We are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets. To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on $\mathbb{R}^2$ observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover $\mathbb{R}^2$, the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite-sample data setting.

翻译：本文旨在建立统计方法，通过随机场超出集的几何特征提供其信息摘要。为此，我们提出一种估计量，用于计算在正则正方形网格上观测到的$\mathbb{R}^2$上随机场超出集的边界长度。该估计量作用于经验可获取的超出区域二值数字图像，并计算超出边界分段线性近似曲线的长度。研究表明，在像素尺寸减小时，无需任何归一化常数，且不假设基础随机场满足高斯性或各向同性，该估计量即具有相合性。在此一般框架下，即使定义域扩展至覆盖$\mathbb{R}^2$，估计误差的量级仍小于定义域边长。对于仿射强混合随机场，当同时考虑多个水平时，该结论可转化为估计量的多元中心极限定理。最后，我们通过多项数值研究考察了所提估计量在有限样本数据场景下的统计性质。