Uncertainty Quantification Protocol

• Uncertainty Quantification Protocol is a systematic framework that characterizes, propagates, and evaluates model uncertainties in high-stakes applications.
• It integrates spatially-conditioned methods, bootstrap ensembles, and split-conformal calibration to offer localized insights tailored for operational decision-making.
• Advanced techniques like Bayesian model averaging and meta-uncertainty assessments enable efficient diagnostics and scalable integration with complex, data-driven models.

Uncertainty Quantification Protocol

Uncertainty quantification (UQ) protocols provide systematic, rigorous frameworks for characterizing, propagating, and evaluating uncertainty in models and predictions, especially in high-stakes or operational contexts where credible bound estimates are essential. Recent advances have introduced specialized protocols tailored to distinct domains, including spatially-conditioned diagnostics in environmental prediction, model-agnostic predictive inference in machine learning, and standardized industrial workflows. This article summarizes leading UQ protocols, detailing their mathematical formulations, algorithmic components, and operational role across scientific and engineering domains.

1. Spatially-Conditioned Protocols: Fire-Centered Evaluation Region (FCER)

The Fire-Centered Evaluation Region (FCER) protocol is designed for operationally relevant uncertainty quantification of boundary-sensitive systems, exemplified by wildfire spread prediction (Funk, 4 May 2026). It addresses the inadequacy of global performance metrics in capturing local, high-consequence uncertainties.

Mathematical Formalism

• Let $\Omega_t \subset \mathbb{R}^2$ denote the ground-truth binary fire mask at time $t$, with boundary $\partial\Omega_t$ defined as those $y \in \mathbb{R}^2$ where each neighborhood of $y$ contains both fire and non-fire pixels.
• Define the distance-to-front:

$$d_\text{front}(x) = \inf_{y\in\partial\Omega_t} \|x - y\|_2.$$

• For buffer width $w$, the spatially-conditioned Fire-Centered Evaluation Region is

$$\text{FCER}_t(w) = \left\{ x \in \mathbb{R}^2 : d_\text{front}(x) \leq w \right\}.$$

In practice, the binary mask $\Omega_t$ is dilated by a disk of radius $w$, and all uncertainty diagnostics are restricted to the dilated band.

UQ Diagnostics within FCER

• Ensemble Variance: Given $t$0 probabilistic forecasts $t$1,

$t$2

• Student Model Uncertainty: A lightweight model outputs $t$3 trained to mimic $t$4, with root-mean-squared-log error (RMSLE) loss:

$t$5

Spatially-restricted UQ Metrics

All diagnostics below are computed only over $t$6:

• Expected Calibration Error (ECE):

$t$7

• Negative Log-Likelihood (NLL):

$t$8

with predictions clipped for numerical stability.

• Continuous Ranked Probability Score (CRPS):

$t$9

Key Findings

The protocol demonstrates that focusing UQ diagnostics within FCER reveals spatially dependent uncertainty behaviors. In wildfire experiments, a distilled single-pass model (DUDES) matches ensemble calibration and surpasses ensemble uncertainty ranking (AUROC, AUPRC) specifically in boundary neighborhoods—regions of maximal operational interest—at significantly reduced inference cost. For example, at $\partial\Omega_t$0 average segmentation distance ($\partial\Omega_t$1 km), the student achieved AUROC $\partial\Omega_t$2 versus ensemble $\partial\Omega_t$3, and AUPRC $\partial\Omega_t$4 versus $\partial\Omega_t$5 (+50% relative gain; Wilcoxon $\partial\Omega_t$6) (Funk, 4 May 2026).

2. Predictability-Computability-Stability (PCS-UQ) Protocol

The PCS-UQ protocol formalizes UQ as a model- and data-driven process aimed at robust, local adaptivity and tight coverage, integrating best practices from the predictability, computability, and stability framework (Agarwal et al., 13 May 2025).

PCS-UQ Algorithm

1. Prediction Check: Split dataset into training/validation. For $\partial\Omega_t$7 candidate models, select the $\partial\Omega_t$8 with lowest validation loss.
2. Bootstrap Universe: For $\partial\Omega_t$9 bootstrap resamples, retrain each selected model; this assesses both inter-sample (data) and inter-model (algorithmic) variability.
3. Preliminary Sets: For each sample, aggregate out-of-bag prediction values across models/bootstraps, compute quantile-based preliminary intervals, half-range, and center.
4. Multiplicative Calibration: Calibrate interval widths globally or via split-conformal logic:
   • For each sample, define the smallest $y \in \mathbb{R}^2$0 such that the true outcome is in $y \in \mathbb{R}^2$1.
   • Globally adjust $y \in \mathbb{R}^2$2 so at least $y \in \mathbb{R}^2$3 of outcomes are covered.
5. Test-time Prediction Interval: For new $y \in \mathbb{R}^2$4, aggregate all model/bootstrap outputs, form interval $y \in \mathbb{R}^2$5.

Split-Conformal Variant

A held-out calibration set is used. The $y \in \mathbb{R}^2$6-th smallest local calibration score $y \in \mathbb{R}^2$7 determines the interval scale, yielding coverage guarantees under exchangeable data.

