This paper explores Uncertainty Quantification (UQ) in SVM predictions, particularly for regression and forecasting tasks. Unlike the Neural Network, the SVM solutions are typically more stable, sparse, optimal and interpretable. However, there are only few literature which addresses the UQ in SVM prediction. At first, we provide a comprehensive summary of existing Prediction Interval (PI) estimation and probabilistic forecasting methods developed in the SVM framework and evaluate them against the key properties expected from an ideal PI model. We find that none of the existing SVM PI models achieves a sparse solution. To introduce sparsity in SVM model, we propose the Sparse Support Vector Quantile Regression (SSVQR) model, which constructs PIs and probabilistic forecasts by solving a pair of linear programs. Further, we develop a feature selection algorithm for PI estimation using SSVQR that effectively eliminates a significant number of features while improving PI quality in case of high-dimensional dataset. Finally we extend the SVM models in Conformal Regression setting for obtaining more stable prediction set with finite test set guarantees. Extensive experiments on artificial, real-world benchmark datasets compare the different characteristics of both existing and proposed SVM-based PI estimation methods and also highlight the advantages of the feature selection in PI estimation. Furthermore, we compare both, the existing and proposed SVM-based PI estimation models, with modern deep learning models for probabilistic forecasting tasks on benchmark datasets. Furthermore, SVM models show comparable or superior performance to modern complex deep learning models for probabilistic forecasting task in our experiments.
Individual treatment effect estimation has gained significant attention in recent data science literature. This work introduces the Double Neural Network (Double-NN) method to address this problem within the framework of extended fiducial inference (EFI). In the proposed method, deep neural networks are used to model the treatment and control effect functions, while an additional neural network is employed to estimate their parameters. The universal approximation capability of deep neural networks ensures the broad applicability of this method. Numerical results highlight the superior performance of the proposed Double-NN method compared to the conformal quantile regression (CQR) method in individual treatment effect estimation. From the perspective of statistical inference, this work advances the theory and methodology for statistical inference of large models. Specifically, it is theoretically proven that the proposed method permits the model size to increase with the sample size $n$ at a rate of $O(n^{\zeta})$ for some $0 \leq \zeta<1$, while still maintaining proper quantification of uncertainty in the model parameters. This result marks a significant improvement compared to the range $0\leq \zeta < \frac{1}{2}$ required by the classical central limit theorem. Furthermore, this work provides a rigorous framework for quantifying the uncertainty of deep neural networks under the neural scaling law, representing a substantial contribution to the statistical understanding of large-scale neural network models.
Conformal prediction methods create prediction bands with distribution-free guarantees but do not explicitly capture epistemic uncertainty, which can lead to overconfident predictions in data-sparse regions. Although recent conformal scores have been developed to address this limitation, they are typically designed for specific tasks, such as regression or quantile regression. Moreover, they rely on particular modeling choices for epistemic uncertainty, restricting their applicability. We introduce $\texttt{EPICSCORE}$, a model-agnostic approach that enhances any conformal score by explicitly integrating epistemic uncertainty. Leveraging Bayesian techniques such as Gaussian Processes, Monte Carlo Dropout, or Bayesian Additive Regression Trees, $\texttt{EPICSCORE}$ adaptively expands predictive intervals in regions with limited data while maintaining compact intervals where data is abundant. As with any conformal method, it preserves finite-sample marginal coverage. Additionally, it also achieves asymptotic conditional coverage. Experiments demonstrate its good performance compared to existing methods. Designed for compatibility with any Bayesian model, but equipped with distribution-free guarantees, $\texttt{EPICSCORE}$ provides a general-purpose framework for uncertainty quantification in prediction problems.
