Probabilistic Forecasting

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9 papers in the last four weeks, up 80% on the four weeks before. 0.1% of all new papers.

Jul 6Week of Sep 21

Latest papers 52

Oct 1, 2026cs.LG

Variational Streaming Flow: Probabilistic Forecasting in Physical Time

Probabilistic forecasting is important for predicting complex dynamical systems because intrinsic randomness and incomplete observations can cause the same observed state to evolve into multiple plausible futures. While flow matching is a flexible approach for probabilistic forecasting, it is computationally expensive. Streaming flow (SF) reformulates this approach to model temporal evolution efficiently by learning a continuous velocity field directly in physical time. However, SF learns a deterministic velocity field. Thus, it provides only a single future trajectory for a given fixed initial state and observation history. To overcome this limitation, we introduce Variational Streaming Flow (VSF). Our approach learns a latent distribution that is conditioned on the dynamics of interest. In turn, this enables probabilistic forecasting. Importantly, we retain the computational efficiency of SF by generating in physical time. Across deterministic and stochastic dynamical systems, VSF demonstrates superior predictive accuracy and distributional fidelity. We demonstrate the advantage for both long-horizon rollouts exceeding 1,000 steps, and settings with bifurcating dynamics. Moreover, VSF can be integrated into existing Joint-Embedding Predictive Architecture (JEPA)-based world models as a plug-and-play predictor to improve temporal dynamics and goal-directed success rate in navigation, motion planning, and manipulation.

Hans Hao-Hsun Hsu, Minseon Gwak, Soon Hoe Lim +2

Georgia Institute of Technology · Lawrence Berkeley National Lab · International Computer Science Institute +2

Probabilistic Forecasting Velocity Field Future Video Prediction Joint-Embedding Predictive Architectures Streaming Single Trajectory

Sep 29, 2026cs.LG

GARDiff: Graph-Aligned Residual Diffusion for Probabilistic Multivariate Time-Series Forecasting

Diffusion models have recently shown strong potential for probabilistic multivariate time-series forecasting by modeling complex conditional distributions. Recent decoupled diffusion frameworks further separate forecasting into deterministic prediction and stochastic residual generation, making it natural to derive dependency graphs from deterministic representations and use them to guide residual diffusion. However, we show that this direct structural transfer is unreliable. Although deterministic-derived graphs encode useful global dependency priors, they exhibit substantial edge-level misalignment with residual dependency structures, introducing inaccurate or redundant conditions during residual generation. This reveals a previously overlooked deterministic-to-residual structural alignment problem in decoupled diffusion forecasting. To address this problem, we propose GARDiff, a Graph-Aligned Residual Diffusion framework for probabilistic multivariate time-series forecasting. Instead of treating deterministic-derived graphs as fixed diffusion conditions, GARDiff progressively adapts them to residual generation. Specifically, GARDiff estimates residual uncertainty to distinguish high- and low-uncertainty regions, enabling uncertainty-aware structural refinement, and further performs timestep-aware edge sparsification during reverse diffusion to evolve graph conditions from broad dependency aggregation to localized residual refinement. Extensive experiments on six real-world benchmarks demonstrate that GARDiff consistently improves probabilistic forecasting performance and uncertainty calibration over strong baselines.

Rui Han, Min Yang, Xu Zhang +3

Shandong University, Jinan, China · Macquarie University, Sydney, Australia · China University of Petroleum (East China), Qingdao, China +1

Residual Conditional Diffusion Model Probabilistic Forecasting Prior Knowledge Integration

Sep 28, 2026cs.LG

Probabilistic electrical power demand forecasting with uncertainty quantification

The majority of research on electricity consumption forecasting has focused on deterministic approaches, which generate a single point estimate for each time step in the forecasting horizon. However, the increasing penetration of renewable energy sources and the growing complexity of modern smart grids have introduced greater variability and uncertainty into power-system demand and operation. Consequently, probabilistic forecasting, which quantifies the uncertainty and variability associated with future electricity demand, is becoming increasingly important for reliable power-system planning and operation. This study presents an empirical comparison of four contemporary probabilistic forecasting models for electricity consumption, highlighting their respective strengths and limitations. We have performed comparision on real-world power systems related datasets. Across all power-consumption zones, NGBoost demonstrates superior probabilistic forecasting performance, achieving the lowest MAE and RMSE while providing well-calibrated uncertainty estimates with high prediction-interval coverage and reasonably narrow intervals. These results indicate that NGBoost offers a more accurate and reliable forecasting framework than Bayesian, Monte Carlo (MC) Dropout, and Gaussian Process Regression (GPR) models for the considered electricity consumption data.

Mahesh Neupane, Pragya Dhungana, Pradip Khatri +2

Interconnection and Numbering Section Nepal Telecommunications Authority Kathmandu, Nepal · Nepal Electricity Authority Kathmandu, Nepal · Department of Global Public Health and Primary Care University of Bergen Bergen, Norway +1

Electricity Price Forecasting Probabilistic Forecasting Power Systems Uncertainty Quantification Calibrated Uncertainty Uncertainty+1

Sep 24, 2026cs.LG

Lightweight Probabilistic Downscaling from a Deterministic Base Model

Climate data downscaling is the task of increasing the spatial resolution of climate data, typically by generating fine-resolution regional climate data from coarse global model output. Recent machine learning (ML) work in the related task of weather forecasting has seen significant improvements due to newly devised training methods and architectural components, but these have not yet benefited downscaling. We adapt two of these methods to create a family of lightweight probabilistic ML downscaling models built on a modified U-Net backbone and evaluate them on the CORDEX-ML-Bench suite for daily maximum temperature and precipitation across three geographic regions: the Alps, New Zealand and South Africa. We find that a two-stage training curriculum, combining deterministic pretraining with probabilistic tuning, transfers well to downscaling, beating the state-of-the-art for RMSE. Our work provides an advancement towards lightweight, probabilistic downscaling models, reducing the current trade-off between computational intensity and distributional fit.

Joseph McLean, Tiffany Vlaar, Sigrid Passano Hellan +1

University of Glasgow · Climate System and Climate Services, NORCE Research AS and Bjerknes Centre for Climate Research, Bergen, Norway

Dynamical Downscaling Probabilistic Forecasting Climate Lightweight

Sep 23, 2026cs.LG

fable.intermittent: benchmarking probabilistic forecasting methods for intermittent time series

Intermittent time series are common in spare-parts demand and retail sales. Since the cost of forecast errors is typically asymmetric, decisions such as inventory control require the full predictive distribution rather than a point forecast. Many probabilistic forecasting methods have been proposed; their implementations, however, are scattered across different software frameworks, making it difficult to compare them systematically. We introduce fable

.

intermittent, an R package that implements several probabilistic forecasting methods for intermittent series within the fable framework. The package allows several models to be fitted and evaluated on a collection of time series through a single, simple forecasting pipeline. We also introduce TWEES, a new exponential smoothing model with a Tweedie predictive distribution. Fitting TWEES requires repeated evaluation of the computationally demanding Tweedie density. We also release the R package tweedieDistr, whose implementation of the Tweedie distribution is substantially faster than the existing one while preserving the same numerical accuracy. We evaluate the methods implemented in fable

.

intermittent on four datasets, also released in the package.

Stefano Damato, Lorenzo Zambon, Giorgio Corani +1

SUPSI, Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA), Lugano, Switzerland

Probabilistic Forecasting Time Series Inventory Control

Sep 22, 2026astro-ph.SR

PROSWIN: Probabilistic Solar Wind Speed Forecasting Using Deep Distributional Regression From Solar Images

Accurately predicting fast solar wind conditions is challenging, as uncertainties are large and unquantified by traditional single-value prediction models. In particular, the risks of high-speed solar wind streams (HSSs), which can cause damage to technological infrastructure, cannot be reliably assessed without probabilistic forecasts. We present PROSWIN, a probabilistic machine learning model that forecasts the hourly solar wind speed (SWS) at Earth with a four-day lead time. The approach combines solar images and magnetograms using a deep neural network coupled to a distributional regression algorithm. Because standard error metrics underweight the relevance of HSS peaks, we further introduce the prediction score, a model-selection metric that jointly rewards timeline and HSS peak accuracy. On 14 years of data, our forecast achieves very well-calibrated uncertainties (<1% average deviation). Using the continuous ranked probability score (CRPS), a metric that assesses distributional accuracy, we obtain a timeline CRPS of 41.0 km/s, an HSS peak CRPS of 45.3 km/s, and a prediction score of 42.3 km/s. We find that the 171 Å channel is an important complement to the typically used 193 Å and 211 Å channels and that the prediction score for model selection improves the applicability of the model. Compared to selected models from the literature, ours is the only one that is accurate for both timeline and HSS peak values, rather than trading one off against the other. These results support the advantages of probabilistic over single-value solar wind models. The introduced methods are also transferable to other forecasting problems.