Scalability

For deep networks, computational bottlenecks are alleviated by:

• Monte Carlo Dropout Ensembles: B stochastic forward passes used as surrogates for full bootstrap retraining.
• Weight Perturbation Ensembles: Add Gaussian noise to weights and aggregate responses accordingly.

Empirical Performance

PCS-UQ delivers coverage at the prescribed level, reduces interval widths by $y \in \mathbb{R}^2$8 over standard conformal methods, and achieves robust subgroup-level coverage, with scalable implementations for large neural architectures (Agarwal et al., 13 May 2025).

3. Industrial/General Engineering UQ Workflow

A five-stage, iterative workflow is adopted in critical infrastructure contexts (e.g., natural gas industry), with explicit taxonomy and propagation steps (Kolade, 2023).

Workflow Steps

1. Define UQ Objectives and Quantities of Interest (QOIs): Specify validated metrics to inform actionable decisions.
2. Identify Sources of Uncertainty: Catalog inputs, parameters, numerics, and model-form; distinguish between aleatory (irreducible), epistemic (reducible), and mixed uncertainties.
3. Characterize Uncertainties: Assign mathematical representations (distributions, intervals) and parameterize using data, elicitation, or expert judgment.
4. Propagate Uncertainties: Use sampling (Monte Carlo, LHS), resampling (bootstrap), polynomial chaos, or optimization-based bounding; if both aleatory and epistemic, nest sampling accordingly.
5. Report Results: Summarize QOI statistics, confidence or prediction intervals, probability-boxes for mixed cases, and contextualize findings.

Example Methods and Formulae

• Law of Total Variance: $y \in \mathbb{R}^2$9
• Confidence Interval with Bootstrap: Use empirical quantiles of bootstrap output statistics.

Application Cases

• Gas Dispersion: Combination of LHS for stochastic meteorological variables and global optimization over epistemic intervals, with results visualized as probability boxes.
• ML Classification (MNIST): Bootstrap retraining for parametric uncertainty, with class-wise error distributions identifying model insufficiency (Kolade, 2023).

4. Inverse UQ and Bayesian Model Averaging

Comprehensive UQ protocols for simulation validation integrate:

• Inverse UQ (Bayesian Calibration): Model parameters are inferred from data via explicit likelihoods accounting for parameter, measurement, model discrepancy (modeled as GP), and code surrogate uncertainties.
• Bayesian Hypothesis Testing: Bayes factors compare calibrated (posterior) and uncalibrated (prior) models using validation data.
• Bayesian Model Averaging (BMA): Prediction is a convex mixture between the calibrated and prior models, weighted by posterior model probabilities computed via Bayes factors (Xie et al., 2021).

5. Meta-Uncertainty for Method Reliability Assessment

Meta-Uncertainty protocols assess the sensitivity and reliability of existing UQ estimators via controlled perturbations. For particle image velocimetry (PIV), synthetic particle addition is used to quantify the rate at which the uncertainty estimator's interquartile range (IQR) grows—defining the meta-uncertainty slope $y$0 per method $y$1. Reliability-weighted combination (inverse slope) fuses estimators into a consensus UQ field with improved calibration and robustness (Rajendran et al., 2020).

6. Cross-domain Core Principles and Best Practices

While each domain adapts the UQ protocol to the structure of its uncertainties and risk functions, certain principles are universal:

• Source Taxonomy: Clearly distinguish between aleatory, epistemic, and numerical/model-form uncertainties.
• Propagation Appropriateness: Match the computational method (sampling, expansions, bounding) to the uncertainty type and model structure.
• Localization and Operational Relevance: Diagnostics must focus on regions/conditions aligned with decision impact (e.g., firefront vs. entire domain).
• Validation and Calibration: All protocols emphasize empirical coverage validation, calibration error measurement, and, where feasible, local adaptation.
• Reporting Standards: Confidence intervals, calibration plots, and sensitivity indices should be interpreted within the context of both model limitations and uncertainty sources.
• Scalability: Modern protocols accommodate high-dimensional parametric settings via ensembles, surrogates, bootstrapping, and scalable software integration.

7. Summary Table: Representative UQ Protocols

Protocol	Domain/Case	Core Diagnostic Region	Key Methods	Calibration Guarantee
FCER	Wildfire, boundary	Fireline neighborhood	Spatially-conditioned ECE, NLL	Empirical, boundary-local
PCS-UQ	ML regression/class.	Ensemble/bootstrap per sample	Prediction screening, bootstraps, split-conformal	Formal (exchangeability)
Industrial UQ	Eng., gas, ML	QOI-defined, full sys	Sampling, bootstrapping, PCE, p-boxes	Empirical, scenario-dependent
Inverse UQ+BMA	Simulation science	Output space	Bayesian calibration, BMA	Bayesian, model-segmented
Meta-Uncertainty	Measurement (PIV)	Local image windows	Perturbation slope, weighted consensus	Empirical, method reliability

For operational and research UQ, protocol selection and adaptation must reflect both the structure of the quantities of interest and the actionable pathways available to practitioners. Protocols such as FCER and PCS-UQ provide paradigms for actionable, targeted uncertainty quantification aligned to modern data and decision ecosystems.