A Survey on Diffusion Models for Time Series and Spatio-Temporal Data

• Diffusion models add noise in a forward phase and remove it in a reverse phase, learning to generate samples aligned with the training-data distribution.Source: §1 · p. 2
• Diffusion models have gained prominence across computer vision, natural language processing, and multimodal learning, with applications spanning image, speech, video, and audio generation.Source: §1 · p. 2
• Temporal data captures the dynamics of most real-world systems, yet deriving universal dynamic laws across data modalities remains challenging.Source: §1 · p. 2
• Diffusion models increasingly appear in time-series and spatio-temporal modeling; Fig. 1 tracks the cumulative number of related papers.Source: Figure 1 · p. 2, §1 · p. 2

• The survey provides a comprehensive, forward-looking review of diffusion models for time-series and spatio-temporal data, including their suitability and the mechanisms behind their advantages.Source: series/spatio-temporal data analysis… · p. 3
• The framework categorizes work as unconditional or conditional diffusion across predictive and generative tasks, providing a roadmap from multiple perspectives.Source: series/spatio-temporal data analysis… · p. 4
• The review covers emerging techniques in both model types, including multimodal conditional generation.Source: series/spatio-temporal data analysis… · p. 4
• The survey identifies current research challenges and highlights promising directions for future exploration.Source: series/spatio-temporal data analysis… · p. 4

2.1 Overview of Time Series and Spatio-Temporal Data

• A time series is a chronologically ordered sequence, either univariate or multivariate across feature channels [97].The section represents univariate series as T values and multivariate series as T time points across D channels.Source: §2.1 · p. 5
• Spatio-temporal data combines temporal and spatial dimensions and includes structures such as graphs, trajectories, and videos [97].The section also treats multivariate time series as a possible form of spatio-temporal data.Source: §2.1 · p. 5
• Spatio-temporal graphs encode locations as nodes and changing traffic volume as weighted edges, while trajectories capture movement paths and attributes over time [33].The section connects trajectory analysis to transportation, wildlife tracking, and mobile network optimization.Source: §2.1 · p. 5
• Time-series diffusion tasks include forecasting, generation, anomaly detection, and imputation.Forecasting may be short- or long-term; generation can simulate scenarios or diversify training data, while imputation fills missing values.Source: §2.1 · p. 6
• Spatio-temporal data analysis applies time-series-like methodologies to application scenarios, including forecasting traffic flow or air quality, generating privacy-compliant synthetic trajectories [305], and anomaly detection in vehicle trajectory analysis.Source: §2.1 · p. 6

2.2 Why Diffusion Model and Its History

• The concept originated in statistical physics, and a 2015 variational-inference training method preceded rapid development in image generation and later use in text-to-image, music, and speech [243].The section says diffusion models began showing potential in these additional fields from 2020.Source: §2.2 · p. 6
• Diffusion models gradually modify data distributions by adding noise and then removing it to generate realistic samples [76].The section describes training as forward diffusion followed by reverse denoising.Source: §2.2 · p. 6
• Diffusion models generate coherent, complex sequences and allow generation to be conditioned and fine-tuned [44].The section also notes flexibility across modalities and possible integration with autoencoders for generation quality and control.Source: §2.2 · p. 6

2.3.1 Denoised Diffusion Probabilistic Models (DDPMs).

• DDPMs [76] use dual Markov chains to add Gaussian noise until data approaches a standard Gaussian, then learn reverse denoising transitions to recover data.Training minimizes a variational negative log-likelihood, with noise prediction used to parameterize denoising; sampling starts from Gaussian noise and iteratively denoises.Source: §2.3.1 · pp. 7–8

2.3.2 Score SDE Formulation.

• Score SDE [228] generalizes DDPM’s discrete steps to a continuous Itô SDE, using a learned time-dependent score for reverse-time generation with SDE or ODE solvers.The forward SDE combines drift and Brownian motion; the probability-flow ODE preserves the SDE’s marginal densities.Source: §2.3.2 · p. 9