Daniel Collin, Yuri Shprits, Luca Chiarabini +3

Space Physics and Space Weather, GFZ Helmholtz Centre for Geosciences, Potsdam, Germany · Department of Electrical Engineering and Computer Science, Technical University of Berlin, Berlin, Germany · Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany +5

Geomagnetic Storms Probabilistic Forecasting Photovoltaic

Sep 21, 2026cs.LG

$t_0$ : A Time-Series Foundation Model for Forecasting with Context

We present

t_0

, a family of open-weights foundation models for forecasting with multivariate context. We release its first two members:

\texttt{t0-alpha}

and

\texttt{t0-beta}

, respectively 102M and 256M parameters. Both condition their forecasts on target history, past covariates, and known-future covariates, without task-specific retraining. Their transformer layers alternate attention along time and across variates. They produce probabilistic forecasts through quantile predictions. Pretraining combines curated public data with synthetic generator families constructed to contain covariate-to-target dependencies. On GIFT-Eval,

\texttt{t0-alpha}

reaches an aggregate CRPS of 0.4941, and

\texttt{t0-beta}

a CRPS of 0.4738 and a MASE of 0.6865, third on both and within 4.0% of the best zero-shot TSFM. On fev-bench they score 42.2 and 46.7 in skill, the latter third again and 2.0 points behind the leader. We analyze

\texttt{t0-alpha}

in depth. Known-future covariates raise its skill by 6.3 percentage points across 30 tasks. The report also examines its calibration, its rollout strategy on long horizons, and its robustness to missing data. On the Victoria electricity-demand benchmark,

\texttt{t0-beta}

is among the most accurate models with a context of nearly a year. In an independent Macrocosm evaluation of hourly ERCOT prices over 29 months, both cut the MAE of the lagged-price baseline by 38%.

Lucas Meyer, Claudio Sole, Huikan Xiang +6

The Forecasting Company · Macrocosm

Time Series Foundation Models Probabilistic Forecasting Wireless Foundation Models Covariate Balancing

Sep 20, 2026stat.ML

TEMPER: Temporal Encoder-Masked Probabilistic Ensemble Regressor for Time-Series Forecasting

Probabilistic forecasting requires accurate central predictions and calibrated uncertainty estimates. This paper presents TEMPER, the Temporal Encoder-Masked Probabilistic Ensemble Regressor, a univariate time-series forecasting algorithm that combines a temporal autoencoder, a differentiable masked neural decision forest, continuous ranked probability score (CRPS) training, and Gaussian-mixture post-processing. The R implementation is built on torch for R and returns horizon-wise density, distribution, quantile, and sampler functions. We evaluate TEMPER on three deterministic synthetic level series with trend, periodic, regime-switching, nonlinear-threshold, and heteroskedastic components. Across 96 rolling-origin forecasts at horizons t + 1, t + 5, t + 20, and t + 60, TEMPER obtains 2.824% mean CRPS normalized by origin level, 3.635% median absolute error, and 68.8% empirical 90% interval coverage after training with a 300-epoch cap and early-stopping patience of 100. A naive persistence bootstrap has the best aggregate CRPS, 2.763%, while TEMPER has the best median absolute error and the best CRPS at t+1 and t+5. The ablation study uses matched series-origin-horizon cells, horizon-wise CRPS deltas, endpoint sensitivity summaries, and a calibration-specific interval study. Relaxing the learned mask improves average CRPS by 0.472 percentage points on the ablation subset, mainly through long-horizon gains. A twofold interval inflation improves held-out coverage from 54.2% to 91.7% and gives the best 90% interval score among tested calibration rules. The results identify calibration, horizon-specific tuning, and component selection as the central research priorities.

Giancarlo Vercellino

Independent researcher

Time Series Forecasting Probabilistic Forecasting Spatial Encoding Ensemble Horizon

Sep 15, 2026eess.SP

Goal-oriented probabilistic forecasting for dynamic PRB allocation in 5G networks

Efficient physical resource block (PRB) allocation in 5G networks requires accurate demand forecasting. Conventional methods minimize symmetric error metrics (MAE, RMSE), ignoring the operational cost asymmetry where under-provisioning (service degradation) is far costlier than over-provisioning (wasted capacity). We propose a goal-oriented probabilistic forecasting framework that aligns model training with the operator's decision-making objectives. Specifically, we train DeepAR and Temporal Fusion Transformer (TFT) models using the Pinball Loss function and derive the optimal allocation quantile from the operator's cost matrix. Evaluation on a real beam-level 5G traffic dataset shows that the proposed approach reduces operational cost compared to MSE-trained baselines while maintaining calibrated uncertainty estimates. The framework enables dynamic PRB allocation that explicitly balances service reliability against resource efficiency.

Oier Larumbe-Lizarraga, Roberto Pereira, Cristian J. Vaca-Rubio

Universidad Isabel I, Burgos, Spain · Keysight Technologies, Barcelona, Spain · Ericsson Research, Stockholm, Sweden

Probabilistic Forecasting

Sep 2, 2026cs.LG

DynG-Diff: A State-Aware Dynamic Guidance Diffusion Framework for Probabilistic Time Series Forecasting

Probabilistic multivariate time series (MTS) forecasting is crucial for modeling complex dynamical systems. However, existing diffusion-based methods rely on task-specific conditional paradigms that lack flexibility and struggle with inherent "information heterogeneity"--the significantly varying noise levels and evolutionary patterns across variables. To address this, we propose DynG-Diff, a variable-sensitive dynamic guidance diffusion framework for probabilistic multivariate time-series forecasting: (1) DynG-Diff adopts a two-stage separated training strategy and uses an unconditional diffusion backbone to model the joint distribution of multivariate time series. (2) DynG-Diff introduces a lightweight state-aware policy network that adaptively infers variable reliability from real-time noisy states and one-step denoising estimates, outputting a dynamic guidance strength matrix. (3) DynG-Diff mathematically formulates this dynamic weight as the local precision of the observation distribution, enabling precise guidance for high-confidence variables during inference while filtering out interference from anomalous noise. Extensive experiments on real-world benchmarks demonstrate competitive probabilistic forecasting performance against state-of-the-art conditional diffusion models and improved robustness under severe observation corruption.The implementation code is available at: https://github.com/TT-20011031/DynG-Diff

Zhente Zhang, Zhengwei Ni, Wei Fan

School of Information and Electronic Engineering (Sussex Artificial Intelligence Institute), Zhejiang Gongshang University, Hangzhou, Zhejiang, China · School of Computer Science, University of Auckland, Auckland, New Zealand

Multivariate Time Series Forecasting Probabilistic Forecasting Diffusion Models

Sep 1, 2026cs.LG

OutageDiT: A Generative Foundation Model for Power Outage Forecasting and Scenario Simulation

Power-outage planning requires scenarios before an event occurs. These scenarios must represent uncertainty in magnitude, timing, and duration while preserving temporal dependence. However, severe events are rare, and data from any single region contain few examples of extreme outage and restoration patterns. To address this challenge, we introduce OutageDiT, a foundation model for generating seven-day outage trajectories at quarter-hour resolution, trained on outage and weather records across the United States. Specifically, a condition encoder processes the historical context and known future covariates once per forecast, and a shallow flow decoder reuses the resulting horizon-aligned states to generate complete trajectories. The resulting samples support point forecasting, uncertainty quantification, and conditional event simulation within one deep generative model. Across outage forecasting benchmarks, OutageDiT improves forecast accuracy and scenario quality over strong baselines and supports zero-shot transfer to held-out regions. Together, these results position conditional outage simulation as a bridge from outage forecasting to operational planning under uncertainty.

Yunqin Zhu, Feng Qiu, Yao Xie

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology · Argonne National Laboratory

Probabilistic Forecasting Data-Driven Forecasting Generative Models Wireless Foundation Models Edit

Sep 1, 2026cs.AI

LEAP: Likelihood Elicitation and Aggregation for LLM-based Probabilistic Forecasting

LLM-based forecasting systems have improved on real-world tasks such as financial markets and sports outcomes, largely through stronger search and tool use. Many systems still ask an LLM to read all collected evidence together and produce the final forecast. We call this design Monolithic Prediction. It can obscure how individual evidence items affect the result and collapse uncertainty across competing outcomes. We propose LEAP (Likelihood Elicitation and Aggregation for Probabilistic forecasting), which reorganizes how collected evidence is used in the prediction stage. LEAP examines each evidence item separately and elicits likelihood parameters that describe its implications for the target. An explicit prior and a deterministic probabilistic model then combine these likelihoods into a posterior distribution. This procedure supports continuous, single-choice, and multi-choice forecasts while preserving reproducible evidence contributions. We build a benchmark covering forecasting, information-seeking, and browsing tasks, and evaluate LEAP on our own agent loop and several agent CLI frameworks. Given the same evidence, LEAP improves most prediction and calibration metrics across models and remains stronger under controlled comparisons of prior access, inference budget, and aggregation.

Yufei Chen, Yiran Zhao, Xiaogang Xu +3

1Shandong University · 2Nanjing University of Aeronautics and Astronautics · School of Software Technology, Zhejiang University +2

Large Language Model Forecasting Probabilistic Forecasting Elicitation Reproducibility

Aug 20, 2026cs.LG

CLaST: Context-aware Contrastive VAE for Probabilistic Time Series Forecasting

Probabilistic forecasting models are widely used for time series forecasting in domains such as energy systems, finance, medicine, and transportation. In recent years, deep generative models have shown strong results on probabilistic forecasting, yet many conventional approaches struggle to capture internal temporal dependencies, leading to latent representations with limited expressive power. To address this limitation, we propose \textit{CLaST}, a VAE framework for probabilistic multivariate time series forecasting. Unlike existing generative models, CLaST learns embeddings that preserve contextual similarity between observations through our contrastive loss function. Experiments across nine widely adopted benchmarks demonstrate that CLaST consistently surpasses strong baseline methods. In short-term forecasting tasks, our approach achieves improvements of up to

16.4\%

in CRPS and

14.4\%

in NMAE over the second-best method. Furthermore, in long-term prediction CLaST attains superior overall performance, exceeding the second-best method by up to

48.6\%

and

25.1\%

in CRPS and NMAE, respectively.

