$K^2$VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting

Probabilistic Time Series Forecasting (PTSF) plays a crucial role in decision-making across various fields, including economics, energy, and transportation. Most existing methods excell at short-term forecasting, while overlooking the hurdles of Long-term Probabilistic Time Series Forecasting (LPTSF). As the forecast horizon extends, the inherent nonlinear dynamics have a significant adverse effect on prediction accuracy, and make generative models inefficient by increasing the cost of each iteration. To overcome these limitations, we introduce $K^2$VAE, an efficient VAE-based generative model that leverages a KoopmanNet to transform nonlinear time series into a linear dynamical system, and devises a KalmanNet to refine predictions and model uncertainty in such linear system, which reduces error accumulation in long-term forecasting. Extensive experiments demonstrate that $K^2$VAE outperforms state-of-the-art methods in both short- and long-term PTSF, providing a more efficient and accurate solution.

Binary Cumulative Encoding meets Time Series Forecasting

Recent studies in time series forecasting have explored formulating regression via classification task. By discretizing the continuous target space into bins and predicting over a fixed set of classes, these approaches benefit from stable training, robust uncertainty modeling, and compatibility with modern deep learning architectures. However, most existing methods rely on one-hot encoding that ignores the inherent ordinal structure of the underlying values. As a result, they fail to provide information about the relative distance between predicted and true values during training. In this paper, we propose to address this limitation by introducing binary cumulative encoding (BCE), that represents scalar targets into monotonic binary vectors. This encoding implicitly preserves order and magnitude information, allowing the model to learn distance-aware representations while still operating within a classification framework. We propose a convolutional neural network architecture specifically designed for BCE, incorporating residual and dilated convolutions to enable fast and expressive temporal modeling. Through extensive experiments on benchmark forecasting datasets, we show that our approach outperforms widely used methods in both point and probabilistic forecasting, while requiring fewer parameters and enabling faster training.

Effective Probabilistic Time Series Forecasting with Fourier Adaptive Noise-Separated Diffusion

We propose the Fourier Adaptive Lite Diffusion Architecture (FALDA), a novel probabilistic framework for time series forecasting. First, we introduce the Diffusion Model for Residual Regression (DMRR) framework, which unifies diffusion-based probabilistic regression methods. Within this framework, FALDA leverages Fourier-based decomposition to incorporate a component-specific architecture, enabling tailored modeling of individual temporal components. A conditional diffusion model is utilized to estimate the future noise term, while our proposed lightweight denoiser, DEMA (Decomposition MLP with AdaLN), conditions on the historical noise term to enhance denoising performance. Through mathematical analysis and empirical validation, we demonstrate that FALDA effectively reduces epistemic uncertainty, allowing probabilistic learning to primarily focus on aleatoric uncertainty. Experiments on six real-world benchmarks demonstrate that FALDA consistently outperforms existing probabilistic forecasting approaches across most datasets for long-term time series forecasting while achieving enhanced computational efficiency without compromising accuracy. Notably, FALDA also achieves superior overall performance compared to state-of-the-art (SOTA) point forecasting approaches, with improvements of up to 9%.

Context-Aware Probabilistic Modeling with LLM for Multimodal Time Series Forecasting

Time series forecasting is important for applications spanning energy markets, climate analysis, and traffic management. However, existing methods struggle to effectively integrate exogenous texts and align them with the probabilistic nature of large language models (LLMs). Current approaches either employ shallow text-time series fusion via basic prompts or rely on deterministic numerical decoding that conflict with LLMs' token-generation paradigm, which limits contextual awareness and distribution modeling. To address these limitations, we propose CAPTime, a context-aware probabilistic multimodal time series forecasting method that leverages text-informed abstraction and autoregressive LLM decoding. Our method first encodes temporal patterns using a pretrained time series encoder, then aligns them with textual contexts via learnable interactions to produce joint multimodal representations. By combining a mixture of distribution experts with frozen LLMs, we enable context-aware probabilistic forecasting while preserving LLMs' inherent distribution modeling capabilities. Experiments on diverse time series forecasting tasks demonstrate the superior accuracy and generalization of CAPTime, particularly in multimodal scenarios. Additional analysis highlights its robustness in data-scarce scenarios through hybrid probabilistic decoding.

Non-stationary Diffusion For Probabilistic Time Series Forecasting

Due to the dynamics of underlying physics and external influences, the uncertainty of time series often varies over time. However, existing Denoising Diffusion Probabilistic Models (DDPMs) often fail to capture this non-stationary nature, constrained by their constant variance assumption from the additive noise model (ANM). In this paper, we innovatively utilize the Location-Scale Noise Model (LSNM) to relax the fixed uncertainty assumption of ANM. A diffusion-based probabilistic forecasting framework, termed Non-stationary Diffusion (NsDiff), is designed based on LSNM that is capable of modeling the changing pattern of uncertainty. Specifically, NsDiff combines a denoising diffusion-based conditional generative model with a pre-trained conditional mean and variance estimator, enabling adaptive endpoint distribution modeling. Furthermore, we propose an uncertainty-aware noise schedule, which dynamically adjusts the noise levels to accurately reflect the data uncertainty at each step and integrates the time-varying variances into the diffusion process. Extensive experiments conducted on nine real-world and synthetic datasets demonstrate the superior performance of NsDiff compared to existing approaches. Code is available at https://github.com/wwy155/NsDiff.

