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.

翻译：本文的主要重点是基于闭环Volterra级数表示对时间序列数据进行近似。Volterra级数展开是表示、分析和综合非线性动力系统的有效工具。然而，该方法的一个主要局限是，随着展开阶数的增加，需要估计的项数呈指数增长，这构成了重大挑战。本文利用再生核希尔伯特空间（RKHS）和多项式核的概念，提出了一种估计平稳非线性时间序列中闭环Volterra级数的实用解决方案。我们通过模拟和实际数据分析展示了所提出的Volterra表示方法的适用性。此外，我们应用了Kolmogorov-Smirnov预测精度（KSPA）检验，首先确定竞争时间序列模型之间估计误差的分布在统计上是否存在显著差异，其次确定基于某种损失函数具有较低误差的估计时间序列是否比来自竞争方法的估计时间序列在随机意义上具有更小的误差。所得结果表明，闭环Volterra方法在误差更小和可解释性更高方面，能够优于ARFIMA、ETS和岭回归方法。