Alexander Marusov, Dmitry Anikin, Petr Sokerin +3

Applied AI Institute Moscow, Russia

Probabilistic Forecasting Time Series Conditional Variational Autoencoder Generative Models

Aug 13, 2026cs.LG

Defensive Boosting for Online Probabilistic Forecasting

We study online probabilistic forecasting of binary outcomes chosen by an adaptive adversary. Given an online learning algorithm for a weak hypothesis class

H

, we would like to efficiently obtain two incomparable guarantees that existing online boosting techniques provide separately. Online gradient boosting competes in Brier score with the best predictor induced by the span of

H

on every sequence, but promises nothing when the span does not contain an accurate predictor. Online weak-to-strong boosting drives classification error to zero under a weak-learning condition, but promises little when that condition fails. We give a simple defensive forecasting algorithm, the Defensive Booster, that obtains both guarantees. On every adaptive sequence, its Brier score is competitive with the best prediction induced by the span of

H

at the same rate as online gradient boosting; simultaneously, whenever the realized transcript satisfies the smooth weak-learning condition, its Brier score and randomized classification error satisfy the same rate guarantee as online classification boosting. This is achieved by operationalizing the "dual view" of boosting: When the algorithm's randomized classification error is persistently high, its mistake weights form a smooth reweighting on which every weak hypothesis has low edge, yielding an ex-post hard-core certificate that the weak-learning condition fails. We also develop a strongly adaptive variant, which satisfies both guarantees on every time interval. The Defensive Booster is very efficient: it accesses just one weak-class learner, whereas the prior online boosting methods we compare against maintain large weak-learner ensembles. Experiments on synthetic and real data streams demonstrate its strong predictive performance (sometimes substantially improving over all prior baselines) coupled with orders-of-magnitude faster runtime.

Georgy Noarov, Aaron Roth

Probabilistic Forecasting Learning-Augmented Algorithms Data-Driven Forecasting Hypothesis Class LLM Defense Mechanisms

Aug 11, 2026cs.LG

Two-stage Odd Residual Flows for Mean-Preserving Probabilistic Time Series Forecasting

Probabilistic forecasting plays an essential role in risk-sensitive decision-making, particularly in long-horizon settings. However, existing approaches often face a fundamental trade-off between distributional flexibility and accurate mean prediction. Traditional parametric methods, such as Mean Variance Estimation (MVE), can suffer from degraded point accuracy when trained under joint Negative Log-Likelihood (NLL) objectives, while modern-flexible generative models, including Normalizing Flows and Diffusion Models, typically rely on costly Monte Carlo sampling and may yield suboptimal mean estimates. To address this limitation, we propose Two-stage Odd Residual Flows (TORF), a framework that decouples mean forecasting from uncertainty estimation. In the first stage, a pre-trained deterministic model is used to produce an accurate mean prediction. In the second stage, a Restricted Normalizing Flow, with strictly odd functions learns flexible residual distributions around the point forecast, guaranteeing mean preservation from the first stage without sampling. Experiments show that TORF achieves state-of-the-art deterministic accuracy (NMAE) while providing strong density estimation performance (CRPS) on short and long-horizon forecasting.

Kiran Madhusudhanan, Christian Klötergens, Lars Schmidt-Thieme +1

Institute of Computer Science University of Hildesheim Hildesheim, Germany

Probabilistic Forecasting Long-Horizon Forecasting

Aug 5, 2026cs.LG

Real-time probabilistic tsunami forecasting via generative AI

Explicit onshore tsunami inundation forecasting can improve public risk awareness, but deterministically predicted inundation boundaries under highly uncertain conditions, such as near-field tsunamis generated by megathrust earthquakes, may falsely imply safety outside the boundaries. Consequently, current warnings primarily target coastal tsunami height, not onshore inundation. Although machine learning enables instant inundation predictions, they remain deterministic, lacking uncertainty quantification. Here, we develop a probabilistic ensemble model based on a conditional diffusion model (a type of generative AI) that reconciles accuracy with calibration. Validated with the 2011 Tohoku-oki earthquake data, our model faithfully tracks the postearthquake uncertainty decreasing over time while accurately predicting inundation depth and extent. Our framework shows that generative AI can shift tsunami forecasting from determinism to probabilism, providing a foundation for next-generation early warning.

Yusuke Oishi, Takashi Furumura, Fumihiko Imamura

Fujitsu Research, Fujitsu Limited, Kawasaki, Japan. · Earthquake Research Institute, The University of Tokyo, Tokyo, Japan. · International Research Institute of Disaster Science, Tohoku University, Sendai, Japan.

Accurate Flood Prediction Probabilistic Forecasting Full Waveform Inversion Generative Artificial Intelligence Early Warning Real-Time Systems

Aug 3, 2026stat.AP

Probabilistic Deep Learning for Drought Forecasting: Role of Internal Climate Variability

Predicting drought risk is essential for anticipating impacts on water resources, agriculture, ecosystems, and climate adaptation planning. Yet drought forecasts remain uncertain because variability can substantially alter regional precipitation and evaporative demand. Treating this variability as unstructured noise ignores the fact that internal variability has spatial, seasonal, and temporal structure and thus contains information that can be used to improve drought forecasting. We propose a deep-learning-based forecasting framework for European drought prediction and extend it with an uncertainty-aware drought bound that explicitly incorporates internal forecast variability from a large climate model ensemble. This bound represents a physically plausible lower-tail trajectory of future drought conditions and marks how severe drought could plausibly become under an unfavourable realisation of internal variability, giving adaptation planning a conservative, risk-averse reference. We compare the proposed bound with a lower bound derived from reanalysis data only and show that our proposed ensemble-informed bound is better calibrated across most regions and seasons. This is specifically true during anomalously dry conditions, when historical reanalysis alone underestimates lower-tail drought risk. Our results show that internal variability should be treated as a forecast quantity in its own right. More broadly, large ensembles provide a practical way to transfer physically plausible climate variability into machine-learning drought forecasts, yielding risk-aware bounds that are more informative for drought assessment under shifting climate conditions.

Henri Funk, Cornelia Gruber, Göran Kauermann +2

Department of Statistics, LMU Munich, Munich, Germany · Department of Geography, LMU Munich, Munich, Germany

Probabilistic Forecasting Global Climate Models

Jul 29, 2026stat.ML

Crossing-Free Probabilistic K-Line Forecasts Without Retraining

Probabilistic K-line forecasting describes uncertainty in four complementary prices, namely open--high--low--close (OHLC). However, it introduces two consistency problems: quantile crossing and K-line crossing. Quantile crossing occurs when a higher-quantile forecast falls below a lower-quantile forecast, while K-line crossing occurs when the forecast low exceeds the open or close, or the forecast high falls below the open or close. Existing solutions generally address only one problem through output reordering, specialized architectures, or penalized training objectives. We propose K-line--Quantile Sequential Projection (KQSP), a parameter-free and training-free reconciliation method applicable to forecasts produced by any model. Compared with other crossing solutions, KQSP preserves predictive accuracy while producing substantially smaller corrections to the original forecasts. To mitigate model bias, we evaluate KQSP using various models, including pretrained foundation models. KQSP reduces both quantile and K-line crossing rates to zero for all test data undertaken. These results show that probabilistic K-line consistency can be enforced independently of forecast generation and without retraining.

Runyao Yu, Yuchen Tao, Yujie Chen +2

London Business School London, United Kingdom · RWTH Aachen University Aachen, Germany · The Chinese University of Hong Kong Shenzhen, China +1

Probabilistic Forecasting Forecast Quantile Regression Crossing Retraining

Jul 25, 2026cs.LG

Foundation Models and Fine-Tuning: Toward a New Generation of Models for Time Series Forecasting

Inspired by recent breakthroughs in large language models for natural language processing, foundation models have emerged as a promising paradigm for zero-shot time series forecasting, enabling accurate predictions on datasets never seen during pre-training. Ranging from tens to hundreds of millions of parameters, these models are pre-trained on vast and diverse collections of time series, learning generalizable representations that support both point and probabilistic forecasting. This approach alleviates the need for dataset-specific model design and manual tuning, offering a unified solution across forecasting problems. In this work, we review the main architectures, pre-training strategies, and optimization methods underpinning these models. We further investigate post-pre-training fine-tuning of selected foundation models to enhance their performance on specific datasets. Our empirical results demonstrate that this step consistently improves forecasting accuracy over the zero-shot baseline.

Morad Laglil, Bertrand Pracca, Emilie Devijver +1

Savoye, 18 bd des Gorgets, 21000, Dijon, France · Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG, Grenoble, 38000, France

Zero-Shot Forecasting Time Series Forecasting Time Series Wireless Foundation Models Probabilistic Forecasting Foundation Model+1

Jul 22, 2026stat.ML

SPECTRA: State-Space Exogenous Context and Temporal-Frequency Resolution Architecture for Probabilistic Energy Forecasting

Modern power systems increasingly require probabilistic forecasts amid interacting uncertainties from renewable intermittency, flexible demand, market volatility, and weather-dependent generation. However, existing methods often treat multi-scale decomposition, exogenous-variable alignment, and probabilistic output as separate steps, obscuring how predictable structures and uncertainty-bearing fluctuations jointly shape the forecast distribution. This paper proposes a state-space exogenous-context and temporal-frequency resolution architecture for general probabilistic energy forecasting. Its central premise is that trend-periodic components primarily determine the baseline trajectory, whereas high-frequency residuals and external perturbations govern the spread and asymmetry of forecast uncertainty. Accordingly, the architecture adaptively separates deterministic and residual streams, aligns exogenous context with both, refines the deterministic backbone through multi-resolution spectral-temporal state-space modeling, and estimates ordered quantile boundaries from their complementary representations. Experiments on load, price, solar, and wind forecasting achieve the best continuous ranked probability score in 14 of 18 settings, reducing average CRPS by 5.74% and upper-tail quantile risk by 7.27% over the strongest baselines. These results support deterministic-stochastic separation as an effective design principle for general probabilistic energy forecasting.