2.3.3 Conditional Diffusion Models.

• Conditional models take class labels or input-related features as conditions, while self-, textual-, graph-, physical-, and task-based guidance supports application-specific generation [201] [57] [63] [121] [209] [6].Source: §2.3.3 · p. 9

2.3.4 Improvements with Diffusion Model and Its Variants.

• Practical models face slow iterative sampling, high-dimensional computational cost, and generalization or integration concerns; forward-process variants include electric-field methods PFGM and PFGM++ [269], and image-transforming Cold Diffusion.PFGM and its extensions use electric-field dynamics and augmented dimensions to improve performance.Source: §2.3.4 · p. 10

2.3.4 Improvements with Diffusion Model and Its Variants

• Training-free sampling, noise-to-data trajectory methods such as DDIM [224] and DPM-Solver, and SDE restart sampling [99] target fewer generation steps.Knowledge distillation transfers insights from larger models to simpler ones [18] [225].Source: §2.3.4 · p. 10
• Combining diffusion models with VAE [170] [190] or GAN [254] [299], and using latent-space inputs [190], are presented as efficiency strategies.Latent-space input can reduce dimensionality and enhance efficiency.Source: §2.3.4 · p. 10
• Improved schedulers, including DDIM [224], DPM-Solvers [165], and UniPC, are described as enabling faster convergence and fewer iterations.The section also lists CMS [225], IDDPM, Euler and Heun [103], LCM [168], RePaintScheduler, TCD [300], and VQD [65].Source: §2.3.4 · p. 10
• Architecture improvements include DDPM’s U-Net, normalization [18] [104], positional attention [109] [199] [292], and transformer-based designs such as ViT [52], DiT, and SiT [172].The section also names SwinTransformer among increasingly used transformer-based architectures.Source: §2.3.4 · p. 10
• Performance-oriented training changes include adjusting diffusion steps [170] [299], using projection methods across diffusion processes [47] [81], and optimizing noise, losses, and matching [109] [114] [86].The section frames these as strategies to improve performance and versatility.Source: §2.3.4 · p. 10
• For bridging Gaussian noise and complex distributions, the section describes SDE/ODE principles, α-blending, Rectified Flow [2] [156], and Schrödinger Bridges [150] [229].These approaches explore ODE creation between distributions and distribution transportation.Source: §2.3.4 · p. 10
• The section cites Stable Diffusion for text-conditioned image generation, ControlNet for image-generation control, and consistency models [225] for state-of-the-art performance without slow iterative generation.It notes OpenAI and Stability AI as institutions that have launched diffusion models.Source: §2.3.4 · p. 11

2.4 Diffusion Model vs Other Generative Models

• VAEs encode input data into a latent space and decode from it, but tend to produce lower-quality data and have limited latent-space expressiveness.VAEs aim to maximize the Evidence Lower Bound (ELBO) of the data log-likelihood.Source: §2.4 · p. 11
• GANs pair a generator that produces samples with a discriminator that distinguishes generated from real data, but training is difficult and prone to instability and mode collapse.Their training involves a zero-sum game between generator and discriminator.Source: §2.4 · p. 11
• Flow-based models transform data through reversible mappings to a simpler distribution, enabling precise likelihood evaluation, but require substantial computation and complex designs.The transformations are invertible, supporting precise likelihood evaluation.Source: §2.4 · p. 11
• Diffusion models generate high-quality, detailed outputs with flexibility and probabilistic foundations, but have long training times and high computational costs that variants have mitigated.The section describes these costs as mitigated by diffusion-model variants, not eliminated.Source: §2.4 · p. 11

Model categories

• The survey divides the literature into unconditioned and conditioned diffusion models focused on time-series and spatio-temporal data.Table 1 summarizes notable papers and distinguishes univariate, multivariate, and spatio-temporal data by color.Source: §3 · p. 12, Table 1 · p. 12

Taxonomy dimensions

• The taxonomy distinguishes probability-based and score-based models and organizes them by predictive versus generative tasks, data types, and applications.Figure 6 presents these classification dimensions.Source: §3 · p. 12, Figure 6 · p. 13

