This paper is concerned with the statistical analysis of matrix-valued time series. These are data collected over a network of sensors (typically a set of spatial locations) along time, where a vector of features is observed per time instant per sensor. Thus each sensor is characterized by a vectorial time series. We would like to identify the dependency structure among these sensors and represent it by a graph. When there is only one feature per sensor, the vector auto-regressive models have been widely adapted to infer the structure of Granger causality. The resulting graph is referred to as causal graph. Our first contribution is then extending VAR models to matrix-variate models to serve the purpose of graph learning. Secondly, we propose two online procedures respectively in low and high dimensions, which can update quickly the estimates of coefficients when new samples arrive. In particular in high dimensional regime, a novel Lasso-type is introduced and we develop its homotopy algorithms for the online learning. We also provide an adaptive tuning procedure for the regularization parameter. Lastly, we consider that, the application of AR models onto data usually requires detrending the raw data, however, this step is forbidden in online context. Therefore, we augment the proposed AR models by incorporating trend as extra parameter, and then adapt the online algorithms to the augmented data models, which allow us to simultaneously learn the graph and trend from streaming samples. In this work, we consider primarily the periodic trend. Numerical experiments using both synthetic and real data are performed, whose results support the effectiveness of the proposed methods.

翻译：本文关注矩阵值时间序列的统计分析。这类数据是通过传感器网络（通常是一组空间位置）随时间收集的，每个时间点每个传感器观测到一个特征向量。因此，每个传感器由向量时间序列刻画。我们旨在识别这些传感器之间的依赖结构，并用图来表示。当每个传感器仅有一个特征时，向量自回归模型已被广泛用于推断格兰杰因果结构，所得图称为因果图。我们的第一个贡献是将VAR模型扩展到矩阵变量模型，以实现图学习目标。其次，我们提出了两种在线算法，分别适用于低维和高维情形，可在新样本到达时快速更新系数估计。特别地，在高维场景下，我们引入了一种新型Lasso型方法，并为其在线学习开发了同伦算法。我们还为正则化参数提供了自适应调优过程。最后，考虑到将AR模型应用于数据通常需要对原始数据进行去趋势处理，但该步骤在在线环境中不可行。因此，我们通过将趋势作为额外参数纳入模型来增强所提出的AR模型，并将在线算法适配到增强数据模型，从而允许从流式样本中同时学习图和趋势。本研究主要考虑周期性趋势。我们使用合成数据和真实数据进行了数值实验，结果支持所提方法的有效性。