Hang Ye, Xinyan Jiang, Yuedong Shi +5

Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China · University of Chinese Academy of Sciences, Beijing 101408, China · School of Physical Science and Technology, Shanghaitech University, Shanghai 201210, China +2

Probabilistic Forecasting Energy Forecasting Battery Energy Storage Systems State Space Models Fluctuation-Dissipation Theorem

Jul 14, 2026cs.LG

Learning-based Probabilistic Load Forecasting with Post-hoc and In-model Uncertainty

Smart-building load forecasters are often trained offline on dense, multivariate, high-frequency data, but deployment may provide only hourly, feature-limited inputs. Missing features must then be reconstructed, and their errors can propagate through the model. If this input uncertainty is not reflected, prediction intervals may become miscalibrated, affecting demand-response scheduling. Our work examines where uncertainty should be placed once inference inputs are reconstructed. We develop a unified one-day-ahead probabilistic forecasting framework that aligns temporal resolution, reconstructs the unavailable inputs, and derives causal features, and we compare a modular post-hoc residual-quantile scheme with an integrated in-model quantile-learning scheme. The comparison uses three mid-scale Deep Learning (DL) backbones: recurrent, hybrid recurrent, and attention-based Temporal Fusion Transformer (TFT) models, under identical inputs, forecasting horizon, preprocessing rules, and training budgets. Results show that uncertainty placement is backbone-dependent. Integrated quantile learning is most reliable with the TFT, yielding 2.2-3.6% MAPE and 28-83W RMSE on the labeled test window, while producing intervals about 5x narrower than the modular intervals at the closest-to-nominal coverage level. Diebold-Mariano tests support the TFT ranking and the mixed behavior of the recurrent backbones. A reconstruction-sensitivity test shows that reconstructed inputs increase the Quantile Score (QS) by 106% while interval width remains nearly unchanged, indicating that the model does not automatically absorb reconstruction-induced uncertainty. Robustness checks against non-DL baselines and seasonal hold-out weeks support this ranking. Our results expose the limits of post-hoc residual quantiles when inference depends on reconstructed inputs.

Sarah Al-Shareeda, Gulcihan Ozdemir, Heung Seok Jeon

aComputer System and Technology Department, Faculty of Computer Science and Information Technology, University of Malaya, Malaysia · bInformatics Institute, Istanbul Technical University, Turkey · cComputer Engineering Department Konkuk University South Korea

Accurate Short-Term Electricity Load Forecasting Probabilistic Forecasting Uncertainty Post-Hoc Recurrent Model

Jul 13, 2026stat.AP

Climate-Invariant Conformal Prediction Intervals for Multi-Horizon Solar and Wind Forecasting

Reliable uncertainty quantification is essential for integrating solar and wind generation into modern power systems, where operators must weigh risk rather than act on point forecasts alone. Existing probabilistic methods, however, often either lack finite-sample validity or require per-site recalibration, so a single model rarely transfers across the diverse climates of a dispersed generation fleet. This paper proposes a heteroscedastic, asymmetric, group-conditional split-conformal framework built on a bootstrap-diverse XGBoost ensemble, producing prediction intervals that adapt in width to local difficulty while retaining distribution-free coverage guarantees. A single fixed specification, with no per-site or per-horizon tuning, is evaluated across four climatologically distinct sites spanning both hemispheres, at horizons of 1 to 12 hours, for both solar irradiance and wind speed. The framework holds near-nominal coverage on both targets and reduces the Interval Score by up to 35% relative to competitive baselines, with the calibration and sharpness of its intervals shown to be properties of the method rather than of site-specific tuning.

Shreedhar Gangwar, Abhinav Bains, Banalaxmi Brahma

B. R. Ambedkar National Institute of Technology, Jalandhar, India · Department of Computer Science and Engineering, Dr. B.R. Ambedkar National Institute of Technology Jalandhar, India

Energy Forecasting Probabilistic Forecasting Offshore Wind Turbine Online Conformal Prediction Wind Uncertainty Quantification+1

Jul 2, 2026cs.LG

QuantFlow: A Federated Mamba-Based Post-Transformer Foundation Model for Time-Series Forecasting

Time-series forecasting supports decisions in finance, en-ergy, transportation, public health, and industrial monitoring. Recent foundation models improve transfer across forecast-ing tasks, but many depend on centralized data and Trans-former attention, which restricts their use for long, high-di-mensional, and privacy-sensitive signals. This paper presents QuantFlow, a probabilistic forecasting framework that com-bines inverted sequence embedding, bidirectional Mamba state-space decoders, quantile regression, and federated learning. Each variable is embedded over the complete ob-servation window, processed in forward and reverse direc-tions, and projected to five conditional quantiles. TSMixup expands temporal diversity through Dirichlet-weighted inter-polation while preserving sequence structure. Experiments cover cryptocurrency, traffic, electricity, Electricity Trans-former Temperature, influenza, and weather data. QuantFlow obtains mean squared errors of 0.2834 on ETTm1 and 0.2218 on Weather, and a 20-client non-IID deployment retains use-ful accuracy after three communication rounds without cen-tralizing raw records. The results indicate that selective state-space modelling is a promising basis for scalable, uncer-tainty-aware, and privacy-conscious time-series prediction, while also revealing limitations on irregular epidemiological signals and long-horizon generalization.

Shah Nawaz Haider, Steve Austin, Arnab Barua +2

Department of Computer Science and Engineering, University of Science and Technology Chittagong · Department of Electrical and Electronic Engineering, University of Science and Technology Chittagong · Faculty of Science, Engineering and Technology, University of Science and Technology Chittagong

Time Series Forecasting Probabilistic Forecasting Foundation Model Long-Horizon Inference

Jul 1, 2026cs.LG

Decision-Aware Training for Sample-Based Generative Models

Sample-based generative models are increasingly used for probabilistic forecasting in high-stakes decision settings, yet their training objectives are blind to the decision maker's cost structure. These models are commonly trained with strictly proper scoring rules, such as the energy score, which allocate their training signal in proportion to data density, with no awareness of where forecast errors are most costly for downstream decisions. We therefore propose decision-aware training for sample-based generative models, augmenting the energy score objective with a differentiable decision loss that directly penalises the cost incurred by acting on the model's forecast. This combined loss is theoretically grounded, as the decision loss is itself a proper scoring rule. We validate our method on one synthetic and two real-world tasks, showing targeted improvements in cost-sensitive regions while retaining full probabilistic forecasts.

Kornelius Raeth, Nicole Ludwig

T ubingen AI Center, University of T ubingen, Germany · University of Augsburg, Germany

Probabilistic Forecasting Generative Models Uncertainty Score Energy-Efficient

Jun 30, 2026cs.LG

TRIE: An Evaluation Framework for Stochastic PDE Surrogates

Many scientific systems exhibit uncertainty from stochastic forcing, unresolved degrees of freedom, or imperfect observations, making reliable surrogate forecasting fundamentally distributional rather than pointwise. For such systems, deterministic neural surrogates fail to capture statistical measures and forecast uncertainty. We introduce TRIE, an evaluation framework for stochastic PDE surrogates that asks whether models reproduce invariant measures, provide trustworthy predictive uncertainty, and scale to efficient probabilistic generation. We demonstrate TRIE on two stationary chaotic spatially extended SPDEs, stochastic Kuramoto--Sivashinsky and stochastic Kolmogorov flow, across 11 parameter values. Our evaluation shows that standard pointwise-trained neural surrogates can produce plausible short rollouts while failing to match long-time statistical structure. Approximate uncertainty methods such as Monte Carlo dropout and heteroscedastic Gaussian likelihoods produce stochastic forecasts, but are often miscalibrated and overconfident under temporal and spatial uncertainty diagnostics. Across these criteria, generative models provide the most consistent performance, accurately capturing invariant measure statistics and achieving the lowest CRPS in all reported probabilistic settings. Finally, we show that latent generative models with automatic dimension discovery retain much of this statistical fidelity while reducing Kolmogorov inference time by roughly

12\times

. We release our code and data at https://github.com/scailab/TRIE-SPDE-Bench to support reproducible evaluation of stochastic PDE forecasting models.