Task types

• Predictive tasks include forecasting and anomaly detection, while generative tasks include time-series imputation and augmentation.Predictive tasks use historical data and patterns to anticipate current or future events.Source: §3 · p. 13

Data modalities and applications

• The reviewed methods address time-series and spatio-temporal data across sectors including healthcare, energy, climate, and traffic.The broader application overview also includes sequential recommendation and audio.Source: §3 · pp. 13–14

Conditioned models

• Conditioned models use labels such as extracted short-term trends or urban flow maps to enhance inference under specific constraints.The survey includes both probability-based and score-based models for predictive and generative conditioned analysis.Source: §3 · pp. 13–14

Survey organization

• The survey proceeds from model categories to tasks, data modalities, and application fields to describe the current landscape and its challenges.The review covers forecasting, generation, anomaly detection, and imputation, and distinguishes time-series from spatio-temporal data.Source: §3 · p. 14

4.1.1 Probability-Based Model.

• DDPM adds noise at discrete intervals and learns to reverse the process, with both training and inference conducted in fixed steps; TimeGrad forecasts across thousands of interconnected dimensions [196].Source: §4.1.1 · p. 14
• DDPM-based spatio-temporal work has grown since 2023, including graph-based imputation from observed data and trajectory forecasting with a GNN-based encoder and important sampling.Source: §4.1.1 · p. 15

4.1.2 Score-Based Model.

• Score-based SDE models represent diffusion continuously, enabling samples at arbitrary times; ScoreGrad and TimeGrad use condition-based SDE score matching for forecasting [271].Source: §4.1.2 · p. 15
• Score-based diffusion supports realistic EHR synthesis and ECG generation [283] [128], while speech-enhancement methods transform speech into time-frequency information treated as image input [256].Source: §4.1.2 · pp. 15–16
• StoRM uses predictive-model guidance with stochastic resampling to produce higher-quality samples under complex, high-SNR conditions using fewer diffusion steps and more streamlined models [125].Its resampling method uses forecasts to guide subsequent diffusion steps.Source: §4.1.2 · p. 16
• DYffusion integrates temporal dynamics into diffusion steps and uses a time-conditioned interpolator with a predictor to simulate forward and reverse processes, balancing performance and efficiency.Source: §4.1.2 · p. 16

4.2.1 Conditional Diffusion Model.

• Conditional diffusion steers generation with supplied information to produce condition-correlated outputs [216], and other guidance includes statistical measures for sensor synthesis and conditional enhancement for speech [306].Source: §4.2.1 · p. 16
• CSDI and MIDM condition diffusion on observed values for imputation, while DiffAD and ImDiffusion use conditional weight-incremental diffusion for imputation and anomaly detection [34].Preserving observed-value information is described as improving generation quality for stable anomaly detection.Source: §4.2.1 · p. 16

4.2.2 Latent Diffusion Model.

• Latent diffusion performs denoising in a compressed space to reduce computation while retaining output quality, with applications spanning precipitation, EEG, trajectories, and multivariate forecasting.LDCast uses latent diffusion for near-term precipitation forecasting, reporting greater training stability, optimized computational demands, and uncertainty representation compared with GAN [124]. CLDM separates deterministic forecasting from predictive-error scenario generation.Source: §4.2.2 · p. 17

4.2.3 Other Variants of Diffusion Models.

• MedDiff uses DDIM with a new sampling strategy to accelerate generation of high-dimensional, large-scale electronic medical records, and DCM is a diffusion-based causal-inference model accelerated by DDIM.Source: §4.2.3 · p. 17

4.3 Denoisers in the Diffusion Model

• Denoisers transform noise into structured signals, and their architecture, inductive biases, and prior-guided designs shape generative capacity, temporal reasoning, and inverse-problem applicability.The section frames denoisers as the diffusion model’s core modules and analyzes their architectures, biases, and use as priors.Source: §4.3 · pp. 17–18, §4.3.1 · p. 18

