  The main focus of this paper is to approximate time series data based on the
closed-loop Volterra series representation. Volterra series expansions are a
valuable tool for representing, analyzing, and synthesizing nonlinear dynamical
systems. However, a major limitation of this approach is that as the order of
the expansion increases, the number of terms that need to be estimated grows
exponentially, posing a considerable challenge. This paper considers a
practical solution for estimating the closed-loop Volterra series in stationary
nonlinear time series using the concepts of Reproducing Kernel Hilbert Spaces
(RKHS) and polynomial kernels. We illustrate the applicability of the suggested
Volterra representation by means of simulations and real data analysis.
Furthermore, we apply the Kolmogorov-Smirnov Predictive Accuracy (KSPA) test,
to determine whether there exists a statistically significant difference
between the distribution of estimated errors for concurring time series models,
and secondly to determine whether the estimated time series with the lower
error based on some loss function also has exhibits a stochastically smaller
error than estimated time series from a competing method. The obtained results
indicate that the closed-loop Volterra method can outperform the ARFIMA, ETS,
and Ridge regression methods in terms of both smaller error and increased
interpretability.
