1/f noise from the nonlinear transformations of the variables

The origin of the low-frequency noise with power spectrum $1\mathrm{/}{f}^{\beta }$ (also known as $1\mathrm{/}f$ fluctuations or flicker noise) remains a challenge. Recently, the nonlinear stochastic differential equations for modeling $1\mathrm{/}{f}^{\beta }$ noise have been proposed and analyzed. Here we use the self-similarity properties of this model with respect to the nonlinear transformations of the variable of these equations and show that $1\mathrm{/}{f}^{\beta }$ noise of the observable may yield from the power-law transformations of well-known standard processes, like the Brownian motion, Bessel and similar stochastic processes. Analytical and numerical investigations of such techniques for modeling processes with $1\mathrm{/}{f}^{\beta }$ fluctuations is presented.