The rapid growth of the wind energy sector underscores the urgent need to optimize turbine operations and ensure effective maintenance through early fault detection systems. While traditional empirical and physics-based models offer approximate predictions of power generation based on wind speed, they often fail to capture the complex, non-linear relationships between other input variables and the resulting power output. Data-driven machine learning methods present a promising avenue for improving wind turbine modeling by leveraging large datasets, enhancing prediction accuracy but often at the cost of interpretability. In this study, we propose a hybrid semi-parametric model that combines the strengths of both approaches, applied to a dataset from a wind farm with four turbines. The model integrates a physics-inspired submodel, providing a reasonable approximation of power generation, with a non-parametric submodel that predicts the residuals. This non-parametric submodel is trained on a broader range of variables to account for phenomena not captured by the physics-based component. The hybrid model achieves a 37% improvement in prediction accuracy over the physics-based model. To enhance interpretability, SHAP values are used to analyze the influence of input features on the residual submodel's output. Additionally, prediction uncertainties are quantified using a conformalized quantile regression method. The combination of these techniques, alongside the physics grounding of the parametric submodel, provides a flexible, accurate, and reliable framework. Ultimately, this study opens the door for evaluating the impact of unmodeled variables on wind turbine power generation, offering a basis for potential optimization.
Probabilistic forecasting in power systems often involves multi-entity datasets like households, feeders, and wind turbines, where generating reliable entity-specific forecasts presents significant challenges. Traditional approaches require training individual models for each entity, making them inefficient and hard to scale. This study addresses this problem using GUIDE-VAE, a conditional variational autoencoder that allows entity-specific probabilistic forecasting using a single model. GUIDE-VAE provides flexible outputs, ranging from interpretable point estimates to full probability distributions, thanks to its advanced covariance composition structure. These distributions capture uncertainty and temporal dependencies, offering richer insights than traditional methods. To evaluate our GUIDE-VAE-based forecaster, we use household electricity consumption data as a case study due to its multi-entity and highly stochastic nature. Experimental results demonstrate that GUIDE-VAE outperforms conventional quantile regression techniques across key metrics while ensuring scalability and versatility. These features make GUIDE-VAE a powerful and generalizable tool for probabilistic forecasting tasks, with potential applications beyond household electricity consumption.
Post-hoc calibration of pre-trained models is critical for ensuring reliable inference, especially in safety-critical domains such as healthcare. Conformal Prediction (CP) offers a robust post-hoc calibration framework, providing distribution-free statistical coverage guarantees for prediction sets by leveraging held-out datasets. In this work, we address a decentralized setting where each device has limited calibration data and can communicate only with its neighbors over an arbitrary graph topology. We propose two message-passing-based approaches for achieving reliable inference via CP: quantile-based distributed conformal prediction (Q-DCP) and histogram-based distributed conformal prediction (H-DCP). Q-DCP employs distributed quantile regression enhanced with tailored smoothing and regularization terms to accelerate convergence, while H-DCP uses a consensus-based histogram estimation approach. Through extensive experiments, we investigate the trade-offs between hyperparameter tuning requirements, communication overhead, coverage guarantees, and prediction set sizes across different network topologies. The code of our work is released on: https://github.com/HaifengWen/Distributed-Conformal-Prediction.
Uncertainty quantification is crucial in time series prediction, and quantile regression offers a valuable mechanism for uncertainty quantification which is useful for extreme value forecasting. Although deep learning models have been prominent in multi-step ahead prediction, the development and evaluation of quantile deep learning models have been limited. We present a novel quantile regression deep learning framework for multi-step time series prediction. In this way, we elevate the capabilities of deep learning models by incorporating quantile regression, thus providing a more nuanced understanding of predictive values. We provide an implementation of prominent deep learning models for multi-step ahead time series prediction and evaluate their performance under high volatility and extreme conditions. We include multivariate and univariate modelling, strategies and provide a comparison with conventional deep learning models from the literature. Our models are tested on two cryptocurrencies: Bitcoin and Ethereum, using daily close-price data and selected benchmark time series datasets. The results show that integrating a quantile loss function with deep learning provides additional predictions for selected quantiles without a loss in the prediction accuracy when compared to the literature. Our quantile model has the ability to handle volatility more effectively and provides additional information for decision-making and uncertainty quantification through the use of quantiles when compared to conventional deep learning models.
