estimate level sets

This work addresses the level-set estimation problem for unknown, expensive-to-evaluate functions—i.e., efficiently identifying the super-level set (region where function values exceed a given threshold) under a limited budget of function evaluations. We propose the first theoretically grounded automatic stopping criterion for this task. Our method employs a Gaussian process surrogate model and introduces a novel acquisition function that integrates statistical confidence bounds within a sequential Bayesian optimization framework. We establish rigorous theoretical guarantees: the algorithm achieves $epsilon$-accuracy with probability at least $1-delta$, and we derive provable lower bounds on performance metrics such as the F-score. Experiments demonstrate that our approach matches the identification accuracy of state-of-the-art methods while substantially reducing redundant evaluations, enabling provably correct, adaptive, and timely termination.

This study addresses the absence of uncertainty quantification in Jacobi set computation for multi-scalar fields by proposing an uncertainty-aware computational framework. Leveraging multivariate normal distribution modeling and analytical propagation mechanisms, this work achieves precise uncertainty quantification within Jacobi set analysis for the first time, complemented by a visualization overlay scheme integrating multidimensional information. Validated through Monte Carlo simulations on fluid dynamics and meteorological ensemble datasets, the proposed method accurately reveals topological structures alongside their associated uncertainties. Consequently, this approach significantly enhances the reliability and interpretability of multi-field visualization, providing novel theoretical support for complex data analysis in scientific domains requiring rigorous uncertainty characterization.

This paper addresses the challenge of estimating response contour lines in computer experiments with mixed quantitative and qualitative inputs. We propose a region-wise collaborative adaptive sequential design method grounded in Gaussian process surrogate modeling. The approach innovatively introduces a dynamic regional partitioning mechanism and a dual-set confidence-bound-driven acquisition function, jointly optimizing sample selection by prioritizing regions near the contour line with high predictive uncertainty. We theoretically establish the validity of the region-adaptive strategy and prove the algorithm’s convergence. Numerical experiments and real-world case studies demonstrate that the proposed method significantly improves both accuracy and efficiency in contour estimation, offering a novel, robust paradigm for contour modeling under mixed-input settings.

In spatial statistics, reliable uncertainty quantification at unobserved locations under complex heterogeneity remains challenging: classical kriging relies on strong Gaussianity and stationarity assumptions, while existing machine learning approaches—including conformal prediction—often neglect spatial dependence, failing to guarantee conditional coverage in finite samples. We propose Spatial Conformal Quantile Regression (SCQR), the first method coupling localized quantile regression with conformal prediction. SCQR establishes theoretical guarantees under mild stationarity and spatial mixing (non-i.i.d.) conditions, jointly improving conditional coverage accuracy and prediction interval sharpness. Evaluated on synthetic and real-world spatial datasets, SCQR strictly achieves nominal coverage, yields narrower intervals, and exhibits stronger spatial coherence compared to both kriging and state-of-the-art spatial conformal methods.

This paper addresses the level set estimation (LSE) problem in manufacturing quality control under input uncertainty—arising from equipment precision variations and operator deviations—and introduces, for the first time, *cost-dependent input-uncertainty LSE*: accurately identifying the region where product performance meets specifications under heterogeneous, multi-precision, and multi-cost inspection devices, while minimizing total inspection cost. We propose a Bayesian optimization–based active learning algorithm that explicitly accounts for input uncertainty, equipped with a cost-weighted acquisition function and supported by theoretical convergence guarantees. Experiments on synthetic benchmarks and real-world industrial datasets demonstrate that our method achieves significantly lower total inspection costs than state-of-the-art approaches, while maintaining high-accuracy level set estimates—thereby bridging theoretical rigor and practical engineering applicability.

为解决标量场数据中特征提取的不确定性表示问题，提出了一种基于Hoeffding不等式的分布无关置信带方法，以提供更准确的不确定性可视化。

本文针对聚类中的不确定性问题，提出基于流形假设的聚簇方法（MBC），通过几何与样本量度结合定义不确定性区间，量化数据固有的模糊性。

This work addresses the problem of target localization in n-dimensional space under unknown but bounded noise in distance measurements. It proposes a set-membership approach based on direct geometric modeling, constructing a non-convex localization set from range measurements to anchor points and precisely characterizing it via intersections of polyhedra and balls. Avoiding conventional semidefinite relaxation, the method employs convex optimization to efficiently compute tight outer approximations—either bounding boxes or ellipsoids—yielding both inner and outer bounds on the true localization set. The resulting set-valued estimate enjoys rigorous theoretical guarantees, features a compact outer approximation structure, and delivers high-accuracy point estimates, outperforming existing methods that rely on cost-function relaxation.

This study addresses the challenge of constructing confidence intervals for the expectation of a target variable within the observed sample in small-area estimation. The authors propose a novel conditional conformal inference method that, under exchangeability or i.i.d. assumptions and within a regression framework, achieves finite-sample valid inference for the in-sample mean—a task previously unattainable with standard conformal prediction, which is inherently designed for out-of-sample prediction. By conditioning appropriately on the observed data, the proposed approach yields confidence intervals that rigorously maintain the prescribed coverage probability even when the underlying regression model is misspecified, thereby extending the applicability of conformal inference to internal estimation problems while preserving its nonparametric reliability guarantees.

This study addresses the efficient estimation of confidence regions for outputs exceeding a prescribed threshold in simulators with random inputs and functional outputs. To this end, the authors propose a surrogate modeling approach that integrates principal component analysis with Gaussian process regression. An innovative active learning strategy based on a max-min criterion is introduced, which, combined with Karhunen–Loève expansion, enables highly efficient sampling and substantially reduces uncertainty in confidence region estimation. The method is validated on three test cases: a synthetic function, the surface pressure coefficient distribution of a hypersonic vehicle, and the glide trajectory of a reusable rocket’s first stage. In all cases, it accurately and efficiently constructs confidence regions, outperforming existing benchmark methods.