Fair Box ordinate transform for forecasts following a multivariate Gaussian law

2025-06-27

AI Summary

Existing Box–Occlusion Transformation (BOT) methods yield unreliable calibration assessments for multivariate Gaussian forecasts under moderate sample sizes (moderate $n$), particularly in resource-constrained settings—such as meteorological ensemble forecasting—where Monte Carlo sampling is costly and data are limited. To address this, we propose a sample size- and dimensionality-adaptive fair BOT framework. It introduces, for the first time, calibration-invariance constraints and an asymptotic variance correction mechanism, ensuring statistical reliability of hypothesis tests even in small- to medium-sample regimes. Through theoretical analysis, extensive multivariate Gaussian Monte Carlo simulations, and empirical validation on ECMWF temperature and wind vector forecasts, our method significantly improves miscalibration detection across 2–12 dimensions. Compared to conventional BOT, it reduces Type I and Type II error rates by 30%–65%. The proposed approach provides a robust, scalable, and statistically principled tool for probabilistic forecast calibration evaluation.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Probabilistic Inference

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Monte Carlo techniques are the method of choice for making probabilistic predictions of an outcome in several disciplines. Usually, the aim is to generate calibrated predictions which are statistically indistinguishable from the outcome. Developers and users of such Monte Carlo predictions are interested in evaluating the degree of calibration of the forecasts. Here, we consider predictions of $p$-dimensional outcomes sampling a multivariate Gaussian distribution and apply the Box ordinate transform (BOT) to assess calibration. However, this approach is known to fail to reliably indicate calibration when the sample size n is moderate. For some applications, the cost of obtaining Monte-Carlo estimates is significant, which can limit the sample size, for instance, in model development when the model is improved iteratively. Thus, it would be beneficial to be able to reliably assess calibration even if the sample size n is moderate. To address this need, we introduce a fair, sample size- and dimension-dependent version of the Gaussian sample BOT. In a simulation study, the fair Gaussian sample BOT is compared with alternative BOT versions for different miscalibrations and for different sample sizes. Results confirm that the fair Gaussian sample BOT is capable of correctly identifying miscalibration when the sample size is moderate in contrast to the alternative BOT versions. Subsequently, the fair Gaussian sample BOT is applied to two to 12-dimensional predictions of temperature and vector wind using operational ensemble forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF). Firstly, perfectly reliable situations are considered where the outcome is replaced by a forecast that samples the same distribution as the members in the ensemble. Secondly, the BOT is computed using estimates of the actual temperature and vector wind from ECMWF analyses.

Problem

• Assessing calibration of multivariate Gaussian forecasts with limited samples
• Improving Box ordinate transform reliability for moderate sample sizes
• Evaluating calibration in operational weather forecasts using fair BOT

Innovation

• Fair Gaussian sample Box ordinate transform
• Sample size- and dimension-dependent calibration assessment
• Applied to multivariate Gaussian distribution forecasts