generating q-Gaussian random variates across a broad range of q values, enhancing simulation capabilities in nonextensive statistical mechanics.

Contribution

The authors develop a simple, versatile numerical method for generating q-Gaussian deviates for q<3, extending previous methods and including MATLAB implementation.

Findings

The method can generate q-Gaussian deviates for a wider q range than previous algorithms.

It accommodates arbitrary width and center of the distribution.

The approach clarifies the transformation related to the distribution's dimensionality.

Abstract

Addendum: The generalized Box-M\"uller algorithm provides a methodology for generating q-Gaussian random variates. The parameter $-\infty<q\leq3$ is related to the shape of the tail decay; $q<1$ for compact-support including parabola $(q=0)$; $1<q\leq3$ for heavy-tail including Cauchy $(q=2)$. This addendum clarifies the transformation $q'=((3q-1)/(q+1))$ within the algorithm is due to a difference in the dimensions d of the generalized logarithm and the generalized distribution. The transformation is clarified by the decomposition of $q=1+2\kappa/(1+d\kappa)$, where the shape parameter $-1<\kappa\leq\infty$ quantifies the magnitude of the deformation from exponential. A simpler specification for the generalized Box- M\"uller algorithm is provided using the shape of the tail decay. Original: The q-Gaussian distribution is known to be an attractor of certain correlated systems, and is the distribution which, under appropriate constraints, maximizes the entropy Sq, basis of nonextensive statistical mechanics. This theory is postulated as a natural extension of the standard (Boltzmann-Gibbs) statistical mechanics, and may explain the ubiquitous appearance of heavy-tailed distributions in both natural and man-made systems. The q-Gaussian distribution is also used as a numerical tool, for example as a visiting distribution in Generalized Simulated Annealing. We develop and present a simple, easy to implement numerical method for generating random deviates from a q-Gaussian distribution based upon a generalization of the well known Box-Muller method. Our method is suitable for a larger range of q values, q<3, than has previously appeared in the literature, and can generate deviates from q-Gaussian distributions of arbitrary width and center. MATLAB code showing a straightforward implementation is also included.