  The origin of the low-frequency noise with power spectrum $1/f^\beta$ (also
known as $1/f$ fluctuations or flicker noise) remains a challenge. Recently,
the nonlinear stochastic differential equations for modeling $1/f^\beta$ 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 $1/f^\beta$ 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
$1/f^\beta$ fluctuations is presented.
