Collaborative Deterministic–Probabilistic Forecasting for Real-World Spatiotemporal Systems

We sincerely thank Reviewer b7LL for the valuable comments. Below, we provide detailed responses to your questions.

Q1. Clarification on the differences with TMDM

Tanks for your feedback. While both CoST and TMDM aim to combine deterministic and diffusion models for improved probabilistic forecasting, we would like to emphasize that the two differ significantly. (We have briefly discussed the differences between CoST and TMDM in Lines 107–108. To better clarify their distinctions, we have now expanded Section 2.3 in the revised manuscript to provide a more detailed comparison.)

1. Decomposition vs. Modulation

(i) CoST adopts a mean–residual decomposition framework: the deterministic model predicts the conditional mean, while the diffusion model learns only the residual distribution. This simplifies the diffusion objective, supported by the law of total variance theory[1], and avoids modeling full temporal dependencies from scratch.

(ii) TMDM, in contrast, uses a modulation approach, where the deterministic output is injected as a prior to guide the diffusion model in generating the entire target sequence. Thus, the diffusion component in TMDM still learns the full complex distribution.

2. Spatiotemporal-Specific Design

CoST targets spatiotemporal forecasting, and introduces a scale-aware diffusion mechanism that computes location-specific fluctuation scales (via FFT) to address spatial heterogeneity. TMDM, as a time-series model, does not include spatial modeling or similar mechanisms.

3. Clarification on Model Configuration

(i) In the deterministic component, both CoST and TMDM adopt well-established architectures from their respective domains. We test STID, ConvLSTM, STNorm, and iTransformer in our framework, while TMDM uses Transformer, Informer, Autoformer, and NSformer.

(ii) For the diffusion model, both models build upon the conditional injection mechanism from CARD [2]. However, their purposes differ: TMDM uses it to incorporate the deterministic model's output into the diffusion process, whereas CoST leverages it to better integrate the customized spatial fluctuation scale Q into the residual learning process, thereby enhancing spatial modeling. Therefore, any architectural similarity reflects a shared technique, not shared objectives.

[1] Introduction to Probability. Athena Scientific, 2008

[2] CARD: Classification and Regression Diffusion Models. NeurIPS, 2022

Q2. Distinction from Decomposition-Based Methods (Leddam & TimeMixer)

We thank the reviewer for their insightful comment and for prompting us to clarify the novelty of our work in relation to prior decomposition-based models such as Leddam and TimeMixer. While we agree that signal decomposition is a well-established concept in time series analysis, we respectfully argue that CoST introduces a fundamentally different paradigm, particularly in its objective, decomposition formulation, and architectural design.

1. Forecasting Objective: Probabilistic vs. Deterministic Forecasting

(i) CoST: CoST is designed for probabilistic forecasting, aiming to model the full predictive distribution $p(y|x)$. This enables robust uncertainty quantification, which is critical in high-stakes, real-world decision-making. Accordingly, we evaluate CoST using distributional metrics such as CRPS, QICE, and IS.

(ii) Leddam and TimeMixer: They are focused exclusively on deterministic forecasting. They are optimized to minimize point-wise errors (e.g., MSE, MAE), and do not generate uncertainty estimates, nor are they evaluated under probabilistic criteria.

2. Decomposition Strategy: Mean-Residual vs. Trend-Seasonal

(i) CoST: We propose a mean-residual decomposition ($x = \mathbb{E}[x] + r$). This is a functional decomposition. We task a powerful deterministic model to capture the conditional mean ($\mathbb{E}[x]$), which represents the predictable, regular patterns. A lightweight diffusion model is then tasked only with learning the distribution of the residual ($r$), which represents the stochastic, unpredictable variations. This division of labor is the cornerstone of our framework.

(ii) Leddam & TimeMixer: Both models use a traditional trend-seasonal decomposition. This is a signal processing technique aimed at disentangling different components of the historical signal to create better features for a single, unified forecasting model.

3. Model Architecture: Collaborative vs. Unified

(i) CoST employs a collaborative architecture, wherein:

• A deterministic model learns the conditional mean, and
• A diffusion model learns the distribution of the residual.

This division of labor not only improves modeling efficiency, but also allows the diffusion model to focus solely on capturing uncertainty, leading to more calibrated forecasts.

(ii) Leddam and TimeMixer: In contrast, Leddam and TimeMixer use a single deterministic model that internally integrates decomposed components (e.g., trend and seasonal signals) as features. These components are not separately modeled or interpreted, and the model ultimately outputs only a point estimate.

Q3. More Baselines.

