Time series data arising in many applications nowadays are high-dimensional. A large number of parameters describe features of these time series. We propose a novel approach to modeling a high-dimensional time series through several independent univariate time series, which are then orthogonally rotated and sparsely linearly transformed. With this approach, any specified intrinsic relations among component time series given by a graphical structure can be maintained at all time snapshots. We call the resulting process an Orthogonally-rotated Univariate Time series (OUT). Key structural properties of time series such as stationarity and causality can be easily accommodated in the OUT model. For Bayesian inference, we put suitable prior distributions on the spectral densities of the independent latent times series, the orthogonal rotation matrix, and the common precision matrix of the component times series at every time point. A likelihood is constructed using the Whittle approximation for univariate latent time series. An efficient Markov Chain Monte Carlo (MCMC) algorithm is developed for posterior computation. We study the convergence of the pseudo-posterior distribution based on the Whittle likelihood for the model's parameters upon developing a new general posterior convergence theorem for pseudo-posteriors. We find that the posterior contraction rate for independent observations essentially prevails in the OUT model under very mild conditions on the temporal dependence described in terms of the smoothness of the corresponding spectral densities. Through a simulation study, we compare the accuracy of estimating the parameters and identifying the graphical structure with other approaches. We apply the proposed methodology to analyze a dataset on different industrial components of the US gross domestic product between 2010 and 2019 and predict future observations.

翻译：在许多现代应用中产生的时间序列数据往往是高维的。大量参数描述了这些时间序列的特征。我们提出了一种新颖的高维时间序列建模方法，该方法通过若干独立的一维时间序列进行正交旋转和稀疏线性变换。采用这种方法，由图形结构给出的分量时间序列之间的任意指定内在关系可以在所有时间快照上得到保持。我们将由此得到的过程称为正交旋转一维时间序列（OUT）。时间序列的关键结构特性（如平稳性和因果性）可以在OUT模型中方便地得到体现。在贝叶斯推断中，我们为独立潜时间序列的谱密度、正交旋转矩阵以及每个时间点上分量时间序列的公共精度矩阵设置了合适的先验分布。利用Whittle近似法构建了基于一维潜时间序列的似然函数。我们开发了一种高效的马尔可夫链蒙特卡洛（MCMC）算法用于后验计算。在建立了一个新的伪后验分布通用后验收敛定理的基础上，我们研究了基于Whittle似然构建的模型参数伪后验分布的收敛性。我们发现，在关于时间依赖性的非常温和条件下（以相应谱密度的光滑性描述），独立观测下的后验压缩率在OUT模型中基本成立。通过模拟研究，我们比较了该方法与其他方法在参数估计精度和图形结构识别方面的表现。我们将所提出的方法应用于分析2010年至2019年间美国国内生产总值各产业组成的数据集，并对未来观测值进行了预测。