从多属性时间序列中学习条件独立图，突破传统单变量方法限制。
On Conditional Independence Graph Learning From Multi-Attribute Gaussian Dependent Time Series
- 基于频域惩罚似然法，统一建模多属性向量时间序列的依赖结构。
- 在高维下实现逆功率谱密度一致收敛与图结构正确恢复。
- 无需复杂条件，适合处理多变量、长程依赖的金融或生物数据。
针对高维多变量高斯时间序列的多属性数据，研究条件独立图（CIG）估计问题。现有方法多基于单属性模型，即每个节点对应一个标量时间序列；而多属性图模型中，每个节点代表一个随机向量或向量时间序列。本文提出一种统一的理论分析框架，基于频域中的惩罚似然目标函数（利用离散傅里叶变换将时域数据转换至频域），采用凸（稀疏组Lasso）与非凸（log-sum、SCAD组正则化）惩罚函数进行图学习。在高维设定下，建立了逆功率谱密度一致收敛（弗罗贝尼乌斯范数意义）、非凸惩罚下的局部凸性及图恢复的充分条件，且无需引入相干性或不可表示性条件。通过贝叶斯信息准则（BIC）评估调参策略，并在合成与真实数据上验证方法有效性。
原文摘要 · Abstract (English)
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.
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