Discovering Latent Causal Graphs from Spatiotemporal Data

We thank the reviewer for a thorough and insightful review.

Predictive Performance Evaluation: Our evaluation is consistent with the literature of causal discovery. As noted or observed in prior works [1, 2, 3] causal discovery/representation learning differs from prediction/forecasting. Causes may not explain a large portion of the effect’s variance. For example, let $X \sim N(0, 1)$ and $Y = 0.2X + E$ with $E \sim N(0, 1)$. X has a significant causal effect on Y yet explains only $\frac{0.2^2}{0.2^2 + 1} \approx 3.8 \\%$ of Y’s variance.

Nevertheless, following the reviewer’s suggestion, we modify SPACY to predict one-step ahead $\widehat{X}^{t+1}$ using ${X}^{t}... {X}^{t-\tau}$. We edit the SCM module to autoregressively predict the latent values $\widehat{Z}^{t+1}$ from ${Z}^{t}... {Z}^{t-\tau}$ following the topological ordering of the inferred causal graph and then decode these to obtain the prediction. Similarly, we use the learned transition prior in LEAP to predict one step ahead. We report the results for our method on both the synthetic (Nonlinear SCM and Nonlinear spatial mapping) and the climate dataset.

Our results indicate that SPACY captures most of the variance in the synthetic dataset. In the climate dataset, where complex exogenous factors dominate, the explained variance, though significant, is lower—consistent with expectations. LEAP shows a similar pattern, with lower performance than SPACY. In summary, while predictive performance is relevant in some contexts, our primary contribution is causal representation learning which we evaluate accordingly.

Non-Isotropic Spatial Factors: Our experiments use isotropic and elliptical anisotropic (Section B.3) spatial kernels, but SPACY’s theoretical framework supports fully anisotropic kernels, as our identifiability guarantees require only mild conditions about the spatial factors. We used RBF kernels since they parsimoniously capture high-level features. However, our framework can also be used with more expressive kernels.

To assess SPACY’s robustness to anisotropy, we conduct an additional ablation study in which the spatial factors are irregular and anisotropic. We introduce anisotropy by applying a sinusoidal warp to the spatial coordinates before using an anisotropic RBF kernel to generate data. We use isotropic RBF kernels with SPACY to model the synthetic data (Nonlinear SCM and Nonlinear spatial mapping setting). Results

Our findings show that even with isotropic RBF kernels, SPACY can approximate each latent variable’s location and scale and recover the causal graph. We will add these results to our paper.

Climate Dataset Claims: We appreciate the reviewer’s concern regarding spatial factor interpretability. However, many key atmospheric processes (the Walker circulation, monsoonal systems, and ENSO teleconnections) exhibit strong localization due to the interplay between heterogeneous boundary conditions (e.g., land-sea contrast, topography) and localized forcings. Despite using isotropic kernels, our method captures these localized relationships, offering greater physical interpretability than traditional approaches (e.g., principal components or EOFs) that may obscure spatial meaning through averaging. Additionally, our framework avoids diffuse spatial factors, linking causal links to identifiable physical processes.

SPACY vs Varimax Interpretability: SPACY’s spatial factors enforce geographic coherence by linking causal variables to interpretable regions, unlike Varimax’s scattered loadings. Visualizations show that many nodes recovered by Varimax are diffuse, uninterpretable, and lack clear physical locations, with clusters (e.g., nodes 42, 44) suggesting similar underlying components.

Precipitation, Temperature

Other Suggestions: We will clarify the notations and synthetic data generation process. To address some concerns:

• Any differentiable temporal causal discovery algorithm can be used with SPACY. We chose Rhino due to its flexibility and strong identifiability guarantees.
• SPACY operates with general metrics defined on $[0, 1]^K$. We can model diverse scenarios, such as the spherical grid with Haversine distance in the climate experiment.
• Following [4], we use a DAGness constraint for the instantaneous matrix (Appendix C.1).

