We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points, we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points, we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step, parameter estimation based on maximum pseudolikelihood methods is tractable. For model selection, we consider maximized log pseudolikelihood functions for models of the radii conditioned on the points. Model checking is performed using global envelopes and corresponding tests in both steps and moreover by comparing observed and simulated tessellation characteristics in the second step. We apply our methodology for a 3D Laguerre tessellation data set representing the microstructure of a polycrystalline metallic material, where simulations under a fitted model may substitute expensive laboratory experiments.

我们提出了一种分析拉盖尔镶嵌数据集的通用统计方法，该数据集被视为标记点过程模型的实现。第一步，对于点，我们使用嵌套的多尺度过程序列，它们构成了成对交互点过程模型的灵活参数类。在第二步中，对于以点为条件的标记/半径，我们考虑各种指数族模型，其中规范充分统计量基于镶嵌特征。对于每一步，基于最大伪似然方法的参数估计是易于处理的。对于模型选择，我们考虑以点为条件的半径模型的最大化对数伪似然函数。模型检查是在两个步骤中使用全局包络和相应的测试进行的，此外，通过比较第二步中观察到的和模拟的镶嵌特征来执行模型检查。我们将我们的方法应用于代表多晶金属材料微观结构的 3D Laguerre 镶嵌数据集，其中拟合模型下的模拟可以替代昂贵的实验室实验。