Estimation of expected Euler characteristic curves of nonstationary smooth random fields
非平稳光滑随机场期望欧拉示性数曲线的估计AI 解读 ↓
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非平稳光滑随机场期望欧拉示性数曲线的估计
- 研究问题
- 如何估计非平稳光滑随机场的期望欧拉示性数曲线及相关几何量，以支持高阈值下对随机场上确界分布的近似？
- 方法与思路
- 作者将固定化欧拉示性数曲线作线性投影以估计 Lipschitz–Killing 曲率，并用高斯乘子自助法处理具有高斯过程极限的统计量。
- 材料陈述的贡献
- 摘要称，曲率估计具有一致性、低方差和计算效率；期望曲线的参数代入估计比非参数曲线平均更有效，并提供模拟和应用。
- 阅读关注点
- 应核对光滑性、流形及高斯极限条件，曲线固定化和乘子自助法的构造，以及效率比较标准与高阈值近似的适用范围。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；一致性与效率的具体条件、模拟数值及近似误差未说明，不能推广到任意随机场。
原文摘要
The expected Euler characteristic (EEC) of excursion sets of a smooth Gaussian-related random field over a compact manifold approximates the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC of a Gaussian-related field is expressed by the Gaussian kinematic formula (GKF) as a finite sum of known functions multiplied by the Lipschitz–Killing curvatures (LKCs) of the generating Gaussian field. This paper proposes consistent estimators of the LKCs as linear projections of “pinned” Euler characteristic (EC) curves obtained from realizations of zero-mean, unit variance Gaussian processes. As observed, data seldom is Gaussian and the exact mean and variance is unknown, yet the statistic of interest often satisfies a CLT with a Gaussian limit process; we adapt our LKC estimators to this scenario using a Gaussian multiplier bootstrap approach. This yields consistent estimates of the LKCs of the possibly nonstationary Gaussian limiting field that have low variance and are computationally efficient for complex underlying manifolds. For the EEC of the limiting field, a parametric plug-in estimator is presented, which is more efficient than the nonparametric average of EC curves. The proposed methods are evaluated using simulations of 2D fields, and illustrated on cosmological observations and simulations on the 2-sphere and 3D fMRI volumes.