In machine learning applications, data are often high-dimensional and intricately related. It is often of interest to find the underlying structure and Granger causal relationships among the data and represent these relationships with directed graphs. In this paper, we study multivariate time series, where each series is associated with a node of a graph, and where the objective is to estimate the topology of a sparse graph that reflects how the nodes of the graph affect each other, if at all. We propose a novel fully Bayesian approach that employs a sparsity-encouraging prior on the hyperparameters. The proposed method allows for nonlinear and multiple lag relationships among the time series. The method is based on Gaussian processes, and it treats the entries of the graph adjacency matrix as hyperparameters. It utilizes a modified automatic relevance determination (ARD) kernel and allows for learning the mapping function from selected past data to current data as edges of a graph. We show that the resulting adjacency matrix provides the intrinsic structure of the graph and answers causality-related questions. Numerical tests show that the proposed method has comparable or better performance than state-of-the-art methods.

中文翻译：

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具有稀疏性约束的高斯过程有向图的拓扑推理


在机器学习应用中，数据通常是高维的且相互错综复杂。人们常常感兴趣的是找到数据之间的底层结构和格兰杰因果关系，并用有向图表示这些关系。在本文中，我们研究多元时间序列，其中每个序列与图的一个节点相关联，其目标是估计稀疏图的拓扑，该拓扑反映图的节点如何相互影响（如果有的话） 。我们提出了一种新颖的完全贝叶斯方法，该方法在超参数上采用稀疏性鼓励先验。所提出的方法允许时间序列之间的非线性和多重滞后关系。该方法基于高斯过程，并将图邻接矩阵的条目视为超参数。它利用修改后的自动相关性确定（ARD）内核，并允许学习从选定的过去数据到当前数据的映射函数作为图的边缘。我们表明，生成的邻接矩阵提供了图的内在结构并回答了与因果关系相关的问题。数值测试表明，所提出的方法具有与最先进的方法相当或更好的性能。