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Enhancing reliability in prediction intervals using point forecasters: Heteroscedastic Quantile Regression and Width-Adaptive Conformal Inference

Carlos Sebastián, Carlos E. González-Guillén, Jesús Juan

TL;DR

This paper tackles the challenge of constructing reliable, width-adaptive prediction intervals for time-series forecasts when only point forecasts are available. It proposes a two-stage pipeline that combines Heteroscedastic Quantile Regression (HQR) to model interval width as a function of prediction difficulty with Width-Adaptive Conformal Inference (WACI) to guarantee coverage uniformly across difficulty levels. The approach is validated on a synthetic benchmark and a real Electricity Price Forecasting (EPF) task, showing improvements in validity and efficiency over standard methods and robustness to changing uncertainty regimes. The results indicate practical benefits for decision-making in industry, offering a model-agnostic framework that preserves finite-sample validity while adapting interval length to predictive uncertainty.

Abstract

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.

Enhancing reliability in prediction intervals using point forecasters: Heteroscedastic Quantile Regression and Width-Adaptive Conformal Inference

TL;DR

This paper tackles the challenge of constructing reliable, width-adaptive prediction intervals for time-series forecasts when only point forecasts are available. It proposes a two-stage pipeline that combines Heteroscedastic Quantile Regression (HQR) to model interval width as a function of prediction difficulty with Width-Adaptive Conformal Inference (WACI) to guarantee coverage uniformly across difficulty levels. The approach is validated on a synthetic benchmark and a real Electricity Price Forecasting (EPF) task, showing improvements in validity and efficiency over standard methods and robustness to changing uncertainty regimes. The results indicate practical benefits for decision-making in industry, offering a model-agnostic framework that preserves finite-sample validity while adapting interval length to predictive uncertainty.

Abstract

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.
Paper Structure (34 sections, 4 theorems, 45 equations, 11 figures, 5 tables, 2 algorithms)

This paper contains 34 sections, 4 theorems, 45 equations, 11 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Context of the problem
  2. Prior work on probabilistic forecasting
  3. General overview
  4. Quantile regression
  5. Quantile Regression Averaging
  6. Conformal Prediction
  7. Conformalized Quantile Regression (CQR)
  8. Adaptive Conformal Inference (ACI)
  9. Our proposal
  10. Heteroscedastic Quantile Regression (HQR)
  11. Width-Adaptive Conformal Inference
  12. Weighting Scheme Considerations
  13. Computational experiments
  14. Evaluation metrics
  15. Mean empirical coverage
  16. ...and 19 more sections

Key Result

Theorem 1

Let $(\bm{x}_i, y_i)_{i=1}^{n+1}$ be exchangeable. The process of conformalizing a conditional mean predictor as described in Algorithm alg:split_conformal produces a prediction interval for the observation $n+1$, $\widehat{C}_{\alpha, n+1}$, such that If, in addition, the scores $\mathcal{S}_{\text{Cal}}$ have a continuous joint distribution, then:

Figures (11)

  • Figure 1: Rolling window mechanism with size equal to 5 time steps. To predict the next time step, only the data from the previous 5 time steps is used to estimate the model parameters.
  • Figure 2: Difference between marginal coverage and conditional coverage in a toy dataset.
  • Figure 3: The joint distribution of two explanatory features is shown on the left. On the right, the expected error for a predictive model is plotted as a function of the two features. One would expect to have a higher error in the unexplored areas of the space, while a lower error would be expected in the very common areas. The plot is for guidance as the model could have good extrapolation properties in some situations.
  • Figure 4: Evolution of $\alpha_t$ in the ACI (orange line) and WACI (blue line) methods. The $\alpha$ used in each iteration per interval length is shown.
  • Figure 5: (a) Comparing the different weight schemes for the WACI algorithm. The exponential decay weight is shown before scaling. (b) The behaviour of the two schemes can be very similar in practice.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Theorem 1: lei2018distribution
  • Theorem 2: romano2019conformalized
  • Theorem 3: gibbs2021adaptive
  • Theorem 4
  • proof