4.3.1 Architecture of the Denoiser.

• U-Net remains the most widely used diffusion-model architecture [76], with skip connections supporting multi-scale representation learning; temporal dilations target sequential dependencies, while attention-based CSDI and PriSTI [152] model long-range dependencies.Source: §4.3.1 · p. 18

4.3.2 Inductive Bias in Temporal Denoisers.

• Temporal denoisers encode continuity, periodicity, and causal dynamics through hierarchical temporal architectures, CNN skip connections, graph convolutions, and transformer attention.Temporal U-Net and WaveGrad capture multi-scale patterns through progressive downsampling and upsampling [31]; graph-based denoisers incorporate structural priors, while temporal transformers combine causal and hierarchical biases.Source: §4.3.2 · p. 18

4.3.3 Denoiser-as-Prior in Temporal Inverse Problems.

• Denoisers can serve alone as learned priors for temporal restoration, avoiding full stochastic sampling in methods that reconstruct missing segments.The Plug-and-Play and Score Prior paradigms use denoisers as proximal operators for iterative reconstruction [101]; ImDiffusion [34] and SADI [89] use denoisers to restore missing segments, reducing computational overhead and separating prior learning from generative sampling.Source: §4.3.3 · p. 18

5.1 Forecasting

• TimeGrad and D3VAE [136] use diffusion for probabilistic time-series forecasting, while DiffSTG models spatio-temporal graph dependencies and uncertainty.Source: §5.1 · pp. 18–19

5.2 Generation

• Diffusion generation spans sequential recommendation, trajectories, and audio, including unconditioned trajectory generation and gradient-based or non-autoregressive audio synthesis.Source: §5.2 · p. 19

5.3 Imputation

• CSDI probabilistically imputes missing time-series and spatio-temporal data, while MIDM, PriSTI [152], TabCSDI [301], and MissDiff address other data settings.Source: §5.3 · pp. 19–20

5.4 Anomaly Detection

• For time-series anomaly detection, DiffAD and ImDiffusion [34] combine imputation techniques with diffusion, while other work targets instability and noise.Source: §5.4 · p. 20

6.1.1 Univariate Time Series.

• Trend and periodicity challenge univariate time-series models; denoisers commonly use 1-D CNNs, while frequency-domain transformations can capture long-term correlations with 2-D CNNs.Source: §6.1.1 · pp. 20–21

6.1.2 Multivariate Time Series.

• Multivariate series treat variates as image channels and commonly use U-Nets; MIDM re-derives the conditional diffusion ELBO to improve observed–missing consistency.Source: §6.1.2 · p. 21

6.2 Spatio-Temporal Data

• Spatio-temporal data are organized as graphs or trajectories, requiring models to capture spatial as well as temporal dependencies.Source: §6.2 · p. 21

6.2.1 Spatio-Temporal Graph.

• Domain-oriented graph models address traffic and climate tasks, including fine-grained urban-flow inference, origin-destination travel estimation, air-quality uncertainty, and rainfall nowcasting.Source: §6.2.1 · pp. 21–22
• Other graph approaches broaden scope: DiffSTG supports multiple tasks, PriSTI and study imputation, and SpecSTG diffuses in graph-projected spectral space for faster inference.Source: §6.2.1 · pp. 21–22

6.2.2 Spatio-Temporal Trajectory.

• Trajectory work chiefly generates data by injecting spatio-temporal correlations into diffusion; examples include unconditioned GPS generation and road-network-constrained generation, with forecasting also emerging.Source: §6.2.2 · p. 22

7.1 Healthcare

• Healthcare applications include synthetic EEG generation for sleep-related uses, dataset augmentation for seizure forecasting [219], realistic EHR synthesis [23] [283], and generative-model enhancement of autism detection.Source: §7.1 · p. 23

7.2 Traffic

• Flow uses denoising diffusion to capture stochastic urban flows and generate accurate, fine-grained forecasts, while DiffTraj generates meaningful trajectory data reflecting real-world movement patterns.Source: §7.2 · p. 23

