qNoise: A generator of non-Gaussian colored noise

J Ignacio Deza, Roberto R. Deza

Published 2018-11-28Version 1

We introduce a software generator for a class of \emph{colored} (self-correlated) and \emph{non-Gaussian} noise, whose statistics and spectrum depend upon only two parameters, $q$ and $\tau$. Inspired by Tsallis' nonextensive formulation of statistical physics, this so-called $q$-distribution is a handy source of self-correlated noise for a large variety of applications. The $q$-noise---which tends smoothly for $q=1$ to Ornstein--Uhlenbeck noise with autocorrelation $\tau$---is generated via a stochastic differential equation, using the Heun method (a second order Runge--Kutta type integration scheme). The algorithm is implemented as a stand-alone library in \texttt{c++}, available as open source in the \texttt{Github} repository. The noise's statistics can be chosen at will, by varying only parameter $q$: it has compact support for $q<1$ (sub-Gaussian regime) and finite variance up to $q=5/3$ (supra-Gaussian regime). Once $q$ has been fixed, the noise's autocorrelation can be tuned up independently by means of parameter $\tau$. This software provides a tool for modeling a large variety of real-world noise types, and is suitable to study the effects of correlation and deviations from the normal distribution in systems of stochastic differential equations which may be relevant for a wide variety of technological applications, as well as for the understanding of situations of biological interest. Applications illustrating how the noise statistics affects the response of a variety of nonlinear systems are briefly discussed. In many of these examples, the system's response turns out to be optimal for some $q\neq1$.