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：本文研究矩阵值时间序列的统计分析方法。此类数据通过传感器网络（通常为空间位置集合）随时间采集，每个传感器在每个时间点观测到一个特征向量，因此每个传感器对应一个向量时间序列。我们旨在识别这些传感器间的依赖结构，并用图进行表示。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关系结构，所得图称为因果图。本文的首要贡献是将VAR模型扩展至矩阵变量模型以实现图学习。其次，我们分别提出了适用于低维和高维场景的两种在线算法，可在新样本到达时快速更新系数估计。特别地，在高维场景中引入了一种新型类Lasso方法，并为其在线学习开发了同伦算法，同时提供了正则化参数的自适应调优方案。最后考虑到实际应用AR模型需对原始数据进行去趋势处理，但在线场景中无法执行该步骤，因此我们将趋势作为额外参数纳入AR模型，并针对增强后的数据模型调整在线算法，从而实现在流式样本中同步学习图结构与趋势。本研究主要考虑周期性趋势。通过合成数据与真实数据的数值实验，验证了所提方法的有效性。