Recent methods in quantile regression have adopted a classification perspective to handle challenges posed by heteroscedastic, multimodal, or skewed data by quantizing outputs into fixed bins. Although these regression-as-classification frameworks can capture high-density prediction regions and bypass convex quantile constraints, they are restricted by quantization errors and the curse of dimensionality due to a constant number of bins per dimension. To address these limitations, we introduce a conformalized high-density quantile regression approach with a dynamically adaptive set of prototypes. Our method optimizes the set of prototypes by adaptively adding, deleting, and relocating quantization bins throughout the training process. Moreover, our conformal scheme provides valid coverage guarantees, focusing on regions with the highest probability density. Experiments across diverse datasets and dimensionalities confirm that our method consistently achieves high-quality prediction regions with enhanced coverage and robustness, all while utilizing fewer prototypes and memory, ensuring scalability to higher dimensions. The code is available at https://github.com/batuceng/max_quantile .
We present a novel approach for predicting the distribution of asset returns using a quantile-based method with Long Short-Term Memory (LSTM) networks. Our model is designed in two stages: the first focuses on predicting the quantiles of normalized asset returns using asset-specific features, while the second stage incorporates market data to adjust these predictions for broader economic conditions. This results in a generalized model that can be applied across various asset classes, including commodities, cryptocurrencies, as well as synthetic datasets. The predicted quantiles are then converted into full probability distributions through kernel density estimation, allowing for more precise return distribution predictions and inferencing. The LSTM model significantly outperforms a linear quantile regression baseline by 98% and a dense neural network model by over 50%, showcasing its ability to capture complex patterns in financial return distributions across both synthetic and real-world data. By using exclusively asset-class-neutral features, our model achieves robust, generalizable results.
Reinforcement learning from human feedback (RLHF) has become a key method for aligning large language models (LLMs) with human preferences through the use of reward models. However, traditional reward models typically generate point estimates, which oversimplify the diversity and complexity of human values and preferences. In this paper, we introduce Quantile Reward Models (QRMs), a novel approach to reward modeling that learns a distribution over rewards instead of a single scalar value. Our method uses quantile regression to estimate a full, potentially multimodal distribution over preferences, providing a more powerful and nuanced representation of preferences. This distributional approach can better capture the diversity of human values, addresses label noise, and accommodates conflicting preferences by modeling them as distinct modes in the distribution. Our experimental results show that QRM outperforms comparable traditional point-estimate models on RewardBench. Furthermore, we demonstrate that the additional information provided by the distributional estimates can be utilized in downstream applications, such as risk-aware reinforcement learning, resulting in LLM policies that generate fewer extremely negative responses. Our code and model are released at https://github.com/Nicolinho/QRM.
Recent advancements in Distributional Reinforcement Learning (DRL) for modeling loss distributions have shown promise in developing hedging strategies in derivatives markets. A common approach in DRL involves learning the quantiles of loss distributions at specified levels using Quantile Regression (QR). This method is particularly effective in option hedging due to its direct quantile-based risk assessment, such as Value at Risk (VaR) and Conditional Value at Risk (CVaR). However, these risk measures depend on the accurate estimation of extreme quantiles in the loss distribution's tail, which can be imprecise in QR-based DRL due to the rarity and extremity of tail data, as highlighted in the literature. To address this issue, we propose EXtreme DRL (EX-DRL), which enhances extreme quantile prediction by modeling the tail of the loss distribution with a Generalized Pareto Distribution (GPD). This method introduces supplementary data to mitigate the scarcity of extreme quantile observations, thereby improving estimation accuracy through QR. Comprehensive experiments on gamma hedging options demonstrate that EX-DRL improves existing QR-based models by providing more precise estimates of extreme quantiles, thereby improving the computation and reliability of risk metrics for complex financial risk management.