7.3 Sequential Recommendation

• Sequential recommendation methods bridge discrete item identities and continuous diffusion data, predict future interaction probabilities [246], or model evolving interests [138].Other approaches include DCDR’s discrete forward and conditional reverse processes [145] and RecFusion’s binomial diffusion for interaction time series [17].Source: §7.3 · pp. 23–24

7.4 Climate and Weather

• Climate and weather applications address precipitation nowcasting and uncertainty [124], wind-resolution enhancement, weather forecasting [30], Earth-system emulation [16], and data assimilation [87].DiffESM is described as a computationally efficient alternative to traditional simulation with improved accuracy [16].Source: §7.4 · p. 24

7.5 Energy and Electricity

• Energy applications include uncertainty-quantifying load forecasting with DiffLoad [252], probabilistic load forecasts from DiffPLF [131], and realistic short-term wind-power scenario generation using conditional latent diffusion models.Source: §7.5 · p. 24

7.6 Audio

• In audio, future research may improve model efficiency, reduce computational demands, and explore untapped applications in audio processing.Source: §7.6 · p. 25

7.7 AIOps

• Diffusion models support cloud operations through anomaly anticipation with Maat [122], realistic traffic generation with NetDiffus [221], and missing-data imputation for failure forecasting.These applications target early anomaly detection, network planning and testing, and improved forecasting despite data gaps.Source: §7.7 · p. 25
• Diffusion-based imputation of missing time-series data is investigated to improve cloud failure forecasting accuracy.The approach addresses gaps in cloud-service data to support better predictive outcomes.Source: §7.7 · p. 25

7.8 Finance

• In finance, a Diffusion Variational Autoencoder approach predicts stock prices while aiming to capture market stochasticity and volatility [112].Source: §7.8 · p. 25
• FinDiff generates synthetic financial tabular data that closely mimics real datasets, addressing data scarcity and potentially supporting training and regulatory compliance without exposing sensitive information.Source: §7.8 · p. 25
• DiffSTOCK uses diffusion for probabilistic relational reasoning in stock forecasting to capture interdependencies among market factors [46].Source: §7.8 · p. 25

Scalability and Efficiency

• Diffusion models remain computationally demanding, limiting their use in resource-constrained settings; proposed directions include lighter architectures, improved sampling, compression, and parallelization.Source: §8 · p. 26

Robustness and Adaptability

• Real-world series contain noise, missing values, anomalies, and distribution shifts, motivating stronger robustness and adaptation across datasets, scenarios, and changing environments.Source: §8 · p. 26

Domain-Generalizable Challenges

• Small, strongly autocorrelated benchmarks can cause overfitting, and generalization to unseen domains remains limited; some benchmarks may approach intrinsic predictability limits and saturation [250].Source: §8 · p. 26

Prior Knowledge Guided Generation

• Current diffusion models insufficiently incorporate domain-specific constraints such as road networks, population dynamics, and thermodynamics into generated data.Source: §8 · p. 26

Multimodal Data Fusion

• Because temporal data often co-occur with text or images, the paper calls for architectures that effectively fuse multimodal information in applications such as finance and healthcare.Source: §8 · p. 26

Cross-Domain Inspirations

• Image-based diffusion advances offer efficiency and controllability insights, but direct transfer is hindered by temporal data properties including causality and non-stationarity.Source: §8 · p. 26

Integration of LLMs and Diffusion Models

• Combining LLMs with diffusion may unite semantic understanding and generative modeling to enhance temporal reasoning and support more comprehensive decision-making.Source: §8 · p. 26

9 Conclusion

• The survey presents a comprehensive overview of diffusion models for time-series and spatio-temporal data analysis.Source: §9 · p. 26
• It distinguishes unconditioned and conditioned models and examines forecasting, generation, imputation, and anomaly detection.Source: §9 · p. 26
• The survey explores application scenarios and offers insights into future opportunities and directions in the field.Source: §9 · p. 26