error analysis

Conventional statistical inference suffers from bias under non-Gaussian measurement errors, violating the classical Gaussian error assumption. Method: This paper proposes a novel debiasing framework grounded in hypercomplex algebra—marking the first application of hypercomplex numbers to measurement error modeling. By explicitly representing and correcting non-Gaussian error structures, the method relaxes restrictive distributional assumptions. It unifies treatment across parametric regression and kernel density estimation, delivering unbiased or nearly unbiased inference under contaminated data. Contribution/Results: Theoretical analysis establishes consistency and asymptotic normality; extensive simulations and real-world sports analytics data demonstrate substantial gains in estimation accuracy, robustness to error distribution misspecification, and practical applicability. The approach offers an interpretable, generalizable paradigm for errors-in-variables problems, advancing beyond traditional moment-based or simulation-extrapolation techniques.

In Gaussian process (GP) regression, systematic input bias errors—e.g., from drifting mobile sensor localization—degrade prediction accuracy; existing approaches require full model retraining upon input correction, incurring prohibitive computational cost. Method: We propose a training-free online GP model refinement method that dynamically corrects for time-varying input biases. Leveraging the differentiability of the squared-exponential kernel, we introduce a second-order Taylor expansion of the GP predictive mean and variance with respect to input perturbations, efficiently computed via pre-estimated Jacobian and Hessian matrices of the kernel. Input bias is estimated in real time using a Kalman filter. Contribution/Results: The method significantly improves both point prediction accuracy and uncertainty quantification quality. In two simulation studies, it achieves millisecond-scale model refinement without retraining, while relaxing the restrictive assumption of zero-mean input noise inherent in conventional GP formulations.

This work addresses the limitations of traditional Gaussian assumptions in accurately representing complex uncertainties, which often lead to information loss and reduced accuracy in multi-stage measurement and control processes. To overcome these challenges, the paper proposes a scalable precision framework based on Gaussian Mixture Models (GMMs), leveraging GMMs as universal approximators of probability density functions. The approach integrates closed-form uncertainty propagation algorithms with memory-efficient computational strategies, thereby transcending the representational constraints of Gaussian methods while maintaining computational tractability. Experimental evaluations in manufacturing and metrology scenarios—such as circular factories—demonstrate that the proposed method significantly enhances the fidelity of uncertainty characterization and propagation, outperforming conventional Gaussian-based techniques.

Quasi-Monte Carlo (QMC) integration achieves high accuracy but lacks reliable error estimation, hindering rigorous uncertainty quantification. Method: Leveraging the empirical observation that randomized QMC (RQMC) estimators approximately follow symmetric distributions, we propose a novel error estimation framework that avoids reliance on the central limit theorem or variance estimation. Our approach integrates classical error bound analysis, RQMC resampling, symmetry testing of the estimator’s distribution, and construction of conservative, controllable confidence intervals. Contribution/Results: The method is both theoretically sound and computationally feasible, preserving QMC’s superior convergence rate while significantly improving the reliability and practicality of error assessment. It establishes a new paradigm for trustworthy uncertainty quantification in high-dimensional numerical integration.

Gaussian process regression (GPR) is often treated as a black-box surrogate model, limiting its pedagogical utility and interpretability in uncertainty quantification (UQ) for beginners. Core UQ tasks—including uncertainty propagation, risk estimation, Bayesian optimization, parameter inference, and sensitivity analysis—require deeper engagement with GPR’s inherent probabilistic structure. Method: This work develops a systematic, pedagogically grounded GPR-based UQ framework that integrates UQ-specific techniques: Bayesian quadrature, active learning, and surrogate-based sensitivity analysis. It emphasizes principled covariance kernel design, Bayesian hyperparameter estimation, and reproducible implementation. Contribution/Results: The framework lowers the barrier to applying GPR in complex UQ scenarios, enhances model transparency and decision reliability, and provides a theoretically rigorous yet practically actionable paradigm for UQ education and research across engineering and scientific disciplines.

This study addresses the limitations of traditional Gaussian process (GP) calibration methods, which neglect intermediate variables in computer experiments, leading to inadequate bias modeling and non-identifiability between the simulator and the discrepancy term. To resolve this, the authors propose a robust GP calibration framework that explicitly incorporates intermediate variables. The approach systematically selects key intermediate variables, constrains the discrepancy term using a scaled Gaussian stochastic process (S-GaSP), and employs space-filling designs to choose constraint points, thereby enabling identifiable joint modeling of the simulator and bias. This work is the first to systematically integrate intermediate variables into GP calibration, substantially improving predictive accuracy and the reliability of uncertainty quantification. Empirical results on nuclear binding energy prediction demonstrate clear superiority over existing baseline methods.

This work addresses the estimation of the mean of discretization errors in numerical solutions of ordinary differential equations by proposing a Bayesian inference framework. The approach models the error as a random variable and constructs a linear Gaussian state-space model by incorporating a Markovian prior informed by classical error analysis and dependent on the solver’s step size. The ensemble Kalman filter is employed to efficiently infer the temporal evolution of the error mean. Theoretical analysis establishes that this prior exhibits a well-defined probabilistic convergence rate as the step size tends to zero. Experimental results on the simple pendulum system and the FitzHugh–Nagumo model demonstrate the method’s effectiveness and practicality in accurately estimating the mean of discretization errors.

This work addresses the challenge of error propagation and unreliable outcomes in noisy quantum computing, arising from both hardware noise and intrinsic stochasticity. It introduces, for the first time, a systematic uncertainty quantification (UQ) framework into quantum computation by formulating the problem as a statistical inference task. By integrating tools from probabilistic modeling, Bayesian inference, stochastic analysis, and sensitivity analysis, the study establishes a novel paradigm for error characterization and algorithm design tailored to noisy intermediate-scale quantum (NISQ) devices. The proposed uncertainty-aware framework is not only scalable but also provides a unified and mathematically rigorous foundation for error verification, characterization, and mitigation strategies.

Current research in quantum error mitigation (QEM) often lacks statistical rigor, rendering performance evaluations vulnerable to biases from parameter selection and hardware timing drift. This work systematically examines evaluation practices across 81 QEM papers through a large-scale literature review, complemented by 132 parameter sweeps, 72 hours of longitudinal hardware experiments, and statistical inference. We quantify, for the first time, the sources of statistical artifacts in QEM benchmarking, revealing that only 25% of studies employ inferential statistics and 42% report merely descriptive uncertainty. Notably, identical zero-noise extrapolation (ZNE) configurations can yield effect size variations exceeding threefold due to differences in execution time. To address these issues, we propose a minimal reporting standard encompassing parameter transparency, robustness checks, drift assessment, and effect size inference, establishing a methodological framework for reliable QEM benchmarking.

Current evaluation of quantum error correction (QEC) decoders relies heavily on Monte Carlo simulations, which suffer from high sample requirements, large variance, and limited ability to characterize robustness. This work proposes an efficient, formal-methods-based evaluation framework that establishes, for the first time, a formal semantics for QEC programs in the Stim format. By integrating structured error-space exploration with constrained polynomial optimization, the framework enables precise quantification of decoder performance across varying physical error rates and robustness against error drift. Experimental results demonstrate that the approach significantly outperforms conventional simulation methods in the low-error-rate regime, achieving higher accuracy while substantially improving evaluation efficiency and stability.