Thank you for the suggestion. In addition to the original six probabilistic forecasting baselines (D3VAE, DiffSTG, TimeGrad, CSDI, DYffusion, NPDiff), we have further included four more probabilistic models (GP, DeepAR, DeepState, and TMDM) as additional baselines in our revised experiments.

To ensure a fairer comparison, we evaluate these models on long-term forecasting tasks, as TMDM’s original experimental setting focuses on longer prediction horizons. Specifically, we also use the ETTh1 and ETTh2 datasets provided by the TMDM paper, and CoST achieves the best performance on both. We conjecture that TMDM performs suboptimally on spatiotemporal data for two reasons: (1) It lacks explicit spatial modeling mechanisms, and (2) Spatiotemporal data often have high spatial dimensionality, which can be challenging for models like TMDM to learn effectively, especially when the dataset size is limited.

Thank you very much for the author's reply. I believe that in terms of innovation, this article still has some limitations.Taking into account the opinions of the other reviewers, I have decided to maintain my score.

We sincerely thank Reviewer pqiK for the valuable comments. Below, we provide detailed responses to your questions.

Q1. Concern about computational cost.

Our method significantly reduces modeling time via mean residual decomposition, which simplifies the learning task by focusing the diffusion model on residuals. This enables the use of a lightweight denoising network, thereby reducing training complexity and time. We have conducted additional efficiency evaluations, with key results summarized below. CoST achieves strong computational efficiency: 2.5 minutes for training and 43 seconds for inference on CrowdBJ dataset, enabling real-world deployment. More results and analysis are in Section 5.4 and Appendix Table 10.

Q2. How CoST performs under extreme cases.

Thank you for the question. CoST handles extreme cases effectively through its mean–residual decomposition: the deterministic model captures regular patterns, while the diffusion model focuses on residual uncertainty, which typically increases during extreme events. This separation enables CoST to better model rare, high-variance deviations. Additionally, the scale-aware diffusion enhances sensitivity to localized fluctuations. As shown in Section 5.7 (Figures 6c, 6f), CoST outperforms baselines in capturing sudden spikes or sharp fluctuations. Since these cases are hard to quantify, we have supplemented the analysis with additional visualizations and qualitative assessment in the appendix.

Q3. Does the probabilistic component always refine the deterministic output?

Thank you for the question. Our diffusion component is not introduced to refine the deterministic output, but to provide probabilistic forecasts by modeling the residual uncertainty. To assess whether any conflict exists between the two components, we compared the number of samples where CoST (with both components) outperforms the diffusion-only variant, using the Interval Score (IS, a metric that evaluates the quality of probabilistic forecasts at the sample level). We find that the vast majority of samples benefit from the addition of the deterministic component, showing improved IS. While a small number of cases experience marginal degradation, overall results confirm that the two components collaborate effectively rather than conflict.

Q4. How can the learned uncertainty estimates be effectively used in downstream decision-making tasks?

Thank you for the question. CoST produces not only a point forecast but also a predictive distribution, which reflects how confident the model is and provides probabilistic bounds (e.g., upper/lower quantiles or intervals) for possible outcomes. This enables risk-aware decision-making:

1. Traffic management: If CoST predicts heavy congestion with high uncertainty in a certain area, traffic control systems can adopt conservative routing policies (e.g., increase buffer time, suggest detours, prepare contingency plans) to reduce risk. In contrast, low-uncertainty predictions can be used more aggressively.

2. Public safety alerts: For crowd events or storms, if the upper bound of predicted density or precipitation crosses a safety threshold—even if the mean doesn’t—early warnings can be issued preemptively, improving preparedness without overreacting.

3. Ride-sharing dispatch: Predictive intervals help platforms assess best- and worst-case demand, allowing for dynamic driver reallocation or pricing that balances service quality and operational cost under uncertainty.

In short, CoST doesn’t just tell you how confident it is; it also gives a range of possible outcomes, like the best and worst cases. This helps decision-makers plan ahead under uncertainty, which is especially important in real-world systems like traffic, public safety, or energy, where acting early can make a big difference.

We sincerely thank Reviewer pFy7 for the valuable comments. Below, we provide detailed responses to your questions.

Q1. Key Differences from CaCaST and DiffCaST.

Response: We thank the reviewer for their insightful feedback and for referencing the important works of DiffCast and CasCast.We agree that the high-level concept of hybridizing deterministic and diffusion models has been explored. . However, we respectfully argue that CoST is not an incremental work but introduces a distinct and novel approach in its core methodology, technical contributions, and demonstrated generality.

