Temporal Latent Variable SCM

• Temporal Latent Variable SCM is a framework for modeling time-evolving causal dependencies in multivariate series with unobserved confounders.
• It employs generative models and variational inference to extract interpretable causal graphs under high-dimensional and spatiotemporal conditions.
• Empirical evaluations reveal superior precision and stability compared to baseline methods, especially under non-Gaussian noise and latent confounding.

A Temporal Latent Variable Structural Causal Model (TLVSCM) is a structural framework for modeling the time-evolving causal dependencies among variables in multivariate time series, in which unobserved (latent) processes play a critical mediating or confounding role. TLVSCMs form a foundational paradigm for extracting interpretable causal graphs from observational temporal data under complicated interference, high dimensionality, and spatiotemporal correlation structures. Their core is a generative graphical model coupling observed and latent states by parameterized, time-dependent structural equations, typically leveraging identifiability results, variational inference, and domain priors for robust learning and scalability in scientific and engineering contexts.

1. Structural Specification of Temporal Latent Variable SCMs

In the general TLVSCM formulation, consider observed time series $X(t)\in\mathbb{R}^m$ and latent time series $Z(t)\in\mathbb{R}^n$. The model posits interdependent update equations, typically of vector autoregressive (VAR) form or extensions to nonlinear autoregressive families:

$$\begin{aligned} X(t) &= (A^{XX} \odot W^{XX}) X(t{-}1) + (A^{XZ} \odot W^{XZ}) Z(t{-}1) + N^X(t), \ Z(t) &= (A^{ZZ} \odot W^{ZZ}) Z(t{-}1) + N^Z(t), \end{aligned}$$

where $A^{\cdot,\cdot}$ are binary adjacency matrices encoding the presence/absence of lagged and cross-block influences, $W^{\cdot,\cdot}$ are real-valued strength matrices, and $N^X(t), N^Z(t)$ are independent noise components (often non-Gaussian mixtures for identifiability) (Cai et al., 13 Nov 2025). The “$\odot$” symbol denotes Hadamard product. Block structure allows observed-to-latent, latent-to-observed, and latent self-transition dependencies.

The model can be extended to the spatiotemporal setting, where observations $X(x, t)$ at spatial location $x \in \mathcal{G}$ are mapped via spatial kernel functions to lower-dimensional latent processes $Z_t\in\mathbb{R}^D$ through factors $s_d(x; \rho_d, \gamma_d)$. The grid-to-latent transformation employs RBF or similar spatial bases, enabling scalable causal inference across large spatial layouts (Wang et al., 8 Nov 2024).

For certain scientific domains, a continuous-time dynamic SCM is defined, with entire paths as random variables and latent confounders (e.g., exogenous noise processes), serving as a bridge between stochastic differential equations and discrete-time SCM inference (Boeken et al., 3 Jun 2024).

2. Identifiability Theory in TLVSCMs

Identifiability refers to the ability to recover the latent causal structure (graph and parameters) uniquely from observed data (up to permissible equivalences). In the finite-lag VAR case with latent AR(1) processes and non-Gaussian innovations, Cai et al. (Cai et al., 13 Nov 2025) show that under assumptions of diagonal latent block, independence, and full-rank cross-moments, the mapping from observed time series to the product $A^{XX}\odot W^{XX}$ is algebraically identifiable. The cross-block (observed-latent) structure is identifiable up to permutation and scaling.

In the infinite-grid spatiotemporal case, SPACY establishes identifiability of both spatial factors and latent processes under the following conditions: linearly independent spatial modes (distinct RBF centers), continuum sampling ($X(x)$ for all $x$), and identity spatial mappings. The main theorems guarantee that if two spatial factor processes produce the same observed distribution, then the factor matrices and latents must coincide up to permutation and invertible transform (Wang et al., 8 Nov 2024).

In nonlinear, history-dependent settings (including temporally latent ICA and nonparametric time-delayed SCMs), identifiability is guaranteed under sufficient non-stationarity, smoothness, and the existence of discriminating regimes such that the mixed derivatives of the conditional log-likelihood span the relevant function space (Yao et al., 2022, Yao et al., 2021, Cai et al., 23 Feb 2025).

3. Variational Inference and Learning Algorithms

TLVSCMs of practical scale make exact posterior inference intractable; hence variational inference frameworks are constructed. The posterior approximation factorizes as $q(A) q(W) q(Z)$ or incorporates additional parameters (e.g., spatial kernels $F$ in SPACY, or regime embeddings). Optimization is driven by the Evidence Lower Bound (ELBO):

$$\mathrm{ELBO} = \mathbb{E}_q[\log p(X|Z,A,W)] + \mathbb{E}_q[\log p(Z|A,W)] + \mathbb{E}_q[\log p(A)] + \mathbb{E}_q[\log p(W)] - \mathbb{E}_q[\log q(A)] - \mathbb{E}_q[\log q(W)] - \mathbb{E}_q[\log q(Z)]$$

with potential sparsity penalties (Cai et al., 13 Nov 2025, Wang et al., 8 Nov 2024).

