Time Series Gaussian Graphical Models (TSGGM)

• TSGGMs are statistical models that capture conditional independence in high-dimensional Gaussian time series via zeros in the inverse spectral density matrix.
• They estimate the conditional independence graph using ℓ₁-regularized Whittle likelihood and ADMM, leveraging frequency-domain analysis for robust inference.
• TSGGMs offer theoretical guarantees and efficient sparse recovery, with applications in biomedical signals like fMRI and financial time series analysis.

A Time Series Gaussian Graphical Model (TSGGM) is a statistical structure that encodes conditional independence relationships among the components of a multivariate, typically high-dimensional, stationary Gaussian time series. These dependencies are mathematically represented by the zeros in the inverse spectral density matrix (SDM) across the frequency domain. TSGGMs generalize Gaussian graphical model selection from independent and identically distributed (i.i.d.) samples to temporally dependent data, enabling recovery of the conditional independence graph (CIG) from finite-length observations under spectral smoothness assumptions rather than strong parametric models (Jung et al., 2014).

1. Mathematical Formulation and Conditional Independence

Consider a $p$-dimensional, zero-mean, stationary Gaussian time series $x[n] = (x_1[n],\dots,x_p[n])^\mathsf{T}$ with autocovariance function $R[m] = \mathbb{E}\{x[n+m]x[n]^\mathsf{T}\}$ and absolutely summable autocovariance ($\sum_m \|R[m]\| < \infty$). The spectral density matrix (SDM) is defined via the Fourier series

$$S(\omega) = \sum_{m=-\infty}^\infty R[m]e^{-j2\pi m\omega}, \quad \omega \in [-\frac{1}{2},\frac{1}{2}],$$

with the constraint $$0 < L \leq \lambda_{\min}[S(\omega)] \leq \lambda_{\max}[S(\omega)] \leq U < \infty$$ for all $\omega$.

In the Gaussian case, the conditional independence graph $G=(V,E)$ is encoded by the pattern of zeros in the inverse SDM $K(\omega) = S(\omega)^{-1}$. The absence of edge $(i,j)$ in $G$ is equivalent to $K_{ij}(\omega) = 0$ for all $\omega$ (Jung et al., 2014). This generalizes the classical GGM setting to time-dependent data: the CIG captures contemporaneous and dynamic associations between scalar processes.

2. Estimation via ℓ₁-Regularized Whittle Likelihood and ADMM

Estimation is formulated as an ℓ₁-regularized maximum (approximate) likelihood problem in the frequency domain. Exact Gaussian likelihood for stationary series is intractable, motivating the Whittle approximation: $$\min_{\Theta(\omega)\succ 0} \int_{-1/2}^{1/2} \text{tr}\{\Theta(\omega)\widehat{S}(\omega)\} - \log\det \Theta(\omega) \ d\omega + \lambda \|\Theta\|_1,$$ where $\widehat{S}(\omega)$ is a spectral estimator (e.g., Blackman-Tukey, multitaper) and $\|\Theta\|_1$ integrates absolute off-diagonal entries across frequencies. The regularization parameter $\lambda > 0$ modulates the sparsity, driving edge selection in the inferred graph.

For practical computation, the frequency domain is discretized: uniform grid $\omega_f = -\frac{1}{2} + \frac{f-1}{F}$, $f = 1, ..., F$, leading to a finite-dimensional optimization problem over $X[f] \approx \Theta(\omega_f)$: $$\min_{X[1],...,X[F]} \frac{1}{F}\sum_{f=1}^F \left[\text{tr}\{X[f]\widehat{S}[f]\} - \log\det X[f]\right] + \lambda\sum_{i,j}\|\mathbf{X}_{ij}[\cdot]\|_2,$$ with positive-definiteness constraints $X[f] \succ 0$ for all $f$. Here, $\|\mathbf{X}_{ij}[\cdot]\|_2$ is the across-frequency group norm, promoting consistent sparsity over all frequencies.

Optimization proceeds via the Alternating Direction Method of Multipliers (ADMM): each iteration involves an eigen-decomposition-based update for $X$, group soft-thresholding for the auxiliary $Z$, and dual variable updates. This decouples the convex subproblems and ensures global convergence for any $\rho > 0$ (Jung et al., 2014).