Probabilistic QoS Metric Forecasting in Delay-Tolerant Networks Using Conditional Diffusion Models on Latent Dynamics

Active QoS metric prediction, commonly employed in the maintenance and operation of DTN, could enhance network performance regarding latency, throughput, energy consumption, and dependability. Naturally formulated as a multivariate time series forecasting problem, it attracts substantial research efforts. Traditional mean regression methods for time series forecasting cannot capture the data complexity adequately, resulting in deteriorated performance in operational tasks in DTNs such as routing. This paper formulates the prediction of QoS metrics in DTN as a probabilistic forecasting problem on multivariate time series, where one could quantify the uncertainty of forecasts by characterizing the distribution of these samples. The proposed approach hires diffusion models and incorporates the latent temporal dynamics of non-stationary and multi-mode data into them. Extensive experiments demonstrate the efficacy of the proposed approach by showing that it outperforms the popular probabilistic time series forecasting methods.

Fixing the Pitfalls of Probabilistic Time-Series Forecasting Evaluation by Kernel Quadrature

Despite the significance of probabilistic time-series forecasting models, their evaluation metrics often involve intractable integrations. The most widely used metric, the continuous ranked probability score (CRPS), is a strictly proper scoring function; however, its computation requires approximation. We found that popular CRPS estimators--specifically, the quantile-based estimator implemented in the widely used GluonTS library and the probability-weighted moment approximation--both exhibit inherent estimation biases. These biases lead to crude approximations, resulting in improper rankings of forecasting model performance when CRPS values are close. To address this issue, we introduced a kernel quadrature approach that leverages an unbiased CRPS estimator and employs cubature construction for scalable computation. Empirically, our approach consistently outperforms the two widely used CRPS estimators.

Probabilistic Functional Neural Networks

High-dimensional functional time series (HDFTS) are often characterized by nonlinear trends and high spatial dimensions. Such data poses unique challenges for modeling and forecasting due to the nonlinearity, nonstationarity, and high dimensionality. We propose a novel probabilistic functional neural network (ProFnet) to address these challenges. ProFnet integrates the strengths of feedforward and deep neural networks with probabilistic modeling. The model generates probabilistic forecasts using Monte Carlo sampling and also enables the quantification of uncertainty in predictions. While capturing both temporal and spatial dependencies across multiple regions, ProFnet offers a scalable and unified solution for large datasets. Applications to Japan's mortality rates demonstrate superior performance. This approach enhances predictive accuracy and provides interpretable uncertainty estimates, making it a valuable tool for forecasting complex high-dimensional functional data and HDFTS.

Foundation Models for Time Series: A Survey

Transformer-based foundation models have emerged as a dominant paradigm in time series analysis, offering unprecedented capabilities in tasks such as forecasting, anomaly detection, classification, trend analysis and many more time series analytical tasks. This survey provides a comprehensive overview of the current state of the art pre-trained foundation models, introducing a novel taxonomy to categorize them across several dimensions. Specifically, we classify models by their architecture design, distinguishing between those leveraging patch-based representations and those operating directly on raw sequences. The taxonomy further includes whether the models provide probabilistic or deterministic predictions, and whether they are designed to work with univariate time series or can handle multivariate time series out of the box. Additionally, the taxonomy encompasses model scale and complexity, highlighting differences between lightweight architectures and large-scale foundation models. A unique aspect of this survey is its categorization by the type of objective function employed during training phase. By synthesizing these perspectives, this survey serves as a resource for researchers and practitioners, providing insights into current trends and identifying promising directions for future research in transformer-based time series modeling.

Mamba time series forecasting with uncertainty propagation

State space models, such as Mamba, have recently garnered attention in time series forecasting due to their ability to capture sequence patterns. However, in electricity consumption benchmarks, Mamba forecasts exhibit a mean error of approximately 8\%. Similarly, in traffic occupancy benchmarks, the mean error reaches 18\%. This discrepancy leaves us to wonder whether the prediction is simply inaccurate or falls within error given spread in historical data. To address this limitation, we propose a method to quantify the predictive uncertainty of Mamba forecasts. Here, we propose a dual-network framework based on the Mamba architecture for probabilistic forecasting, where one network generates point forecasts while the other estimates predictive uncertainty by modeling variance. We abbreviate our tool, Mamba with probabilistic time series forecasting, as Mamba-ProbTSF and the code for its implementation is available on GitHub (https://github.com/PessoaP/Mamba-ProbTSF). Evaluating this approach on synthetic and real-world benchmark datasets, we find Kullback-Leibler divergence between the learned distributions and the data--which, in the limit of infinite data, should converge to zero if the model correctly captures the underlying probability distribution--reduced to the order of $10^{-3}$ for synthetic data and $10^{-1}$ for real-world benchmark, demonstrating its effectiveness. We find that in both the electricity consumption and traffic occupancy benchmark, the true trajectory stays within the predicted uncertainty interval at the two-sigma level about 95\% of the time. We end with a consideration of potential limitations, adjustments to improve performance, and considerations for applying this framework to processes for purely or largely stochastic dynamics where the stochastic changes accumulate, as observed for example in pure Brownian motion or molecular dynamics trajectories.