Learning Conditional Independence Differential Graphs From Time-Dependent Data

TL;DR

<3-5 sentence high-level summary> The paper tackles learning differences in conditional independence graphs for two time-series Gaussian graphical models by modeling the structure in the frequency domain via inverse PSD differences. It introduces a penalized D-trace loss formulated in the frequency domain, accommodating complex data and time dependence, and solves it with an ADMM algorithm augmented by an LLA scheme for non-convex penalties. The authors establish high-dimensional consistency and graph-recovery guarantees (for convex and log-sum penalties) and demonstrate substantial performance gains over i.i.d.-based baselines on synthetic data, with real-data application to financial time series yielding sparse, interpretable differential graphs. Overall, the work advances differential graph learning for time-dependent data and provides practical, theoretically grounded tools for detecting changes in dynamic dependency structure.

Abstract

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.

Figures (4)

• Figure 1: True $\log_{10} (\sum_{f=0:0.01:5} | [{\bm S}_x^\diamond(f)]_{ij} | )$ (left), $\log_{10} (\sum_{f=0:0.01:5} | [{\bm S}_y^\diamond(f)]_{ij} | )$ (middle), and $\log_{10} (\sum_{f=0:0.01:5} | [({\bm S}_y^\diamond(f))^{-1} - ({\bm S}_x^\diamond(f))^{-1}]_{ij} | )$ (right), $i,j \in [120]$, for the AR model, for a single Monte Carlo run: $p=120$ nodes.
• Figure 2: True $\log_{10} (\sum_{f=0:0.01:5} | [{\bm S}_x^\diamond(f)]_{ij} | )$ (left), $\log_{10} (\sum_{f=0:0.01:5} | [{\bm S}_y^\diamond(f)]_{ij} | )$ (middle), and $\log_{10} (\sum_{f=0:0.01:5} | [({\bm S}_y^\diamond(f))^{-1} - ({\bm S}_x^\diamond(f))^{-1}]_{ij} | )$ (right), $i,j \in [120]$, for the MA model, for a single Monte Carlo run: $p=120$ nodes.
• Figure 3: ROC curves: "DTS-FD, log-sum" is the proposed approach with log-sum penalty, "DTS-FD, lasso" is the proposed approach with lasso penalty, and "IID, lasso" is the time-domain approach of Jiang2018 (also Yuan2017) with lasso penalty. TPR=true positive rate, TNR=true negative rate.
• Figure 4: Differential graphs comparing financial time series (S&P 97 stocks share prices) over period Jan. 2, 2013 to Jan. 14, 2015 with that over period Dec. 17, 2015 to Jan. 1, 2018 (each series with 512 samples): (a) time-domain IID model with lasso penalty Jiang2018 (IID, lasso), (b) time-domain IID model with log-sum penalty Tugnait2025a (IID, log-sum), (c) proposed freq-domain approach with group lasso penalty (FD-DTS, lasso), (d) proposed freq-domain approach with group log-sum penalty (FD-DTS, log-sum). In the freq-domain approaches we used $M=2$ ($m_t=63$, $K=127$). In the figures the thickness of the lines reflects the strength of the connection (determined by $\| \hat{\tilde{\bm \Delta}}^{(ij)} \|$).