连续索引图模型
Continuously Indexed Graphical Models
AI总结：
针对连续时间或空间索引的高斯过程，本文提出基于再生核性质的无向图刻画，并给出有限分辨率下一致恢复图结构的方法、收敛速率及有限样本保证。
摘要：
令 $X = \{X_{u}\}_{u \in U}$ 为由集合 $U$ 索引的实值高斯过程。它可以被视为一个无向图模型，其中每个随机变量 $X_{u}$ 都充当一个顶点。我们利用 $X$ 的协方差并通过其再生核性质来刻画该图。与文献中的其他刻画不同，我们的刻画不限制索引集 $U$ 为有限或可数集，因此可用于对连续时间/空间中随机过程的内在依赖结构进行建模。因此，这种刻画并不是通过逆协方差的零元素来表述的。这给从 $X$ 的独立实现样本中恢复依赖结构（也称为结构估计）的问题带来了新的挑战。我们提出了一种规避这些问题的方法，其目标是以有限分辨率恢复底层图；该分辨率可以任意精细，并且仅受可用样本量限制。只要图在适当意义下具有足够的正则性，就可证明该恢复是一致的。我们推导了相应的收敛速率和有限样本保证。通过一项模拟研究和两项数据分析说明了我们的方法。
英文摘要：
Let $X = \{X_{u}\}_{u \in U}$ be a real-valued Gaussian process indexed by a set $U$. It can be thought of as an undirected graphical model with every random variable $X_{u}$ serving as a vertex. We characterize this graph in terms of the covariance of $X$ through its reproducing kernel property. Unlike other characterizations in the literature, our characterization does not restrict the index set $U$ to be finite or countable, and hence can be used to model the intrinsic dependence structure of stochastic processes in continuous time/space. Consequently, this characterization is not in terms of the zero entries of an inverse covariance. This poses novel challenges for the problem of recovery of the dependence structure from a sample of independent realizations of $X$, also known as structure estimation. We propose a methodology that circumvents these issues, by targeting the recovery of the underlying graph up to a finite resolution, which can be arbitrarily fine and is limited only by the available sample size. The recovery is shown to be consistent so long as the graph is sufficiently regular in an appropriate sense. We derive corresponding convergence rates and finite sample guarantees. Our methodology is illustrated by means of a simulation study and two data analyses.
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