1. Fundamentally Different Methodological Formulation.

(i) CoST introduces a principled Mean-Residual Decomposition, which is theoretically motivated by variance reduction. This simplifies the learning task for the diffusion model by having it only capture the residual distribution, whose variance is smaller than that of the original data. (ii) DiffCast, in contrast, applies a residual diffusion on top of a deterministic model to refine the deterministic output, but does not explicitly formulate or isolate the conditional mean. This does not decouple uncertainty estimation cleanly and still burdens the diffusion model with learning complex spatiotemporal structures.

(iii) CasCast follows a cascaded pipeline, where a deterministic model first predicts coarse global structures and a diffusion model generates refined results in a latent space. However, the diffusion model operates on full future states rather than isolated residuals.

2. Generality and Scope of Application.

(i) We validate CoST as a general-purpose framework on 10 diverse, real-world datasets spanning four different domains (climate, energy, urban systems, and communication). This demonstrates a far broader applicability and generality compared to DiffCast and CasCast, which are specialized models designed and tested for the specific task of precipitation nowcasting on radar data.

(ii) CoST, is fundamentally focused on delivering well-calibrated probabilistic forecasts and uncertainty quantification for general spatiotemporal systems. In contrast, DiffCast and CasCast are primarily designed to enhance the visual fidelity of precipitation nowcasting. They use their diffusion components differently: DiffCast corrects a deterministic output via an additive residual, while CasCast refines it through a guided generation process in a latent space. CoST's goal is not just to produce a sharper image, but to learn the entire predictive distribution.

3. Handling Spatial Heterogeneity via Scale-Aware Diffusion.

CoST introduces a scale-aware diffusion mechanism, guided by customized spatial fluctuation statistics derived from Fourier analysis. This explicitly addresses spatial heterogeneity, a feature not present in either DiffCast or CasCast.

We're grateful to the reviewer for bringing these two very insightful works to our attention. We have now included a specific subsection in Section 2 Related Work, to elaborate on these methods that combine diffusion with deterministic models.

Q2.More Baselines

Response: We chose not to emphasize DiffCast as a strong baseline because our approach fundamentally differs. While DiffCast uses diffusion components as a corrector for its backbone network, our method leverages diffusion for probabilistic distribution modeling, which is central to our work. Furthermore, DiffCast's performance was poor on our benchmarks（we report a subset of results below）, likely due to its domain-specific architecture (U-Net, PhyDNet) tailored for precipitation nowcasting, which is suboptimal for the general sequence dynamics emphasized in our datasets.

Q3. Minor typo in line 199. reprent --> represent.

Response: we've corrected it and thoroughly reviewed the manuscript for any other errors.

Q4. Clarification about the experiment setting used in Figure 8 (Ablation Study).

Response: To address your query, it's important to understand that we constructed these model variants by progressively removing key components, rather than simply removing a single module in isolation. Here's a breakdown of what "(w/o s)" and "(w/o q)" signify:

(w/o s): This variant means we do not use the scale-aware diffusion process to integrate the prior information from the customized fluctuation scale. Instead, the customized fluctuation scale is simply provided as a conditional input to the denoising network.

(w/o q): This variant builds upon the "(w/o s)" setting. In addition to not using the scale-aware diffusion process, we further remove the customized fluctuation scale as a conditional input to the denoising network. This means the model operates without any information about the customized fluctuation scale, neither through the scale-aware diffusion process nor as a direct conditional input.

We hope this explanation clarifies the experimental setup for our ablation study. We've revised the description in the ablation study section to make it clearer.

We sincerely thank Reviewer kEBZ for the valuable comments. Below, we provide detailed responses to your questions.

Q1. Lack of multivariate scoring rules.

Response: Thank you for your question. We address your concern from two perspectives below:

1. Following Established Precedent: Our evaluation protocol aligns with the common practice adopted in prior influential works[1-4]. They have also only used univariate evaluation metrics, without incorporating multivariate assessments.

[1] Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting. ICML 2021

[2] CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation. NeurIPS 2021

[3] Transformer-Modulated Diffusion Models for Probabilistic Multivariate Time Series Forecasting. ICLR 2024

[4] Dynamical Diffusion: Learning Temporal Dynamics with Diffusion Models. ICLR 2025

2. Adopting Suggestion and Adding New Experiments: Nevertheless, we fully agree that multivariate scoring rules are crucial for a comprehensive evaluation. To address this, we have already conducted supplementary experiments using these metrics on both the baselines you mentioned and selected models from our paper. Due to time constraints, we report a subset of results below.

We will add more results and a discussion on multivariate evaluation to the revised manuscript's experiments section. We believe this addition enhances our work's rigor and helps establish a more complete benchmark for future research in this area.