Bharat Srikishan, Javier E. Santos, Nikhil Muralidhar +1

Stevens Institute of Technology · Los Alamos National Laboratory

Surrogate Models Probabilistic Forecasting

Jun 23, 2026stat.ML

Model selection with proper scoring rules on data sets of time series: prefer the mean scaled score

We study the problem of model selection among probabilistic forecasting models evaluated on datasets of multiple time series. The performance of a model on a single time series is quantified by the average value (score) of a proper scoring rule over a test set, but extending model selection to data sets of time series requires aggregating these scores. Common approaches either rely on scaling scores and averaging them (mean scaled score) or avoid scaling by using alternative statistics such as mean ranks or win rates. However, these approaches can yield conflicting conclusions. We show that such discrepancies arise from the skewness of the distribution of the scores, which is particularly pronounced when test sets are short. The skewness can cause non-mean criteria (e.g., mean rank, median, win rate) to select misspecified models. In contrast, the mean score is immune from this problem. We further show that, as the size of the test sets increases, all aggregation criteria converge to the same model selection decision, mitigating these discrepancies. Our experiments on intermittent demand time series, including data from the M5 competition, highlight the importance of sufficiently large test sets; the mean scaled score appears to be the more reliable approach, also because empirically we found its decision to remain consistent when different scaling factors are adopted.

Giorgio Corani, Stefano Damato, Dario Azzimonti +1

SUPSI, Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA), Lugano, Switzerland

Model Selection Probabilistic Forecasting Time Series Skewness Sequential Scaling

Jun 11, 2026cs.LG

Navigating the Safety-Fidelity Trade-off: Massive-Variate Time Series Forecasting for Power Systems via Probabilistic Scenarios

Probabilistic forecasting models are increasingly deployed on multivariate systems with distinct channel physics and operational constraints, but existing benchmarks evaluate neither property at scale. Public canonical multivariate benchmarks cap out at 2,000 channels, while power-system benchmarks either lack temporal structure or probabilistic evaluation. We introduce PowerPhase, a probabilistic forecasting benchmark built on six transmission grids ranging from 2,000 to 36,964 jointly forecasted channels, more than an order of magnitude beyond popular canonical multivariate benchmarks. Each target trajectory is the output of an AC power-flow solve, and PowerPhase ships with constraint-aware metrics, including Safety_mBrier, NECV, and CVaR-alpha, that complement CRPS and Distortion. Across eight baselines and three seeds, distributional accuracy and constraint satisfaction rank models differently, a trade-off we term safety-fidelity. We further propose PowerForge, a scenario-based quantile forecaster with type-specific decoding heads and a causal bridge between variable groups, which achieves the best average rank on every grid.

Kaijie Xu, Anqi Wang, Xilin Dai

ZJU-UIUC Institute Zhejiang University Hangzhou, China

Power Systems Probabilistic Forecasting Multivariate Time Series Forecasting Probabilistic Model

Jun 11, 2026cs.LG

Reliability of Probabilistic Emulation of Physical Systems

Two dominant approaches have emerged for generating probabilistic forecasts of physical systems: generative models, such as diffusion or flow matching; and ensembles of deterministic models with stochasticity injected, trained using the continuous ranked probability score (CRPS) loss. While both approaches have demonstrated strong predictive accuracy, the reliability of their uncertainties has not been systematically assessed. We address this gap by developing a framework to evaluate both approaches across diverse 2D spatiotemporal physical systems, under matched model size and computational budget. We assess the reliability of probabilistic emulation by inspecting the empirical coverage of predictive intervals, while also considering accuracy and computational efficiency metrics. CRPS-trained ensembles typically achieve more reliable uncertainties on both single-step prediction and autoregressive rollouts, demonstrating better coverage than the standard alternative of training generative models in a latent space. Moreover, the CRPS approach offers significantly faster inference. When generative models are trained in ambient rather than a compressed latent space, which is often infeasible for high-dimensional problems, they exhibit comparable coverage to CRPS-trained ensembles, though with substantially larger inference latency. In contrast, when CRPS-trained ensembles are trained in latent space they do not show a marked degradation in coverage with respect to ambient space. Both generative models and CRPS-trained ensembles demonstrate good predictive accuracy. To facilitate future research and application, we release AutoCast, a modular framework implementing both generative models and CRPS-trained ensembles, alongside AutoSim, a flexible dataset generation package for rapid prototyping.

Sam F. Greenbury, Radka Jersakova, Paolo Conti +4

The Alan Turing Institute · Autodesk Research · PhysicsX +3

Probabilistic Forecasting Emulators Generative Models Ensemble Spatiotemporal Inference Latency

Jun 8, 2026cs.LG

Investigating Calibration Challenges in Probabilistic Electricity Price Forecasting

As renewable energy integration increases market volatility, probabilistic electricity price forecasting has become essential for effective risk management. However, current-proper-scoring rules often prioritize forecast sharpness at the expense of calibration, leading to overconfident and statistically unreliable uncertainty estimates. This work highlights the critical gap between theoretical scoring and practical calibration, demonstrating that models can become mere proxies for deterministic forecasts when reliability is neglected. We conclude that future research must shift toward calibration-aware objectives and architectures to ensure the distributional integrity of energy market forecasts.

Jan Niklas Lettner, Hadeer El Ashhab, Benjamin Schäfer

Institute for Automation and Applied Informatics

Electricity Price Forecasting Probabilistic Forecasting Challenges

Jun 8, 2026stat.ML

Report the Floor: A Training-Free Conformal Interval Is a Mandatory Baseline for Probabilistic Time-Series Forecasting

Probabilistic forecasters are increasingly learned, yet the baselines they are compared against are often weak or omitted. We show that the simplest possible conformal interval - a last-value point forecast wrapped in a finite-sample split-conformal residual quantile, with no parameters and no training - is a far stronger baseline than its near-total absence from recent learned-forecasting and conformal-time-series comparisons would suggest. In one-step-ahead online forecasting across 2,217 real series from nine public sources (Monash, LOTSA, the LTSF traffic/electricity/weather suites, METR-LA, BOOM, nips/probts), this ConformalNaive interval decisively beats the naive value-quantile baselines, the entire NPTS family (NPTS 73%, SeasonalNPTS 64% of series), and the published Conformal Seasonal Pools (CSP) method (71% of series, bootstrap 95% CI [69,73], paired Wilcoxon p approx 7.6e-135); it is on par with the simpler learned conformal predictors (RCI, quantile regression; median relative Winkler within 2%) and is beaten only by the adaptive-online and ensemble methods (SPCI, ACI, AgACI), which track distribution shift and lead by 9-33% relative Winkler. It is also better calibrated than a trained neural forecaster: on the six datasets that introduced DeepNPTS, the trivial floors cover the truth 84-85% of the time at a nominal 95%, versus DeepNPTS's 66%. At multi-step seasonal horizons the picture inverts: the random-walk floor is the weakest method and the seasonal pool (CSP) wins - a boundary we map. Finally we give ConformalNaive+, a one-line, training-free, horizon-adaptive selector that attains the better of two complementary floors at every horizon with restored coverage. We argue the matching conformal naive floor must be a mandatory baseline whenever a learned probabilistic forecaster claims gains.

Valery Manokhin

Independent researcher.

Probabilistic Forecasting Online Conformal Prediction Conformal Selection Report

Jun 1, 2026stat.ML

ProbRes: Volatility Learning for Probabilistic Time-Series Forecasting

Probabilistic time series forecasting has attracted increasing attention in financial applications due to the need to quantify risk and uncertainty in future observations. We propose ProbRes, a post-hoc probabilistic calibration method that explicitly learns and incorporates volatility dynamics into probabilistic forecasting, enabling effective handling of heteroskedastic data. During training, ProbRes employs two architecture-agnostic modules to separately model the conditional mean and conditional volatility. At the inference stage, it generates predictive distributions by resampling normalized residuals. ProbRes is applicable to both univariate and multivariate time series and remains robust under a wide range of error distributions, including non-Gaussian innovations with conditional heteroskedasticity. Theoretical results demonstrate ProbRes's validity and experiments on both synthetic and real-world datasets show that ProbRes accurately captures predictive distributions and produces well-calibrated prediction intervals.

Tingting Wang, Yunyi Zhang, Benyou Wang

School of Data Science, The Chinese University of Hong Kong, Shenzhen

Probabilistic Forecasting Volatility Heteroskedasticity Time Series

May 27, 2026cs.LG

Stabilizing distribution-free probabilistic forecasts

Multi-step-ahead forecasts are often updated as new observations become available, since shorter forecast horizons typically improve forecast quality. However, such improvements come at the cost of forecast instability, i.e., variability in forecasts for the same target period. This instability can trigger costly changes to plans formulated based on the forecasts and may erode trust in the forecasting system. In this work, we integrate forecast stability alongside forecast quality into the training of distribution-free probabilistic time-series forecasting models, allowing us to control this trade-off. We propose a method for generating stabilized forecasted conditional quantile functions using regression splines parameterized by a neural network. This approach enables joint optimization of quality and stability, as it allows us to directly penalize dissimilarities arising from forecast updates. Furthermore, it allows assigning varying importance to stabilizing different parts of the forecast distributions (e.g., central parts vs. tails) to focus on the parts most relevant for the intended downstream use (e.g., the upper tail for inventory management). We empirically evaluate the proposed method on two datasets with different statistical properties and show that it can effectively reduce forecast instability without a substantial loss in forecast quality, and that it can target stabilization effort toward specific parts of the forecast distributions.

