  We investigate how various linear and nonlinear transformations affect the
scaling properties of a signal, using the detrended fluctuation analysis (DFA).
Specifically, we study the effect of three types of transforms: linear,
nonlinear polynomial and logarithmic filters. We compare the scaling properties
of signals before and after the transform. We find that linear filters do not
change the correlation properties, while the effect of nonlinear polynomial and
logarithmic filters strongly depends on (a) the strength of correlations in the
original signal, (b) the power of the polynomial filter and (c) the offset in
the logarithmic filter. We further investigate the correlation properties of
three analytic functions: exponential, logarithmic, and power-law. While these
three functions have in general different correlation properties, we find that
there is a broad range of variable values, common for all three functions,
where they exhibit identical scaling behavior. We further note that the scaling
behavior of a class of other functions can be reduced to these three typical
cases. We systematically test the performance of the DFA method in accurately
estimating long-range power-law correlations in the output signals for
different parameter values in the three types of filters, and the three
analytic functions we consider.
