Graph Signal Processing (GSP) provides a powerful framework for analysing complex, interconnected systems by modelling data as signals on graphs. Recent advances in GSP have enabled the learning of graph structures from observed signals, but these methods often struggle with time-varying systems and real-time applications. Adaptive filtering techniques, while effective for online learning, have seen limited application in graph topology estimation from a GSP perspective. To this end, we introduce AdaCGP, an online algorithm for adaptive estimation of the Graph Shift Operator (GSO) from multivariate time series. The GSO is estimated from an adaptive time-vertex autoregressive model through recursive update formulae designed to address sparsity, shift-invariance and bias. Through simulations, we show that AdaCGP performs consistently well across various graph topologies, and achieves improvements in excess of 82% for GSO estimation compared to baseline adaptive vector autoregressive models. In addition, our online variable splitting approach for enforcing sparsity enables near-perfect precision in identifying causal connections while maintaining low false positive rates upon optimisation of the forecast error. Finally, AdaCGP's ability to track changes in graph structure is demonstrated on recordings of ventricular fibrillation dynamics in response to an anti-arrhythmic drug. AdaCGP is shown to be able to identify the stability of critical conduction patterns that may be maintaining the arrhythmia in an intuitive way, together with its potential to support diagnosis and treatment strategies.

翻译：图信号处理（GSP）通过将数据建模为图上的信号，为分析复杂互联系统提供了强大框架。GSP的最新进展使得从观测信号中学习图结构成为可能，但这些方法在处理时变系统和实时应用时常面临困难。自适应滤波技术虽能有效用于在线学习，但从GSP视角进行图拓扑估计的应用仍有限。为此，我们提出AdaCGP——一种从多元时间序列中自适应估计图移位算子（GSO）的在线算法。GSO通过递归更新公式从自适应时-顶点自回归模型估计得出，该公式专为处理稀疏性、平移不变性和偏差而设计。仿真实验表明，AdaCGP在各种图拓扑结构中均表现稳定，与基线自适应向量自回归模型相比，其GSO估计精度提升超过82%。此外，我们用于强制稀疏性的在线变量分裂方法在优化预测误差的同时，能以接近完美的精度识别因果连接，并保持较低误报率。最后，AdaCGP跟踪图结构变化的能力在心室颤动动态响应抗心律失常药物的记录数据中得到验证。实验证明，AdaCGP能以直观方式识别可能维持心律失常的关键传导模式的稳定性，并具备辅助诊断与治疗策略制定的潜力。