Jente Van Belle, Honglin Wen, Wouter Verbeke +1

Faculty of Economics and Business, KU Leuven, Belgium · Department of Electrical Engineering, Shanghai Jiao Tong University, China · Dyson School of Design Engineering, Imperial College London, United Kingdom +3

Probabilistic Forecasting Short-Horizon Forecasting Forecast Quantile Regression

May 26, 2026cs.LG

Aligning LLMs with Human Uncertainty: A Beta-Bernoulli Calibrator for LLM Forecasting

Probabilistic forecasting estimates the likelihood of uncertain future events. To improve LLM forecasting, existing methods typically learn from binary outcomes to output verbalized forecasts. However, while aggregated human forecasts contain rich information in both the crowd probability estimate and the degree of agreement among forecasters, how to utilize these signals remains underexplored. To address this, we propose the Beta-Bernoulli Calibrator (BBC), which converts an initial point estimate forecast from any model into a distribution over event likelihood, using supervision from both binary outcomes and human forecasts. BBC models event likelihood

p \sim \text{Beta}(α, β)

and outcome

y \sim \text{Bernoulli}(p)

, with the mean as the calibrated point forecast and the variance as the epistemic uncertainty. Our results show that BBC generally provides better calibrated and more accurate forecasts than both traditional post-hoc calibration methods and models fine-tuned specifically for forecasting, while remaining lightweight and having good generalization. We also show that the epistemic uncertainty captured by BBC is a more reliable predictor of forecasting error than verbalized confidence.

Hui Dai, Ryan Teehan, Parsa Torabian +1

Agentic Learning AI Lab, New York University · Chronologies AI

Large Language Model Forecasting Probabilistic Forecasting Large Language Model Uncertainty Probability Large Language Model Alignment Human Agency

May 22, 2026cs.LG

Valid and Expressive Copulas for Irregular Multivariate Time Series

We introduce CopFITi, a copula model for probabilistic forecasting of irregular multivariate time series (IMTS). Our model combines the expressivity of normalizing flows for univariate marginals with the consistency and flexibility of a Gaussian Mixture Copula for the joint dependency structure. Our experiments show that copula-based approaches, which decouple the marginals from the joint, yield better marginal models than architectures that directly fit the full joint. With CopFITi, we propose the first IMTS copula that is marginalization-consistent by construction and establish a new state of the art in joint IMTS density modeling.

Christian Klötergens, Tom Hanika, Lars Schmidt-Thieme +1

Institute of Computer Science University of Hildesheim Hildesheim, Germany

Gaussian Copula Multivariate Time-Series Probabilistic Forecasting Validation

May 22, 2026cs.LG

Parametric Prior Mapping Framework for Non-stationary Probabilistic Time Series Forecasting

Effectively modeling non-stationary dynamics in probabilistic multivariate time series(MTS) forecasting requires balancing expressiveness with robustness. Existing parametric approaches benefit from strong inductive biases but lack flexibility, whereas deep generative models struggle to capture complex temporal dependencies without extensive data and computation. We introduce Parametric Prior Mapping (PPM), a framework that injects parametric structural priors into a generative modeling process. Specifically, PPM utilizes a parametric estimator to derive a dynamic, adaptive prior that guides the learning of a complex predictive distribution via a learnable mapping. This design allows the model to retain the efficiency of parametric methods while exploiting the expressive power of generative models. Trained with a hybrid objective, PPM yields precise forecasts with well-calibrated uncertainty estimates. Empirical results show that PPM outperforms existing baselines in handling non-stationary data, offering a superior trade-off between accuracy and computational efficiency. The code is available at https://github.com/ljl8336/PPM.

Jinglin Li, Jun Tan, QI Fang +1

School of Computer Science and Engineering, Central South University, Changsha, China.

Probabilistic Forecasting Multivariate Time-Series Non-Stationarity Generative Models

May 22, 2026cs.LG

PaP-NF: Probabilistic Long-Term Time Series Forecasting via Prefix-as-Prompt Reprogramming and Normalizing Flows

Time series forecasting plays a central role in many real-world applications and has been extensively studied. Most existing approaches rely on deterministic models. However, real-world environments exhibit inherently uncertain and complex future behaviors, making single-point predictions insufficient. This highlights the need for probabilistic forecasting methods that can quantify and represent uncertainty. In this work, we propose PaP-NF, a probabilistic forecasting framework that aligns continuous time series representations with a frozen large language model (LLM) using a Prefix-as-Prompt mechanism, and conditions a normalizing flow decoder on the global context extracted by the LLM. The quality of the resulting predictive distributions is evaluated using the Continuous Ranked Probability Score (CRPS), a standard metric in probabilistic forecasting. Across a variety of long-term forecasting benchmarks, PaP-NF robustly captures multi-modal uncertainty while maintaining competitive point forecasting accuracy. The official implementation is available at: https://github.com/democracy04/PaP-NF

Minju Kim, Youngbum Hur

Department of Industrial Engineering, Inha University, Incheon, Republic of Korea

Probabilistic Forecasting Time Series Frozen Language Model

May 20, 2026physics.geo-ph

Neural Negative Binomial Regression for Weekly Seismicity Forecasting: Per-Cell Dispersion Estimation and Tail Risk Assessment

Standard approaches to forecasting the weekly number of earthquakes on a spatial grid rely on the Poisson distribution with a single global dispersion assumption. We show that this assumption is systematically violated in seismic data from Central Asia (2010-2024), where a likelihood-ratio test with boundary correction strongly rejects the Poisson hypothesis (p < 10^{-179}). The main contribution of this work is the EarthquakeNet architecture, which provides an endogenous per-cell estimate of the overdispersion parameter alpha via a neural network (spatial embeddings + MLP), without explicit spatial covariance specification. In contrast to existing negative binomial regression approaches in seismological forecasting, which typically assume a single global alpha, the proposed per-cell formulation allows the model to identify spatial heterogeneity in seismic clustering and to construct probabilistic risk-aware alerts via quantiles of the predicted distribution. A walk-forward evaluation (2018-2023) over four systems shows an 8.6 percent reduction in mean pinball deviation (MPD) relative to a negative binomial GLM baseline. The strongest improvements are observed in the tail regime (Y >= 5), where the continuous ranked probability score (CRPS) of the proposed model is 12.5 percent lower than that of the baseline, indicating improved calibration in extreme-event forecasting.

Alim Igilik

[physics.geo-ph]

Full Waveform Inversion Probabilistic Forecasting Poisson Dispersion Logistic Regression

May 19, 2026cs.LG

DeRegiME: Deep Regime Mixtures for Probabilistic Forecasting under Distribution Shift

We introduce DeRegiME -- Deep Regime Mixture of Experts -- a direct multi-horizon probabilistic forecaster that separates latent uncertainty regimes from the underlying signal and softly assigns each forecast location to learned recurring regimes using a sparse variational Gaussian process (GP) whose nonstationary regime-mixing kernel and Student-t likelihood combine per-regime sub-kernels and noise processes via a shared gate. This yields a single sparse-GP posterior, not a mixture of GP experts. DeRegiME addresses a key limitation of neural forecasters: point forecasts discard residual uncertainty, and probabilistic heads -- whether single marginals, uninterpreted mixtures, quantile sets, or diffusion samples -- rarely expose the regime structure of the residual. Yet distribution shift in noisy heteroskedastic time series may be abrupt, gradual, or horizon-dependent and often appears in residual uncertainty rather than the conditional mean. DeRegiME yields an interpretable mean-residual-noise decomposition with a direct-sum feature-space representation that anchors regimes as clusters of residual similarity whose transitions surface as implicit changepoints. The effective number of regimes is pruned by the stick-breaking gate. We prove kernel validity and predictive-density propriety, and across ten benchmarks and three encoder grids DeRegiME improves negative log predictive density (NLPD) by 20.3% over the strongest encoder-matched baseline, a DeepAR/GluonTS-style dynamic Student-t head, with parallel gains on CRPS (3.0%) and MSE (4.7%). Improvements are consistent across all datasets, which span abrupt, gradual, and seasonal shifts.

Kieran Wood, Stefan Zohren, Stephen J. Roberts

Machine Learning Research Group, University of Oxford · Oxford-Man Institute, University of Oxford

Probabilistic Forecasting Distribution Shifts Gaussian Process

May 14, 2026cs.LG

PRB-RUPFormer: A Recursive Unified Probabilistic Transformer for Residual PRB Forecasting

Accurate forecasting of residual Physical Resource Blocks (PRBs) is critical for proactive network slice provisioning, energy-efficient operation, and spectrum-aware decision making in cellular systems, where residual PRBs serve as a practical proxy for short- and medium-term spectrum availability. Existing PRB prediction methods typically rely only on historical PRB values and are trained independently per carrier or sector, limiting their ability to capture cross-carrier dependencies and providing no measure of forecast uncertainty. Moreover, point forecasts alone are insufficient for robust spectrum-aware control under highly variable traffic conditions. This paper proposes PRB-RUPFormer, a recursive unified probabilistic Transformer for residual PRB forecasting. The proposed model jointly processes multivariate KPI time series using temporal, seasonal, and carrier-aware embeddings, preserving inter-metric temporal coupling during recursive rollout and stabilizing long-horizon forecasting. A single shared model is trained across all carriers and sectors of an eNB, enabling efficient learning of joint traffic dynamics with low computational overhead. Forecast uncertainty is captured through quantile-based prediction intervals, providing confidence-aware estimates of future PRB availability. Evaluations on six months of commercial LTE network data from multiple U.S. locations demonstrate median MAE below 0.05 and hit probabilities above 0.80 for both one-day and seven-day recursive forecasts. These probabilistic predictions directly support spectrum-aware RAN functions such as dynamic carrier activation, congestion avoidance, and proactive spectrum sharing, making the proposed framework well-suited for dynamic spectrum access scenarios.