3. Statistical Guarantees and Sample Complexity

Theoretical performance guarantees are provided under specific assumptions:

• Spectral boundedness: $$L \leq \lambda_{\min}[S(\omega)] \leq \lambda_{\max}[S(\omega)] \leq U$$.
• Smoothness: the autocovariance function has finite weighted moment, ensuring the spectral estimator is uniformly consistent.
• Sparsity: the true graph has at most $s$ edges.
• Restricted strong convexity: small sub-blocks of $S(\omega)$ have eigenvalue at least $\varphi/s$.
• Minimal signal strength: smallest nonzero group-norm of $K_{ij}[\cdot]$ exceeds $\rho_{\min}$.

Setting $\lambda = C_1 \varphi \rho_{\min}/[s(2U+1)^2]$ and sample size $$N \geq C_2(2U+1)^6 s^2 \varphi^{-2} \rho_{\min}^{-2}\|w\|_1^2\log(p^2 N/\delta)$$, the probability that the recovered graph support matches the true edge set is at least $1 - \delta$. The estimator achieves exact support recovery under sufficiently large $N$ and appropriately chosen $\lambda$ (Jung et al., 2014).

4. Computational Complexity

• Spectral density estimation (e.g., Blackman-Tukey) has complexity $O(p^2 N L)$, with $L$ the window length.
• Each ADMM iteration requires $O(F p^3)$ for eigen-decomposition and $O(F p^2)$ for thresholding.
• Total runtime is $O(p^2 N L + T F p^3)$, where $T$ is the number of ADMM steps.
• The choice of frequency bins $F$ trades off between spectral smoothing bias and computational cost.

5. Empirical Performance and Applications

Synthetic experiments demonstrate the model selection accuracy of the Whittle-gLASSO approach:

• For $p=64$, with a star graph (4 edges), filtered white noise ($L=1$, $U=10$, small autocovariance moment), sample sizes $N=128, 256$, $F = 4$ frequency bins, and $T = 10$ ADMM iterations, ROC curves indicate near-perfect recovery for well-tuned $\lambda$.
• The approach outperforms frequency-domain neighborhood regression when the spectral density matrix is sufficiently smooth.
• The group-regularized approach recovers the correct conditional independence graph from limited samples.

This methodology is directly applicable to high-dimensional biomedical time series (e.g., fMRI, gene expression) and financial data where temporal dependence and complex correlation structures are present.

6. Context, Extensions, and Relation to Prior Work

The general TSGGM framework extends classic graphical LASSO methodology (originally designed for i.i.d. samples [Friedman et al., JMLR 2008]) to time-dependent multivariate processes. The key extension is the use of the inverse spectral density function to encode conditional independence over time. The theoretical foundation relies on Dahlhaus (2000) for equivalence between edge absence and frequency-uniform zeros in the inverse SDM, under sufficient regularity.

Unlike parametric VAR-based TSGGMs, the presented approach is nonparametric, relying solely on spectral smoothness rather than explicit autoregressive structure. The methods are robust to model misspecification, provided spectral properties are regular enough.

Related literature explores piecewise-constant graphical models (GFGL) for changepoint detection (Gibberd et al., 2015), compressive high-dimensional inference under limited samples (Jung et al., 2013), and Bayesian neighborhood selection for temporally-coupled GGMs (Lin et al., 2015). These extensions allow structured changepoints, joint spatial-temporal estimation, and further relaxation of temporal dependence assumptions.

7. Limitations and Practical Notes

• The nonparametric spectral approach depends on sufficient regularity and stationarity; substantial deviations may impede consistent estimation.
• Choice of frequency discretization and regularization requires careful cross-validation or information criterion tuning.
• Computational cost, dominated by cubic scaling in $p$ and number of frequency bins, can be significant in ultra-high-dimensional settings.
• Comparing with time-domain approaches or neighborhood selection, the strength of the Whittle-gLASSO is its ability to leverage frequency-domain decorrelation, yielding more accurate graphs when data are suitably stationary and spectrally smooth (Jung et al., 2014).

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The above framework establishes Time Series Gaussian Graphical Models (TSGGMs) as a well-founded, computationally tractable tool for high-dimensional time-dependent graphical model selection, supported by strong theoretical and empirical results (Jung et al., 2014).