This study delves into the domain of dynamical systems, specifically the forecasting of dynamical time series defined through an evolution function. Traditional approaches in this area predict the future behavior of dynamical systems by inferring the evolution function. However, these methods may confront obstacles due to the presence of missing variables, which are usually attributed to challenges in measurement and a partial understanding of the system of interest. To overcome this obstacle, we introduce the autoregressive with slack time series (ARS) model, that simultaneously estimates the evolution function and imputes missing variables as a slack time series. Assuming time-invariance and linearity in the (underlying) entire dynamical time series, our experiments demonstrate the ARS model's capability to forecast future time series. From a theoretical perspective, we prove that a 2-dimensional time-invariant and linear system can be reconstructed by utilizing observations from a single, partially observed dimension of the system.

翻译：本研究深入探讨动态系统领域，特别是通过演化函数定义的动态时间序列预测问题。传统方法通过推断演化函数来预测动态系统的未来行为，但由于测量困难和系统部分理解导致的变量缺失，这些方法常面临障碍。为克服这一难题，我们提出具有松弛时间序列的自回归模型（ARS模型），该模型能同时估计演化函数并将缺失变量作为松弛时间序列进行插补。在（底层）完整动态时间序列满足时不变性和线性性的假设下，实验表明ARS模型具备预测未来时间序列的能力。理论层面，我们证明通过利用系统中单个部分观测维度的观测数据，可以重构一个二维时不变线性系统。