Saad Masrur, Yuxuan Jiang, Matti Hiltunen +2

AT&T RAN Technology, Bedminster, NJ · Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC

Probabilistic Forecasting Mobile Networks Transformer Architectures

May 11, 2026cs.LG

DeepLévy: Learning Heavy-Tailed Uncertainty in Highly Volatile Time Series

Modeling uncertainty in heavy-tailed time series remains a critical challenge for deep probabilistic forecasting models, which often struggle to capture abrupt, extreme events. While Lévy stable distributions offer a natural framework for modeling such non-Gaussian behaviors, the intractability of their probability density functions severely limits conventional likelihood-based inference. To address this, we introduce DeepLévy, a neural framework that learns mixtures of Lévy stable distributions by minimizing the discrepancy between empirical and parametric characteristic functions. DeepLévy incorporates a mixture mechanism that adaptively learns context-dependent weights and parameters over multiple Lévy components, enabling flexible multi-horizon uncertainty modeling. Evaluations on both real and synthetic datasets demonstrate that DeepLévy outperforms state-of-the-art deep probabilistic forecasting approaches in tail risk metrics, especially under extreme volatility.

Yang Yang, Du Yin, Hao Xue +1

University of New South Wales Australia NSW Sydney · The Hong Kong University of Science and Technology (Guangzhou)

Probabilistic Forecasting Long-Tailed Distribution Volatility Time Series Uncertainty Deepseek

May 11, 2026cs.CV

PixelFlowCast: Latent-Free Precipitation Nowcasting via Pixel Mean Flows

Precipitation nowcasting aims to forecast short-term radar echo sequences for extreme weather warning, where both prediction fidelity and inference efficiency are critical for real-world deployment. However, diffusion-based models, despite their strong generative capability, suffer from slow inference due to multi-step sampling trajectories, limiting their practical usability. Conditional Flow Matching (CFM) improves efficiency via straightened trajectories, but relies on latent space compression, which inevitably discards high-frequency physical details and degrades fine-grained prediction quality. To address these limitations, we propose PixelFlowCast, a two-stage probabilistic forecasting framework that achieves both high-efficiency and high-fidelity prediction without latent compression. Specifically, in the first stage, a deterministic model first produces coarse forecasts to capture global evolution trends. In the subsequent stage, the proposed KANCondNet extracts deep spatiotemporal evolution features to provide accurate conditional guidance. Based on this, a latent-free, few-step Pixel Mean Flows (PMF) predictor employs an

x

-prediction mechanism to generate high-quality predictions, effectively preserving fine-grained structures while maintaining fast inference. Experiments on the publicly available SEVIR dataset demonstrate that PixelFlowCast outperforms existing mainstream methods in both prediction accuracy and inference efficiency, particularly for long sequence forecasting, highlighting its strong potential for real-world operational deployment.

Yufeng Zhu, Chunlei Shi, Yongchao Feng +1

Precipitation Nowcasting Rainfall Probabilistic Forecasting Spatiotemporal Radar Free

Apr 29, 2026stat.ME

Recipes for Calibration Checks in Safety-Critical Applications

Safety-critical prediction systems, such as autonomous vehicles, weather forecasters, and medical monitors, commonly rely on probabilistic forecasters. These forecasters make predictions about possible future outcomes, and their quality and robustness needs to be validated and certified. Often, only accuracy -- the mean of the predictions -- is evaluated against true outcomes. However, for safety-critical scenarios and decision making under uncertainty, the full distributional properties of the forecasts should be checked: do the observed prediction errors actually follow the forecasted probability distributions? To this end, we introduce a framework for calibration checks: statistical tests that validate distributional properties of forecasts when measured over many samples. In order to support ease-of-use in real-world operations, these checks produce a single accept/reject decision for data collected from a forecaster. This contrasts typical calibration calculations which produce one or multiple continuous calibration scores and require expertise to implement in a validation workflow. We further support operationalization by introducing modifications to calibration testing that (a) reject only overconfident predictions, allowing for pessimistic or cautious predictions in safety-critical settings, and (b) tolerate small, operationally acceptable deviations even for large numbers of validation samples. We organize the calibration checking process into a modular pipeline comprising four steps: (i) the data model, (ii) the chosen metric, (iii) the hypothesis formulation, and (iv) the testing procedure. Each step consists of independently swappable components, thereby supporting a large variety of possible use-cases and trade-offs. We demonstrate the applicability of the framework on two complementary example problems, weather forecasting and robot pose estimation.

Romeo Valentin

Department of Aeronautics and Astronautics Stanford University Stanford, CA 94305, USA

Probabilistic Forecasting Prediction Error Safety-Critical Scenarios Recipe

Apr 20, 2026cs.LG

Barrier-enforced multi-objective optimization for direct point and sharp interval forecasting

This paper proposes a multi-step probabilistic forecasting framework using a single neural-network based model to generate simultaneous point and interval forecasts. Our approach ensures non-crossing prediction intervals (PIs) through a model structure design that strictly satisfy a target coverage probability (PICP) while maximizing sharpness. Unlike existing methods that rely on manual weight tuning for scalarized loss functions, we treat point and PI forecasting as a multi-objective optimization problem, utilizing multi-gradient descent to adaptively select optimal weights. Key innovations include a new PI loss function based on an extended log-barrier with an adaptive hyperparameter to guarantee the coverage, a hybrid architecture featuring a shared temporal model with horizon-specific submodels, and a training strategy. The proposed loss is scale-independent and universally applicable; combined with our training algorithm, the framework eliminates trial-and-error hyperparameter tuning for balancing multiple objectives. Validated by an intra-day solar irradiance forecasting application, results demonstrate that our proposed loss consistently outperforms those in current literature by achieving target coverage with the narrowest PI widths. Furthermore, when compared against LSTM encoder-decoder and Transformer architectures--including those augmented with Chronos foundation models--our method remains highly competitive and can be seamlessly adapted to any deep learning structure.

Worachit Amnuaypongsa, Yotsapat Suparanonrat, Pana Wanitchollakit +1

Department of Electrical Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand · Department of Computer Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, Thailand

Probabilistic Forecasting Energy Forecasting Loss Function Multi-Objective Optimization Time Series Foundation Models

Apr 19, 2026stat.ML

Forecast Sports Outcomes under Efficient Market Hypothesis: Theoretical and Experimental Analysis of Odds-Only and Generalised Linear Models

Converting betting odds into accurate outcome probabilities is a fundamental challenge in order to use betting odds as a benchmark for sports forecasting and market efficiency analysis. In this study, we propose two methods to overcome the limitations of existing conversion methods. Firstly, we propose an odds-only method to convert betting odds to probabilities without using historical data for model fitting. While existing odds-only methods, such as Multiplicative, Shin, and Power exist, they do not adjust for biases or relationships we found in our betting odds dataset, which consists of 90014 football matches across five different bookmakers. To overcome these limitations, our proposed Odds-Only-Equal-Profitability-Confidence (OO-EPC) method aligns with the bookmakers' pricing objectives of having equal confidence in profitability for each outcome. We provide empirical evidence from our betting odds dataset that, for the majority of bookmakers, our proposed OO-EPC method outperforms the existing odds-only methods. Beyond controlled experiments, we applied the OO-EPC method under real-world uncertainty by using it for six iterations of an annual basketball outcome forecasting competition. Secondly, we propose a generalised linear model that utilises historical data for model fitting and then converts betting odds to probabilities. Existing generalised linear models attempt to capture relationships that the Efficient Market Hypothesis already captures. To overcome this shortcoming, our proposed Favourite-Longshot-Bias-Adjusted Generalised Linear Model (FL-GLM) fits just one parameter to capture the favourite-longshot bias, providing a more interpretable alternative. We provide empirical evidence from historical football matches where, for all bookmakers, our proposed FL-GLM outperforms the existing multinomial and logistic generalised linear models.

Kaito Goto, Naoya Takeishi, Takehisa Yairi

Research Center for Advanced Science and Technology, The University of Tokyo, Japan

Betting Probabilistic Forecasting Generalized Linear Model

Apr 16, 2026cs.LG

Assessing the Performance-Efficiency Trade-off of Foundation Models in Probabilistic Electricity Price Forecasting

Large-scale renewable energy deployment introduces pronounced volatility into the electricity system, turning grid operation into a complex stochastic optimization problem. Accurate electricity price forecasting (EPF) is essential not only to support operational decisions, such as optimal bidding strategies and balancing power preparation, but also to reduce economic risk and improve market efficiency. Probabilistic forecasts are particularly valuable because they quantify uncertainty stemming from renewable intermittency, market coupling, and regulatory changes, enabling market participants to make informed decisions that minimize losses and optimize expected revenues. However, it remains an open question which models to employ to produce accurate forecasts. Should these be task-specific machine learning (ML) models or Time Series Foundation Models (TSFMs)? In this work, we compare four models for day-ahead probabilistic EPF (PEPF) in European bidding zones: a deterministic NHITS backbone with Quantile-Regression Averaging (NHITS+QRA) and a conditional Normalizing-Flow forecaster (NF) are compared with two TSFMs, namely Moirai and ChronosX. On the one hand, we find that TSFMs outperform task-specific deep learning models trained from scratch in terms of CRPS, Energy Score, and predictive interval calibration across market conditions. On the other hand, we find that well-configured task-specific models, particularly NHITS combined with QRA, achieve performance very close to TSFMs, and in some scenarios, such as when supplied with additional informative feature groups or adapted via few-shot learning from other European markets, they can even surpass TSFMs. Overall, our findings show that while TSFMs offer expressive modeling capabilities, conventional models remain highly competitive, emphasizing the need to weigh computational expense against marginal performance improvements in PEPF.

