SSL based on persistent homology (PH), a mathematical tool for modeling topological features of data that persist across multiple scales.

Electricity price forecasting has become a critical tool for decision-making in energy markets, particularly as the increasing penetration of renewable energy introduces greater volatility and uncertainty. Historically, research in this field has been dominated by point forecasting methods, which provide single-value predictions but fail to quantify uncertainty. However, as power markets evolve due to renewable integration, smart grids, and regulatory changes, the need for probabilistic forecasting has become more pronounced, offering a more comprehensive approach to risk assessment and market participation. This paper presents a review of probabilistic forecasting methods, tracing their evolution from Bayesian and distribution based approaches, through quantile regression techniques, to recent developments in conformal prediction. Particular emphasis is placed on advancements in probabilistic forecasting, including validity-focused methods which address key limitations in uncertainty estimation. Additionally, this review extends beyond the Day-Ahead Market to include the Intra-Day and Balancing Markets, where forecasting challenges are intensified by higher temporal granularity and real-time operational constraints. We examine state of the art methodologies, key evaluation metrics, and ongoing challenges, such as forecast validity, model selection, and the absence of standardised benchmarks, providing researchers and practitioners with a comprehensive and timely resource for navigating the complexities of modern electricity markets.

Precise probabilistic forecasts are fundamental for energy risk management, and there is a wide range of both statistical and machine learning models for this purpose. Inherent to these probabilistic models is some form of uncertainty quantification. However, most models do not capture the full extent of uncertainty, which arises not only from the data itself but also from model and distributional choices. In this study, we examine uncertainty quantification in state-of-the-art statistical and deep learning probabilistic forecasting models for electricity price forecasting in the German market. In particular, we consider deep distributional neural networks (DDNNs) and augment them with an ensemble approach, Monte Carlo (MC) dropout, and conformal prediction to account for model uncertainty. Additionally, we consider the LASSO-estimated autoregressive (LEAR) approach combined with quantile regression averaging (QRA), generalized autoregressive conditional heteroskedasticity (GARCH), and conformal prediction. Across a range of performance metrics, we find that the LEAR-based models perform well in terms of probabilistic forecasting, irrespective of the uncertainty quantification method. Furthermore, we find that DDNNs benefit from incorporating both data and model uncertainty, improving both point and probabilistic forecasting. Uncertainty itself appears to be best captured by the models using conformal prediction. Overall, our extensive study shows that all models under consideration perform competitively. However, their relative performance depends on the choice of metrics for point and probabilistic forecasting.

Vital signs, such as heart rate and blood pressure, are critical indicators of patient health and are widely used in clinical monitoring and decision-making. While deep learning models have shown promise in forecasting these signals, their deployment in healthcare remains limited in part because clinicians must be able to trust and interpret model outputs. Without reliable uncertainty quantification -- particularly calibrated prediction intervals (PIs) -- it is unclear whether a forecasted abnormality constitutes a meaningful warning or merely reflects model noise, hindering clinical decision-making. To address this, we present two methods for deriving PIs from the Reconstruction Uncertainty Estimate (RUE), an uncertainty measure well-suited to vital-sign forecasting due to its sensitivity to data shifts and support for label-free calibration. Our parametric approach assumes that prediction errors and uncertainty estimates follow a Gaussian copula distribution, enabling closed-form PI computation. Our non-parametric approach, based on k-nearest neighbours (KNN), empirically estimates the conditional error distribution using similar validation instances. We evaluate these methods on two large public datasets with minute- and hour-level sampling, representing high- and low-frequency health signals. Experiments demonstrate that the Gaussian copula method consistently outperforms conformal prediction baselines on low-frequency data, while the KNN approach performs best on high-frequency data. These results underscore the clinical promise of RUE-derived PIs for delivering interpretable, uncertainty-aware vital sign forecasts.

Conformal prediction is a popular method to construct prediction intervals for black-box machine learning models with marginal coverage guarantees. In applications with potentially high-impact events, such as flooding or financial crises, regulators often require very high confidence for such intervals. However, if the desired level of confidence is too large relative to the amount of data used for calibration, then classical conformal methods provide infinitely wide, thus, uninformative prediction intervals. In this paper, we propose a new method to overcome this limitation. We bridge extreme value statistics and conformal prediction to provide reliable and informative prediction intervals with high-confidence coverage, which can be constructed using any black-box extreme quantile regression method. The advantages of this extreme conformal prediction method are illustrated in a simulation study and in an application to flood risk forecasting.

Uncertainty Quantification (UQ) is crucial for deploying reliable Deep Learning (DL) models in high-stakes applications. Recently, General Type-2 Fuzzy Logic Systems (GT2-FLSs) have been proven to be effective for UQ, offering Prediction Intervals (PIs) to capture uncertainty. However, existing methods often struggle with computational efficiency and adaptability, as generating PIs for new coverage levels $(ϕ_d)$ typically requires retraining the model. Moreover, methods that directly estimate the entire conditional distribution for UQ are computationally expensive, limiting their scalability in real-world scenarios. This study addresses these challenges by proposing a blueprint calibration strategy for GT2-FLSs, enabling efficient adaptation to any desired $ϕ_d$ without retraining. By exploring the relationship between $α$-plane type reduced sets and uncertainty coverage, we develop two calibration methods: a lookup table-based approach and a derivative-free optimization algorithm. These methods allow GT2-FLSs to produce accurate and reliable PIs while significantly reducing computational overhead. Experimental results on high-dimensional datasets demonstrate that the calibrated GT2-FLS achieves superior performance in UQ, highlighting its potential for scalable and practical applications.

Predicting missing segments in partially observed functions is challenging due to infinite-dimensionality, complex dependence within and across observations, and irregular noise. These challenges are further exacerbated by the existence of two distinct sources of variation in functional data, termed amplitude (variation along the $y$-axis) and phase (variation along the $x$-axis). While registration can disentangle them from complete functional data, the process is more difficult for partial observations. Thus, existing methods for functional data prediction often ignore phase variation. Furthermore, they rely on strong parametric assumptions, and require either precise model specifications or computationally intensive techniques, such as bootstrapping, to construct prediction intervals. To tackle this problem, we propose a unified registration and prediction approach for partially observed functions under the conformal prediction framework, which separately focuses on the amplitude and phase components. By leveraging split conformal methods, our approach integrates registration and prediction while ensuring exchangeability through carefully constructed predictor-response pairs. Using a neighborhood smoothing algorithm, the framework produces pointwise prediction bands with finite-sample marginal coverage guarantees under weak assumptions. The method is easy to implement, computationally efficient, and suitable for parallelization. Numerical studies and real-world data examples clearly demonstrate the effectiveness and practical utility of the proposed approach.