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 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 , 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.

中文翻译：

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二维各向异性随机场像素化偏移集的周长估计

我们有兴趣创建统计方法，通过其偏移集的几何形状提供随机场的信息摘要。为此，我们引入了一个估计器，用于估计随机场的偏移集的周长。在规则的方形瓷砖上观察到。所提出的估计器作用于偏移区域的凭经验可访问的二进制数字图像，并计算偏移边界的分段线性近似的长度。随着像素大小的减小，估计器是一致的，不需要任何归一化常数，并且既没有对基础随机场强加高斯性假设，也没有强加各向同性。在这个通用框架中，即使域增长到覆盖，估计误差的阶数小于域的边长。对于仿射、强混合随机场，当同时考虑多个级别时，这转化为我们的估计器的多元中心极限定理。最后，我们进行了多项数值研究，以研究有限样本数据设置中所提出的估计器的统计特性。