DiffSTG is excluded for ETTh2 datasets as they lack adjacency graphs.

Q2. More Baselines

Response: Thank you for your valuable feedback. As suggested, we've added GP, DeepAR, and DeepStateSpace as new baselines. Part of the results is shown below, and CoST still achieves the best overall performance. In the revised version, we will provide full experimental results and discussions of these works in the Related Work section.

Q3. Justification for using a diffusion model.

Response: Thank you for raising this important point. We provide our justification for using diffusion models from four key perspectives:

1. Complexity of real-world spatiotemporal systems. These systems(e.g., traffic, climate) often exhibit complex dependencies and inherently multimodal, stochastic behavior. Many studies have shown that diffusion models benefit both short-term and long-term forecasting in such systems[1-3].

[1] Non-autoregressive Conditional Diffusion Models for Time Series Prediction. ICML 2023

[2] DiffSTG: Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models. SIGSPATIAL 2023

[3] Transformer-Modulated Diffusion Models for Probabilistic Multivariate Time Series Forecasting. ICLR 2024

2. Limitation of GPs and SSMs. GPs are limited to unimodal predictive distributions due to their Gaussian assumptions, which restrict their ability to model multiple plausible futures. Classical SSMs assume linear dynamics and Gaussian noise, and even their nonlinear variants (e.g., EKF, particle filters) typically produce poorly calibrated or unimodal predictions and are sensitive to tuning.
3. Empirical evidence. Our study covers both short-term and long-term forecasting. We have added empirical comparisons with GP and DeepStateSpace (SSM-based) in both settings. As shown in Table Q2 (short-term) and Tables 8–9 (long-term), diffusion models consistently outperform these baselines.
4. Advancing the frontier. Despite higher computational costs, exploring diffusion models in this context pushes the boundaries of spatiotemporal modeling and offers promising new capabilities.

Q4. Ablation studies on conditional mean.

Response: Thank you for the insightful suggestion. We conduct an ablation study using the historical average as the mean component. As shown in the results below, conditioning on simple historical statistics does provide some improvement over a single diffusion model. However, using a learned deterministic model (STID) to generate the conditional mean yields better performance.

Q5. How was the "ideal" multimodal distribution in Figure 4 derived?

Response: Thank you for the question. We agree it is challenging to obtain a full target distribution in real-world forecasting, as only one future outcome per instance is observed. However, spatiotemporal systems often exhibit multimodal behavior due to complex, latent conditions. For example, traffic volume at a specific road segment can vary drastically under different latent conditions (e.g., normal vs. accident scenarios). A particular time and location may yield distinct values depending on unobserved events, leading to multimodal behavior even if we can only observe one outcome at a time. In Figure 4, the “ideal” distribution is not a true conditional target but an empirical marginal distribution at a fixed location across time, used to qualitatively illustrate the multimodal nature of the data. While it does not represent a ground-truth future distribution, it helps highlight whether the model captures uncertainty beyond a single mode. We will clarify this in the revised version.

Q6. How was the CRPS calculated?

Response: The CRPS is calculated using a sample-based approach. We follow prior work [1], which conducted empirical analysis on spatiotemporal forecasting tasks and found that using 50 samples is sufficient to obtain a stable and reliable estimate of CRPS. Additionally, we perform an ablation study on our dataset to evaluate the impact of the number of samples on CRPS values. The results show that increasing the sample size beyond 50 yields negligible improvements.

[1] DiffSTG: Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models. SIGSPATIAL 2023

Thank you for addressing my questions. I appreciate that the authors have adopted the Energy Score (ES) as a benchmark metric and that the primary goal of the paper is multivariate forecasting. However, my methodological concerns largely remain.

First, as noted, the simple historical average already provides substantial improvement, suggesting that the deterministic component is heavily driven by the periodic nature of the underlying time series. In this case, the deterministic mean serves a proper trend term that can be pre-specified outside the diffusion model (no need for joint learning). Also, I suspect that incorporating a stronger mean/deterministic model could further enhance performance—for example, by introducing low-frequency Fourier components or dynamic modes as the mean function.

In addition, some points remain unclear regarding the role of the deterministic mean. Such a mean is likely to help when the target variable exhibits a unimodal distribution (e.g., a dominant periodic signal with random perturbations). However, long-term forecasting often involves multimodal distributions, which are more challenging. For instance, as traffic prediction is mentioned in the rebuttal, a 2‑hour‑ahead forecast is often either congested or free-flowing, with few intermediate states. In such cases, relying on the mean may not provide meaningful predictive or interpretative value.

Given these, I will keep my current score.