Key technical features:

• Sampling of latent adjacency matrices $A$ employs Concrete/Gumbel-Softmax relaxations for gradient flow.
• Learnable priors (e.g., expert knowledge in $p(A_{ij})$) are incorporated to encode or restrict graph topology.
• Differentiable acyclicity constraints (e.g., NOTEARS) ensure DAG induction where required (Wang et al., 8 Nov 2024).

In high-dimensional settings with spatiotemporal structure, per-time encoders (MLPs, ConvNets) are used for $q_\phi(Z|X)$, and spatial factor parameters are updated via reparameterization (Wang et al., 8 Nov 2024).

Sparsity-promoting regularization (e.g., $\ell_1$ penalty on $A$) and domain adaptation constraints (alignment penalties) further stabilize learning under distribution shift (Cai et al., 23 Feb 2025).

4. Causal Graph Recovery and Sparsity Constraints

Estimation of the latent causal graph (adjacency, strength, and directionality) centers on masking non-active graph entries via $A \odot W$ and promoting sparsity through both prior design and $\ell_1$-norm penalties. Inference procedures threshold expected values of edge probabilities to construct point estimates of the causal graph (Cai et al., 13 Nov 2025).

For spatial-temporal models, gradient-based acyclicity constraints are enforced to recover DAGs on latent components, and sparsity of connections is ensured via explicit regularizers and priors (Wang et al., 8 Nov 2024).

Alignment across domains or environments (for domain adaptation or transfer) is achieved by encouraging common support patterns in the Jacobians of the inferred transition function, using alignment-specific penalties (Cai et al., 23 Feb 2025).

5. Empirical Evaluation and Benchmarks

TLVSCMs exhibit competitive or superior performance on both synthetic and real-world time series datasets where unmeasured confounders or spatially correlated effects are significant. Key empirical results from Cai et al. (Cai et al., 13 Nov 2025):

• Achieve Precision ≈ $0.92$–$0.97$, F1 ≈ $0.65$–$0.70$ for moderate $m\leq 20$, $T\geq 1000$ on synthetic data, exceeding comparator methods (e.g., CLH-NV, VAR-LiNGAM).
• Retain high precision and F1 as graph sparsity, sample size, node count, or latent variable ratio increases, where state-of-the-art baselines fall below $F1<0.40$.
• On simulated fMRI and financial data, maintain significantly higher precision and stability under increasing latent ratio.
• On large climate datasets with spatiotemporal gridding (e.g., global temperature fields), SPACY recovers interpretable large-scale spatial modes and their temporal lags, matching known climate teleconnection structures (Wang et al., 8 Nov 2024).

These models outperform constraint-based and functional-model comparators particularly under high latent confounding and non-Gaussian innovations.

6. Extensions: Spatiotemporal, Continuous-Time, and Hawkes Models

Advanced TLVSCMs extend to:

• Spatiotemporal data: Latent processes are mapped to grid observations by learned spatial factors defined as RBF or kernel modes, enabling scalable end-to-end learning for large spatial domains. Identifiability is framed in terms of spatial mode independence and invertibility (Wang et al., 8 Nov 2024).
• Continuous-time latent SCMs (Dynamic SCMs): The framework is generalized to represent entire trajectory segments as random processes, supporting integration with stochastic differential equations. Such models utilize $\sigma$-separation instead of $d$-separation to capture confounding and Markov properties in the continuous-time domain (Boeken et al., 3 Jun 2024).
• Discretized Hawkes processes: Discrete-time approximations provide necessary and sufficient identifiability conditions using rank constraints, supporting recovery of both observed and latent causal networks in point-process event data (Jin et al., 15 Aug 2025).

7. Limitations and Theoretical Considerations

Identifiability guarantees are conditional on assumptions regarding noise non-Gaussianity or non-stationarity, independence, and genericity of model parameters (e.g., no deterministic relations, sufficiently variable regime structure) (Cai et al., 13 Nov 2025, Wang et al., 8 Nov 2024). For highly nonlinear systems, practical identifiability relies on observable signals preserving the relevant conditional independencies and on structural assumptions such as the diagonal latent block or invertibility/independence in the observation map.

The variational-inference–based algorithms require careful regularization to avoid spurious solutions due to local minima or overfitting, particularly in high-dimensional latent spaces or where expert priors are unavailable or mis-specified.

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References:

• "Temporal Latent Variable Structural Causal Model for Causal Discovery under External Interferences" (Cai et al., 13 Nov 2025)
• "Discovering Latent Causal Graphs from Spatiotemporal Data" (Wang et al., 8 Nov 2024)
• "Dynamic Structural Causal Models" (Boeken et al., 3 Jun 2024)
• "Causal Structure Learning in Hawkes Processes with Complex Latent Confounder Networks" (Jin et al., 15 Aug 2025)