基于矩阵值时间序列的在线图拓扑学习
Online Graph Topology Learning from Matrix-valued Time Series
AI总结：
针对矩阵值时间序列的传感器依赖结构识别问题，将VAR模型扩展为矩阵变量模型，提出高低维场景下的在线图学习方法及带周期性趋势的扩展模型，经实验验证有效。
摘要：
本研究聚焦于矩阵值时间序列的统计分析，这类数据通常由分布在空间位置的传感器网络随时间采集得到。每个传感器在每个时间点记录一个特征向量，对应生成一条向量时间序列。研究目标是识别这些传感器之间的依赖结构，并用图的形式表示。当每个传感器仅观测到一个特征时，向量自回归（VAR）模型常被用于推断格兰杰因果关系，最终得到因果图。\n本研究的第一项贡献是将VAR模型扩展为适用于图学习的矩阵变量模型。此外，针对低维和高维场景分别提出了两种在线处理流程，能够在新样本到达时快速更新系数估计值。在高维设置下，提出了一种新型的Lasso类方法，并开发了用于在线学习的同伦算法，同时还提供了正则化参数的自适应调优流程。\n鉴于自回归模型应用于数据时通常需要去趋势处理，而这在在线场景中无法实现，因此研究对提出的AR模型进行了扩展，将趋势作为额外参数纳入模型，其中重点关注周期性趋势。在线算法也适配了这些扩展数据模型，能够从流式样本中同时学习图结构和趋势。基于合成数据和真实数据的数值实验均证明了所提方法的有效性。
英文摘要：
The focus is on the statistical analysis of matrix-valued time series, where data is collected over a network of sensors, typically at spatial locations, over time. Each sensor records a vector of features at each time point, creating a vectorial time series for each sensor. The goal is to identify the dependency structure among these sensors and represent it with a graph. When only one feature per sensor is observed, vector auto-regressive (VAR) models are commonly used to infer Granger causality, resulting in a causal graph. The first contribution extends VAR models to matrix-variate models for the purpose of graph learning. Additionally, two online procedures are proposed for both low and high dimensions, enabling rapid updates of coefficient estimates as new samples arrive. In the high-dimensional setting, a novel Lasso-type approach is introduced, and homotopy algorithms are developed for online learning. An adaptive tuning procedure for the regularization parameter is also provided. Given that the application of auto-regressive models to data typically requires detrending, which is not feasible in an online context, the proposed AR models are augmented by incorporating trend as an additional parameter, with a particular focus on periodic trends. The online algorithms are adapted to these augmented data models, allowing for simultaneous learning of the graph and trend from streaming samples. Numerical experiments using both synthetic and real data demonstrate the effectiveness of the proposed methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
