非平稳光滑高斯随机场的期望欧拉特征曲线估计
Estimation of Expected Euler Characteristic Curves of Nonstationary Smooth Gaussian Random Fields
AI总结：
针对非平稳高斯场Lipschitz-Killing曲率难估计导致期望欧拉特征曲线应用受限的问题，提出基于固定观测欧拉特征曲线线性投影的一致估计及乘子bootstrap修正方法，提升估计效率并拓展适用范围，经模拟和实际场景验证有效。
摘要：
高斯随机场游程集的期望欧拉特征（expected Euler characteristic, EEC）曲线用于近似高阈值下其最大值的分布。作为游程阈值的函数，EEC由高斯运动学公式（Gaussian kinematic formula, GKF）表示为场的Lipschitz-Killing曲率（Lipschitz-Killing curvatures, LKCs）的线性函数，LKCs仅依赖于场的定义域和协方差函数。目前，其在非平凡定义域上的非平稳高斯场中的应用受到限制，因为这种情况下LKCs难以估计。本文提出LKCs的一致估计量，作为“固定”观测欧拉特征曲线的线性投影，并得到EEC曲线的线性参数估计量，在重复观测下其效率高于非参数估计量。乘子bootstrap修正降低了估计量的方差，并且能够估计满足泛函中心极限定理（CLT）的非高斯场极限场的LKCs和EEC。通过二维场模拟对所提方法进行了评估，并在三维fMRI脑激活图阈值处理以及2-球面宇宙学模拟中进行了示例验证。
英文摘要：
The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian random field is used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold, the EEC is expressed by the Gaussian kinematic formula (GKF) as a linear function of the Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the domain and covariance function of the field. So far its use for non-stationary Gaussian fields over non-trivial domains has been limited because in this case the LKCs are difficult to estimate. In this paper, consistent estimators of the LKCs are proposed as linear projections of "pinned" observed Euler characteristic curves and a linear parametric estimator of the EEC curve is obtained, which is more efficient than its nonparametric counterpart for repeated observations. A multiplier bootstrap modification reduces the variance of the estimator, and allows estimation of LKCs and EEC of the limiting field of non-Gaussian fields satisfying a functional CLT. The proposed methods are evaluated using simulations of 2D fields and illustrated in thresholding of 3D fMRI brain activation maps and cosmological simulations on the 2-sphere.
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