Beyond Point Forecasts: A Survey on Probabilistic Forecasting for Time Series and Spatiotemporal Data

Donia Besher; Rajdeep Pathak; Madhurima Panja; Tanujit Chakraborty

stat.MLSep 11, 2026

Beyond Point Forecasts: A Survey on Probabilistic Forecasting for Time Series and Spatiotemporal Data

Authors: Donia Besher, Rajdeep Pathak, Madhurima Panja, Tanujit Chakraborty

Organizations: SAFIR, Sorbonne University Abu Dhabi, United Arab Emirates · SCAI, Sorbonne Université, Paris, France

Abstract

Probabilistic forecasting is central to decision-making under uncertainty, yet its methodological landscape has become increasingly fragmented across temporal and spatiotemporal forecasting, statistical modeling, machine learning, and deep generative modeling. This survey develops a unified perspective by organizing probabilistic forecasting methods according to where and how uncertainty is introduced into the forecasting pipeline. Our taxonomy connects model-agnostic approaches including ensembles and distribution-free calibration, with model-intrinsic approaches spanning Bayesian modeling, parametric predictive distributions, distributional regression, and modern generative models, and further examines the emerging role of time series foundation models. Beyond methodological synthesis, we identify the assumptions, computational demands, and forms of uncertainty represented by different paradigms, and translate these distinctions into data-driven and domain-specific guidance for method selection. We complement the survey with a cross-paradigm empirical study on univariate, multivariate, and spatiotemporal forecasting tasks. The results reveal that no single uncertainty-quantification paradigm dominates across settings. Calibration, sharpness, predictive accuracy, and computational efficiency can lead to substantially different model preferences, while expressive generative models and zero-shot foundation models exhibit markedly different accuracy-efficiency trade-offs. Lastly, we identify unresolved challenges surrounding uncertainty in evolving dependency structures, physics-informed predictive distributions, forecasting extreme events, handling count-valued, directional, and continuous-time series, and the development of unified software resources. Our survey provides both a conceptual framework and a practical roadmap for probabilistic forecasting research.

Figures & tables

Figure 2: Overview of model-agnostic uncertainty quantification strategies applied to a base forecasting model, comprising residual-based pre-control limits, ensemble-based predictive distributions, and distribution-free calibration.

Figure 3: Illustration of rolling-window conformal prediction for temporal data. Panel

(a)

shows the data split into training (green), calibration (yellow), and test (red) sets. Panel

(b)

displays the conformal scores

S_{\tilde{t}}

computed on the calibration set, with the empirical quantile marked as a dashed black line. Panel

(c)

demonstrates the sliding window mechanism where only the most recent scores (highlighted in blue) contribute to computing the conformal quantile. Panel

(d)

shows the resulting prediction intervals.

Figure 4: Overview of Bayesian methods for probabilistic forecasting, covering classical Bayesian models, Bayesian nonparametric methods, and Bayesian neural networks. Posterior predictive distributions are obtained through exact inference when tractable or through approximate inference methods.

Figure 5: Overview of distributional regression approaches, including quantile regression, engression, and loss-driven interval estimation.

Figure 6: Overview of generative models for probabilistic forecasting.

Figure 7: Overview of diffusion models, including DDPM, score-based, and latent diffusion models.

Figure 8: Evolution of time series foundation models for probabilistic forecasting.

Figure 10: Overview of probabilistic forecasting applications across diverse domains alongside their primary methodological approaches. The associations are illustrative rather than exclusive.

Figure 11: Forecasts and prediction intervals from representative models obtained on COVID-19 cases in Limburg, Belgium. The point predictions and prediction intervals are shown with a representative color in the legend; actuals are color-coded following the taxonomy in Table 1 .

Figure 12: MCB test results for the empirical evaluation based on CRPS for (A) Univariate Air Passengers, (B) Multivariate PEMS-SF, and (C) Spatiotemporal Belgium COVID-19 datasets.

Figure 13: PICP (top) and Winkler score (bottom) attained by the models on the Belgium COVID-19 dataset across 11 spatial nodes.

Figure 14: (A) Training time, (B) inference time, and (C) total time taken by the representative models on the spatiotemporal Belgium COVID-19 dataset.

Probabilistic Forecasting

Probabilistic forecasting predicts distributions or events with quantified uncertainty to support reliable decisions under uncertainty.

