  In this paper, we propose a novel data-driven approach for removing trends
(detrending) from nonstationary, fractal and multifractal time series. We
consider real-valued time series relative to measurements of an underlying
dynamical system that evolves through time. We assume that such a dynamical
process is predictable to a certain degree by means of a class of recurrent
networks called Echo State Network (ESN), which are capable to model a generic
dynamical process. In order to isolate the superimposed (multi)fractal
component of interest, we define a data-driven filter by leveraging on the ESN
prediction capability to identify the trend component of a given input time
series. Specifically, the (estimated) trend is removed from the original time
series and the residual signal is analyzed with the multifractal detrended
fluctuation analysis procedure to verify the correctness of the detrending
procedure. In order to demonstrate the effectiveness of the proposed technique,
we consider several synthetic time series consisting of different types of
trends and fractal noise components with known characteristics. We also process
a real-world dataset, the sunspot time series, which is well-known for its
multifractal features and has recently gained attention in the complex systems
field. Results demonstrate the validity and generality of the proposed
detrending method based on ESNs.