Jan Niklas Lettner, Hadeer El Ashhab, Veit Hagenmeyer +1

Karlsruhe Institute of Technology · Karlsruhe, Germany

Electricity Price Forecasting Probabilistic Forecasting Time Series Foundation Models Battery Energy Storage Systems Capability-Speed Trade-Offs Foundation Model+1

Mar 11, 2026stat.ML

Trustworthy Predictive Distributions for Tail Events with Semiparametric Diagnostic Transport Maps

Machine learning forecast systems are moving beyond point predictions to full predictive distributions for future outcomes y conditional on complex inputs x. However, these distributions are often locally miscalibrated, especially for high-stakes tail events where accurate uncertainty quantification is most needed to establish trust in models. Local miscalibration occurs because training data often lack examples of low-frequency events. The goal of this paper is to describe a simple, yet flexible framework that, at deployment, produces interpretable diagnostics and a robust correction mechanism of predictive distributions when train examples are limited. With this goal in mind, we introduce a semiparametric version of the Local Amortized Diagnostic and Reshaping (LADaR) framework that posits a covariate-dependent parametric model for a diagnostic transport map regressed nonparametrically on inputs to describe how to correct tail probabilities across the feature space to match calibration data. These maps provide the user with local, real-time diagnostics and a reshaped predictive distribution that can be related back to physical processes in the input space. We apply these semiparametric diagnostic transport maps to short-term tropical cyclone intensity forecasting to detect evolutionary modes linked to local miscalibration in the National Hurricane Center's forecasts and improve predictions for severe weather hazards.

Elizabeth Cucuzzella, Rafael Izbicki, Ann B. Lee

Department of Statistics and Data Science, Carnegie Mellon University · Department of Statistics, Federal University of Sao Carlos

Probabilistic Forecasting Tropical Cyclones Distributions

Mar 7, 2026stat.ML

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

Epidemiological Models Probabilistic Forecasting Spatiotemporal Spatiotemporal Representations

Mar 4, 2026stat.ME

An intuitive rearranging of the Yates covariance decomposition for probabilistic verification of forecasts with the Brier score

Proper scoring rules are essential for evaluating probabilistic forecasts. We propose a simple algebraic rearrangement of the Yates covariance decomposition of the Brier score into three independently non-negative terms: a variance mismatch term, a correlation deficit term, and a calibration-in-the-large term. This rearrangement makes the optimality conditions for perfect forecasting transparent: the optimal forecast must simultaneously match the variance of outcomes, achieve perfect positive correlation with outcomes, and match the mean of outcomes. Any deviation from these conditions results in a positive contribution to the Brier score.

Bruno Hebling Vieira

Methods of Plasticity Research, Department of Psychology, University of Zurich, Zurich, Switzerland · Methods of Plasticity Research, Department of Psychology, University of Zurich, Zurich,2026 Switzerland

Probabilistic Forecasting Forecast Data-Driven Forecasting Uncertainty Score Covariance Provable Guarantees+1

Feb 13, 2026cs.LG

Probabilistic Wind Power Forecasting with Tree-Based Machine Learning and Weather Ensembles

Accurate production forecasts are essential for the integration of renewable energy sources into the power grid. This paper illustrates how to obtain probabilistic forecasts of wind power generation using gradient boosting trees and an ensemble of weather forecasts. To this end, we perform a comparative analysis across three state-of-the-art probabilistic prediction methods-conformalized quantile regression, natural gradient boosting and conditional diffusion models-all of which can be combined with tree-based machine learning. The methods are validated using four years of data for all Belgian offshore wind farms. We benchmark the models against the power curve and a calibrated wake model as well as a probabilistic method using stochastic variational Gaussian process regression. The tree-based models significantly reduce the mean absolute error in comparison to the deterministic baselines. Additionally, all three methods outperform the Gaussian process baseline in probabilistic skill, while two out of the three also improve point forecast accuracy. The conditional diffusion model attains the best performance, with improvements of 5% in mean absolute error and 12% in continuous rank probability score compared to the probabilistic baseline. Last, the results indicate an average improvement in point forecast accuracy of 17% by using an ensemble of weather forecasts instead of a single provider.

Max Bruninx, Diederik van Binsbergen, Timothy Verstraeten +2

Vrije Universiteit Brussel, Brussels, Belgium · Norwegian University of Science and Technology, Trondheim, Norway

Offshore Wind Turbine Probabilistic Forecasting Gaussian Process Decision Trees

Feb 4, 2026cs.LG

Probabilistic NDVI Forecasting from Sparse Satellite Time Series and Weather Covariates

Short-term forecasting of vegetation dynamics is a key enabler for data-driven decision support in precision agriculture. Normalized Difference Vegetation Index (NDVI) forecasting from satellite observations, however, remains challenging due to sparse and irregular sampling caused by cloud masking, as well as the heterogeneous climatic conditions under which crops evolve. In this work, we propose a probabilistic forecasting framework for field-level NDVI prediction under sparse, irregular clear-sky acquisitions. The architecture separates the encoding of historical NDVI and meteorological observations from future exogenous covariates, fusing both representations for multi-step quantile prediction. To address irregular revisit patterns and horizon-dependent uncertainty, we introduce a temporal-distance weighted quantile loss that aligns the training objective with the effective forecasting horizon. In addition, we incorporate cumulative and extreme-weather feature engineering to capture delayed meteorological effects relevant to vegetation response. Experiments on European satellite data show that the proposed approach outperforms statistical, deep learning, and time-series baselines on both pointwise and probabilistic evaluation metrics. Ablation studies confirm that target history is the primary driver of performance, with meteorological covariates providing additional gains in the full multimodal setting. The code is available at https://github.com/arco-group/ndvi-forecasting.

Irene Iele, Giulia Romoli, Daniele Molino +4

Unit of Artificial Intelligence and Computer Systems, Universit`a Campus Bio-Medico di Roma, Italy · Department of Diagnostics and Intervention, Biomedical Engineering and Radiation Physics, Ume˚a University, Sweden · Department of Naval, Electrical, Electronics and Telecommunications Engineering, University of Genoa, Italy

Normalized Difference Vegetation Index Probabilistic Forecasting Precision Agriculture Forest Biomass Time Series

Oct 10, 2025cs.LG

CRPS-LAM: Probabilistic Regional Weather Forecasting with Continuous Ranked Probability Score

Limited-Area Models (LAMs) enable weather forecasting over regional domains at higher resolutions than what is computationally feasible for global models. At such high resolutions, machine learning approaches for weather prediction increasingly rely on ensemble methods to produce probabilistic forecasts. However, existing machine learning LAMs are not scalable due to relying on computationally costly diffusion models or inefficient graph neural networks. We tackle this by introducing a new hybrid CNN/GNN architecture, tailored to the LAM weather forecasting problem. Using this architecture, we construct the DET-LAM deterministic model, producing LAM forecasts both more efficiently and accurately than its graph-based competitor. We then tackle the ensemble forecasting problem, by using this architecture as a backbone for the generative model CRPS-LAM. CRPS-LAM is trained using a Continuous Ranked Probability Score (CRPS) objective, enabling efficient training and sampling in a single forward pass. This yields a speedup of

\approx \times 39

compared to diffusion-based baselines. We evaluate our approach on regional domains in northern Europe, demonstrating that CRPS-LAM produces skillful and well-calibrated forecasts across a range of atmospheric variables.

Erik Larsson, Joel Oskarsson, Tomas Landelius +1

Linköping University · ETH AI Center · SMHI & Linköping University

Probabilistic Forecasting Small Reasoning Model Graph Neural Networks Probability

Jun 29, 2025cs.LG

Double-Diffusion: Balancing Speed, Accuracy, and Uncertainty in Probabilistic Forecasting for Urban Sensor Networks

Urban sensor networks need forecasts that are accurate, carry useful uncertainty, and refresh fast enough to act on as new readings arrive. These goals conflict: deterministic models give no distribution, while diffusion forecasters model uncertainty but denoise from pure noise over many steps. We present Double-Diffusion, which integrates a closed-form graph-heat prior into a denoising diffusion model. The prior is a parameter-free low-pass forecast over the sensor graph, and it serves two roles: it is the residual target the model generates, and it conditions the denoiser. The reverse process therefore starts near the prior and denoises a short warm-started chain instead of synthesizing from pure noise; the name denotes this composition, a graph diffusion feeding a denoising diffusion. A compact denoiser, DD-Net, is trained as a Denoising Diffusion Probabilistic Model (DDPM) in the Resfusion warm-start formulation, so generation refines the prior over a short truncated chain rather than synthesizing from pure noise; a graph-spectral read-out of the prior residual sets its switchable spatial filter per domain from training data alone. On four real-world air quality and traffic networks, Double-Diffusion attains the best CRPS of all probabilistic methods on every dataset and stays competitive in point accuracy with the strongest baselines, at a fraction of the sampling cost of from-noise diffusion. The code is available at: https://github.com/teddyicare/Double-Diffusion

Hanlin Dong, Arian Prabowo, Hao Xue +4

University of New South Wales Sydney, Australia · University of Maryland College Park, United States · The Hong Kong University of Science and Technology (Guangzhou) Guangzhou, China

Probabilistic Forecasting Diffusion Dynamics Denoising Diffusion Probabilistic Model Uncertainty