平稳多元时间序列的条件指定图建模
Conditionally specified graphical modeling of stationary multivariate time series
AI总结：
针对非高斯或混合型多元时间序列，本文提出CEStGM，用指数族条件分布和交互核编码过程级条件独立性，并建立Markov性质、几何混合性及近似Gibbs采样方法。
摘要：
图模型被广泛用于概括多元数据中的条件关系。在许多涉及多元时间序列的应用中，人们希望学习一种交互图：把每一条单独的时间序列视为图的节点，两个节点之间存在边表示在给定其他序列条件下的条件依赖性。通常，偏协方差被用作条件依赖性的度量。然而，在许多应用中，结果可能不是高斯的，并且/或者可能是不同结果的混合。对于这类时间序列，使用偏协方差作为条件依赖性的度量可能具有局限性。本文提出一大类时间序列模型，其专门设计用于在参数中简洁地编码整个过程范围内的条件独立性。对于时间序列中的每个单变量分量，我们使用指数族中的一种分布来建模其条件分布。我们提出了过程范围相容性的概念，在该概念下，这些条件指定可以被拼接起来，以形成定义良好的严格平稳多元时间序列。我们将这种构造称为条件指数平稳图模型（conditionally exponential stationary graphical model，CEStGM）。CEStGM的核心量是一个正核，我们称之为交互核。这类正核算子的谱性质构成了本文工作的核心技术基础。我们利用交互核的Hammersley-Clifford型分解，建立了CEStGM的过程范围局部和全局Markov性质。此外，我们研究了CEStGM的各种概率性质，并证明它是几何混合的。我们还开发了一种近似Gibbs采样器，用于模拟CEStGM的样本路径。
英文摘要：
Graphical models are ubiquitous for summarizing conditional relations in multivariate data. In many applications involving multivariate time series, it is of interest to learn an interaction graph that treats each individual time series as nodes of the graph, with the presence of an edge between two nodes signifying conditional dependence given the others. Typically, the partial covariance is used as a measure of conditional dependence. However, in many applications, the outcomes may not be Gaussian and/or could be a mixture of different outcomes. For such time series using the partial covariance as a measure of conditional dependence may be restrictive. In this article, we propose a broad class of time series models which are specifically designed to succinctly encode process-wide conditional independence in its parameters. For each univariate component in the time series, we model its conditional distribution with a distribution from the exponential family. We develop a notion of process-wide compatibility under which such conditional specifications can be stitched together to form a well-defined strictly stationary multivariate time series. We call this construction a conditionally exponential stationary graphical model (CEStGM). A central quantity underlying CEStGM is a positive kernel which we call the interaction kernel. Spectral properties of such positive kernel operators constitute a core technical foundation of this work. We establish process-wide local and global Markov properties of CEStGM exploiting a Hammersley-Clifford type decomposition of the interaction kernel. Further, we study various probabilistic properties of CEStGM and show that it is geometrically mixing. An approximate Gibbs sampler is also developed to simulate sample paths of CEStGM.
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