Accurate energy demand forecasting is crucial for sustainable and resilient energy development. To meet the Net Zero Representative Concentration Pathways (RCP) $4.5$ scenario in the DACH countries, increased renewable energy production, energy storage, and reduced commercial building consumption are needed. This scenario's success depends on hydroelectric capacity and climatic factors. Informed decisions require quantifying uncertainty in forecasts. This study explores a non-parametric method based on \emph{reproducing kernel Hilbert spaces (RKHS)}, known as kernel quantile regression, for energy prediction. Our experiments demonstrate its reliability and sharpness, and we benchmark it against state-of-the-art methods in load and price forecasting for the DACH region. We offer our implementation in conjunction with additional scripts to ensure the reproducibility of our research.
This paper introduces Conformal Thresholded Intervals (CTI), a novel conformal regression method that aims to produce the smallest possible prediction set with guaranteed coverage. Unlike existing methods that rely on nested conformal frameworks and full conditional distribution estimation, CTI estimates the conditional probability density for a new response to fall into each interquantile interval using off-the-shelf multi-output quantile regression. By leveraging the inverse relationship between interval length and probability density, CTI constructs prediction sets by thresholding the estimated conditional interquantile intervals based on their length. The optimal threshold is determined using a calibration set to ensure marginal coverage, effectively balancing the trade-off between prediction set size and coverage. CTI's approach is computationally efficient and avoids the complexity of estimating the full conditional distribution. The method is theoretically grounded, with provable guarantees for marginal coverage and achieving the smallest prediction size given by Neyman-Pearson . Extensive experimental results demonstrate that CTI achieves superior performance compared to state-of-the-art conformal regression methods across various datasets, consistently producing smaller prediction sets while maintaining the desired coverage level. The proposed method offers a simple yet effective solution for reliable uncertainty quantification in regression tasks, making it an attractive choice for practitioners seeking accurate and efficient conformal prediction.
Accurate precipitation forecasts have a high socio-economic value due to their role in decision-making in various fields such as transport networks and farming. We propose a global statistical postprocessing method for grid-based precipitation ensemble forecasts. This U-Net-based distributional regression method predicts marginal distributions in the form of parametric distributions inferred by scoring rule minimization. Distributional regression U-Nets are compared to state-of-the-art postprocessing methods for daily 21-h forecasts of 3-h accumulated precipitation over the South of France. Training data comes from the M\'et\'eo-France weather model AROME-EPS and spans 3 years. A practical challenge appears when consistent data or reforecasts are not available. Distributional regression U-Nets compete favorably with the raw ensemble. In terms of continuous ranked probability score, they reach a performance comparable to quantile regression forests (QRF). However, they are unable to provide calibrated forecasts in areas associated with high climatological precipitation. In terms of predictive power for heavy precipitation events, they outperform both QRF and semi-parametric QRF with tail extensions.
Constructing prediction intervals for time series forecasting is challenging, particularly when practitioners rely solely on point forecasts. While previous research has focused on creating increasingly efficient intervals, we argue that standard measures alone are inadequate. Beyond efficiency, prediction intervals must adapt their width based on the difficulty of the prediction while preserving coverage regardless of complexity. To address these issues, we propose combining Heteroscedastic Quantile Regression (HQR) with Width-Adaptive Conformal Inference (WACI). This integrated procedure guarantees theoretical coverage and enables interval widths to vary with predictive uncertainty. We assess its performance using both a synthetic example and a real world Electricity Price Forecasting scenario. Our results show that this combined approach meets or surpasses typical benchmarks for validity and efficiency, while also fulfilling important yet often overlooked practical requirements.
Predicting edge weights on graphs has various applications, from transportation systems to social networks. This paper describes a Graph Neural Network (GNN) approach for edge weight prediction with guaranteed coverage. We leverage conformal prediction to calibrate the GNN outputs and produce valid prediction intervals. We handle data heteroscedasticity through error reweighting and Conformalized Quantile Regression (CQR). We compare the performance of our method against baseline techniques on real-world transportation datasets. Our approach has better coverage and efficiency than all baselines and showcases robustness and adaptability.
