通过频域差异谱密度建模时序数据的条件独立变化，更准确捕捉动态依赖关系。
Learning Conditional Independence Differential Graphs From Time-Dependent Data
- 在频域使用惩罚D-迹损失函数，结合威廷格微积分处理时序依赖性。
- 对合成与真实数据测试中，对数和惩罚法显著优于Lasso与传统i.i.d.方法。
- 适合研究具有时间相关性的高维动态系统结构变化，如金融或生物信号。
本文研究两个具有相似结构的时间序列高斯图模型（TSGGM）之间条件独立图（CIG）差异的估计问题，其中TSGGM结构由时间序列的逆功率谱密度（IPSD）编码。现有工作多针对独立同分布（i.i.d.）数据，通过估计精度矩阵差异刻画条件依赖变化；而本工作首次在频域中估计两组时序数据的IPSD差异，以揭示其潜在依赖结构演化。所提方法考虑数据的时间依赖性，采用基于威廷格微积分的惩罚D-迹损失函数，结合凸（组Lasso）与非凸（log-sum、SCAD组正则化）惩罚项，并提出交替方向乘子法（ADMM）求解。理论分析给出了高维情形下一致性和图恢复的充分条件。实验表明，在合成数据上，对数和惩罚估计器的F1分数显著优于基于Lasso的时序方法，后者又显著优于传统的i.i.d. Lasso方法。
原文摘要 · Abstract (English)
Estimation of differences in conditional independence graphs (CIGs) of two time series Gaussian graphical models (TSGGMs) is investigated where the two TSGGMs are known to have similar structure. The TSGGM structure is encoded in the inverse power spectral density (IPSD) of the time series. In several existing works, one is interested in estimating the difference in two precision matrices to characterize underlying changes in conditional dependencies of two sets of data consisting of independent and identically distributed (i.i.d.) observations. In this paper we consider estimation of the difference in two IPSDs to characterize the underlying changes in conditional dependencies of two sets of time-dependent data. Our approach accounts for data time dependencies unlike past work. We analyze a penalized D-trace loss function approach in the frequency domain for differential graph learning, using Wirtinger calculus. We consider both convex (group lasso) and non-convex (log-sum and SCAD group penalties) penalty/regularization functions. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the objective function. 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) and graph recovery. Both synthetic and real data examples are presented in support of the proposed approaches. In synthetic data examples, our log-sum-penalized differential time-series graph estimator significantly outperformed our lasso based differential time-series graph estimator which, in turn, significantly outperformed an existing lasso-penalized i.i.d. modeling approach, with $F_1$ score as the performance metric.
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