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Mar 7, 2026

Deep Generative Spatiotemporal Engression for Probabilistic Forecasting of Epidemics

Accurate and reliable forecasting of epidemic incidences is critical for public health preparedness, yet it remains a challenging task due to complex nonlinear temporal dependencies and heterogeneous spatial interactions. Often, point forecasts generated by spatiotemporal models are unreliable in assigning uncertainty to future epidemic events. Probabilistic forecasting of epidemics is therefore crucial for providing the best or worst-case scenarios rather than a simple, often inaccurate, point estimate. We present deep spatiotemporal engression methods to generate accurate and reliable probabilistic forecasts on low-frequency epidemic datasets. The proposed methods act as distributional lenses, and out-of-sample probabilistic forecasts are generated by sampling from the trained models. Our frameworks encapsulate lightweight deep generative architectures, wherein uncertainty is quantified endogenously, driven by a pre-additive noise component during model construction. We establish geometric ergodicity and asymptotic stationarity of the spatiotemporal engression processes under mild assumptions on the network weights and pre-additive noise process. Comprehensive evaluations across six epidemiological datasets over three forecast horizons demonstrate that the proposal consistently outperforms several temporal and spatiotemporal benchmarks in both point and probabilistic forecasting. Additionally, we explore the explainability of the proposal to enhance the models' practical application for informed, timely public health interventions.

Rajdeep Pathak, Tanujit Chakraborty

SAFIR, Sorbonne University Abu Dhabi, United Arab Emirates · SCAI, Sorbonne Université, Paris, France

Epidemic Forecasting Spatiotemporal Forecasting Probabilistic Time Series Forecasting Spatial Epidemic Modeling

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Mar 23, 2026

Noise Titration: Exact Distributional Benchmarking for Probabilistic Time Series Forecasting

Modern time series forecasting is evaluated almost entirely through passive observation of single historical trajectories, rendering claims about a model's robustness to non-stationarity fundamentally unfalsifiable. We propose a paradigm shift toward interventionist, exact-statistical benchmarking. By systematically titrating calibrated Gaussian observation noise into known chaotic and stochastic dynamical systems, we transform forecasting from a black-box sequence matching game into an exact distributional inference task. Because the underlying data-generating process and noise variance are mathematically explicit, evaluation can rely on exact negative log-likelihoods and calibrated distributional tests rather than heuristic approximations. To fully leverage this framework, we extend the Fern architecture into a probabilistic generative model that natively parameterizes the Symmetric Positive Definite (SPD) cone, outputting calibrated joint covariance structures without the computational bottleneck of generic Jacobian modeling. Under this rigorous evaluation, we find that state-of-the-art zero-shot foundation models behave consistently with the context-parroting mechanism, failing systematically under non-stationary regime shifts and elevated noise. In contrast, Fern explicitly captures the invariant measure and multivariate geometry of the underlying dynamics, maintaining structural fidelity and statistically sharp calibration precisely where massive sequence-matching models collapse.

Qilin Wang

Independent Researcher

Probabilistic Time Series Forecasting Forecasting Benchmarks Non-Stationary Time Series Forecasting Chaotic Time-Series Prediction

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Mar 25, 2026

Beyond Static Uncertainty: Modeling Temporal Uncertainty Dynamics for Probabilistic Time Series Forecasting

Real-world time series exhibit temporally structured uncertainty: volatility clusters in turbulent regimes, dissipates in stable periods, and shifts abruptly around structural breaks. Yet many probabilistic forecasting methods estimate predictive uncertainty as an independent per-step quantity, leaving the evolution and persistence of volatility regimes under-modeled. We formalize this missing dimension as temporal uncertainty dynamics and instantiate it in the Volatility Dynamics Variational Autoencoder (VolDy-VAE), a non-autoregressive generative forecaster with a location-scale decoder. VolDy-VAE combines a location path for mean prediction with a recurrent scale path that transfers and evolves a volatility hidden state from the look-back window to the forecasting horizon, enabling temporally coherent predictive variances. This design yields an adaptive attenuation mechanism: high-variance observations receive lower influence on the location estimate while their uncertainty is preserved through explicit scale predictions. We further provide a simplified regime-switching analysis showing that, when variances are known or consistently estimated, the volatility-aware objective reduces to inverse-variance weighting, whereas MSE-based estimators remain unbiased but statistically inefficient. Experiments on nine benchmarks show that VolDy-VAE improves forecasting accuracy and uncertainty calibration over competitive probabilistic and point-forecasting baselines while maintaining low inference latency; plug-in studies further indicate that the VolDy principle can benefit GAN, Koopman VAE, and Transformer backbones. The source code is publicly available at https://github.com/wangyijunlyy/VolDy-VAE.

Yijun Wang, Qiyuan Zhuang, Larysa Marchanka +1

Department of Computer Science, Southeast University, Nanjing, China · Francisk Skorina Gomel State University, Belarus

Probabilistic Time Series Forecasting Uncertainty Quantification Uncertainty Calibration Variational Autoencoders

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Abstract

Probabilistic Forecasting

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