References

[1] Shmueli, Gt. "To explain or to predict?." (2010)

[2] Lee, SY, et al. "Assessment of the predictive power of a causal variable: An application to the Head Start impact study." (2022)

[3] Chauhan, R.S, et al. "Causation versus prediction in travel mode choice modeling." (2025)

[4] Gong, W., et al. "Rhino: Deep Causal Temporal Relationship Learning with History-dependent Noise." (2022)

We thank the reviewer for their careful review and for pointing us to the relevant prior work [1]. We will revise the manuscript to include this citation in the discussion of RBF kernels.

We envision SPACY being applicable across various fields—for instance, in neuroscience for analyzing brain imaging data, or in epidemiology for identifying patterns in disease spread. We plan to include a discussion highlighting these potential applications, while leaving their in-depth exploration for future work.

References

[1] Sennesh, Eli, et al. "Neural topographic factor analysis for fmri data." Advances in Neural Information Processing Systems 33 (2020): 12046-12056.

We thank the reviewer for their careful review and thoughtful feedback.

Noise at latent vs. observation space: We clarify that there are two separate additive noises at latent and observation space respectively. The Gaussian noise assumption applies only to the observation space, not the latent SCM. The identifiability guarantees in Theorems 1–2 do not require explicit assumptions on the noise type in the latent SCM. Empirically, we validated our method on data with non-Gaussian and history-dependent noise (see experiments in Section 5). We will emphasize this distinction in the revised manuscript for better clarity.

Furthermore, the Gaussian assumption in the observation space can be relaxed quite easily (see Step 1 in Appendix B.2.2 from [1]). We can also relax the assumption of Gaussian noise in Theorems 1 and 2 using the aforementioned technique in the updated manuscript.

Causal Structure Identification: Theorems 1 and 2 focus on establishing the identifiability of the latent variables from observational data. Once the latents are identified, any causal discovery algorithm with identifiability guarantees (such as Rhino, which we used in our paper) can recover the causal graph, since the causal relationships are encoded in the conditional independence relationships of the latent variables. We will clarify this in the Theory section of our paper.

Choice of hyperparameters: We thank the reviewer for highlighting this practical concern. Our experiments (Section 5.3) demonstrate that overparameterization the latent dimension D does not degrade performance, allowing users to err on the side of overestimation. To provide systematic guidelines for using our model, we will:

1. Add a recommendation in Section 5.3 to select D larger than the anticipated latent dimensionality.
2. Include an additional experiment with a plot demonstrating the performance trend based on different levels of over-parameterization. The results indicate that overparameterization does not significantly degrade the performance of causal discovery and latent representation inference and, in fact, offers flexibility when there is uncertainty about the true latent dimensionality.

References

[1] Khemakhem, Ilyes, et al. "Variational autoencoders and nonlinear ica: A unifying framework." International conference on artificial intelligence and statistics. PMLR, 2020.

We thank the reviewer for their insightful feedback.

We agree that both DCM and SPACY formulate the problem of inferring causal relationships between latent variables as a multi-level Bayesian model. However, they differ significantly in their assumptions and approaches.

DCMs aim to infer the causal relationships of interactions in a dynamical system with potentially cyclic relationships. In contrast, SPACY uses the structural causal model [1] framework, where the causal graph is constrained to be acyclic. Moreover, DCM assumes that the parameters of the forward model (i.e. the relationship between the latent and observable variables) are known apriori. SPACY, on the other hand, infers the parameters of the forward model, as well as the correspondence between the observed variables and latent variables by fitting neural networks to the observed data. This latent-space approach enables SPACY to explore causal relationships without relying on prior mechanistic assumptions or predefined dynamical models, making it suitable for application to high-dimensional data with unknown dynamics. Algorithmically, DCM uses the EM approach for inferring the structure, while SPACY uses Variational Inference.

We will expand the related work section by including the references listed by the reviewer, and the above discussion. We will also fix the capitalization issue in the references.

References [1] Pearl, Judea. "Causal inference." Causality: objectives and assessment (2010): 39-58.