This paper introduces a $K$-function for assessing second-order properties of inhomogeneous random measures generated by marked point processes. The marks can be geometric objects like fibers or sets of positive volume, and the presented $K$-function takes into account geometric features of the marks, such as tangent directions of fibers. The $K$-function requires an estimate of the inhomogeneous density function of the random measure. We introduce parametric estimates for the density function based on parametric models that represent large scale features of the inhomogeneous random measure. The proposed methodology is applied to simulated fiber patterns as well as a three-dimensional data set of steel fibers in concrete.
翻译:本文介绍了用于评估通过标记点过程产生的不相容随机测量的二阶特性的美元功能。标记可以是纤维或正体体积组等几何物体,而列报的美元功能则考虑到标记的几何特征,如纤维的正向。美元功能要求对随机测量的不相形密度函数进行估计。我们根据代表不相形随机测量的大规模特征的参数模型对密度函数进行参数估计。拟议的方法适用于模拟纤维模式以及混凝土钢纤维的三维数据集。