Constructing valid prediction intervals rather than point estimates is a well-established approach for uncertainty quantification in the regression setting. Models equipped with this capacity output an interval of values in which the ground truth target will fall with some prespecified probability. This is an essential requirement in many real-world applications where simple point predictions' inability to convey the magnitude and frequency of errors renders them insufficient for high-stakes decisions. Quantile regression is a leading approach for obtaining such intervals via the empirical estimation of quantiles in the (non-parametric) distribution of outputs. This method is simple, computationally inexpensive, interpretable, assumption-free, and effective. However, it does require that the specific quantiles being learned are chosen a priori. This results in (a) intervals that are arbitrarily symmetric around the median which is sub-optimal for realistic skewed distributions, or (b) learning an excessive number of intervals. In this work, we propose Relaxed Quantile Regression (RQR), a direct alternative to quantile regression based interval construction that removes this arbitrary constraint whilst maintaining its strengths. We demonstrate that this added flexibility results in intervals with an improvement in desirable qualities (e.g. mean width) whilst retaining the essential coverage guarantees of quantile regression.
This paper explores the use of XGBoost for composite quantile regression. XGBoost is a highly popular model renowned for its flexibility, efficiency, and capability to deal with missing data. The optimization uses a second order approximation of the loss function, complicating the use of loss functions with a zero or vanishing second derivative. Quantile regression -- a popular approach to obtain conditional quantiles when point estimates alone are insufficient -- unfortunately uses such a loss function, the pinball loss. Existing workarounds are typically inefficient and can result in severe quantile crossings. In this paper, we present a smooth approximation of the pinball loss, the arctan pinball loss, that is tailored to the needs of XGBoost. Specifically, contrary to other smooth approximations, the arctan pinball loss has a relatively large second derivative, which makes it more suitable to use in the second order approximation. Using this loss function enables the simultaneous prediction of multiple quantiles, which is more efficient and results in far fewer quantile crossings.
Implicit Q-learning (IQL) serves as a strong baseline for offline RL, which learns the value function using only dataset actions through quantile regression. However, it is unclear how to recover the implicit policy from the learned implicit Q-function and why IQL can utilize weighted regression for policy extraction. IDQL reinterprets IQL as an actor-critic method and gets weights of implicit policy, however, this weight only holds for the optimal value function. In this work, we introduce a different way to solve the implicit policy-finding problem (IPF) by formulating this problem as an optimization problem. Based on this optimization problem, we further propose two practical algorithms AlignIQL and AlignIQL-hard, which inherit the advantages of decoupling actor from critic in IQL and provide insights into why IQL can use weighted regression for policy extraction. Compared with IQL and IDQL, we find our method keeps the simplicity of IQL and solves the implicit policy-finding problem. Experimental results on D4RL datasets show that our method achieves competitive or superior results compared with other SOTA offline RL methods. Especially in complex sparse reward tasks like Antmaze and Adroit, our method outperforms IQL and IDQL by a significant margin.
This work introduces the Supervised Expectation-Maximization Framework (SEMF), a versatile and model-agnostic approach for generating prediction intervals with any ML model. SEMF extends the Expectation-Maximization algorithm, traditionally used in unsupervised learning, to a supervised context, leveraging latent variable modeling for uncertainty estimation. Through extensive empirical evaluation of diverse simulated distributions and 11 real-world tabular datasets, SEMF consistently produces narrower prediction intervals while maintaining the desired coverage probability, outperforming traditional quantile regression methods. Furthermore, without using the quantile (pinball) loss, SEMF allows point predictors, including gradient-boosted trees and neural networks, to be calibrated with conformal quantile regression. The results indicate that SEMF enhances uncertainty quantification under diverse data distributions and is particularly effective for models that otherwise struggle with inherent uncertainty representation.