Estimation of the conditional independence graph (CIG) of high-dimensional multivariate Gaussian time series from multi-attribute data is considered. Existing methods for graph estimation for such data are based on single-attribute models where one associates a scalar time series with each node. In multi-attribute graphical models, each node represents a random vector or vector time series. In this paper we provide a unified theoretical analysis of multi-attribute graph learning for dependent time series using a penalized log-likelihood objective function formulated in the frequency domain using the discrete Fourier transform of the time-domain data. We consider both convex (sparse-group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. We establish sufficient conditions in a high-dimensional setting for consistency (convergence of the inverse power spectral density to true value in the Frobenius norm), local convexity when using non-convex penalties, and graph recovery. We do not impose any incoherence or irrepresentability condition for our convergence results. We also empirically investigate selection of the tuning parameters based on the Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.

翻译：本文研究了从多属性数据中估计高维多变量高斯时间序列的条件独立图问题。现有针对此类数据的图估计方法主要基于单属性模型，即每个节点对应一个标量时间序列。在多属性图模型中，每个节点代表一个随机向量或向量时间序列。本文通过时域数据的离散傅里叶变换，在频域构建惩罚对数似然目标函数，对依赖时间序列的多属性图学习进行了统一的理论分析。我们同时考虑了凸惩罚（稀疏群组套索）与非凸惩罚（对数求和与SCAD群组惩罚）函数。在高维设定下，我们建立了以下充分条件：逆功率谱密度在Frobenius范数下收敛于真实值的一致性、使用非凸惩罚时的局部凸性以及图结构恢复。我们的收敛结果无需任何不相干性或不可表示性条件。此外，我们基于贝叶斯信息准则对调参选择进行了实证研究，并通过合成数据与真实数据的数值算例验证了所提方法。