We study a particular marked three-dimensional point process sample that represents a Laguerre tessellation. It comes from a polycrystalline sample of aluminium alloy material. The ‘points’ are the cell generators while the ‘marks’ are radius marks that control the size and shape of the tessellation cells. Our statistical mark correlation analyses show that the marks of the sample are in clear and plausible spatial correlation: the marks of generators close together tend to be small and similar and the form of the correlation functions does not justify geostatistical marking. We show that a simplified modelling of tessellations by Laguerre tessellations with independent radius marks may lead to wrong results. When we started from the aluminium alloy data and generated random marks by random permutation we obtained tessellations with characteristics quite different from the original ones. We observed similar behaviour for simulated Laguerre tessellations. This fact, which seems to be natural for the given data type, makes fitting of models to empirical Laguerre tessellations quite difficult: the generator points and radius marks have to be modelled simultaneously. This may imply that the reconstruction methods are more efficient than point-process modelling if only samples of similar Laguerre tessellations are needed. We also found that literature recipes for bandwidth choice for estimating correlation functions should be used with care.

中文翻译：

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Laguerre 曲面细分的相关半径标记：案例研究

我们研究了代表 Laguerre 镶嵌的特定标记的 3D 点过程样本。它来自铝合金材料的多晶样品。“点”是单元生成器，而“标记”是控制曲面细分单元大小和形状的半径标记。我们的统计标记相关性分析表明，样本的标记具有清晰合理的空间相关性：靠近的生成器的标记往往较小且相似，并且相关函数的形式不能证明地统计标记是正确的。我们表明，通过具有独立半径标记的 Laguerre 曲面细分对曲面细分的简化建模可能会导致错误的结果。当我们从铝合金数据开始并通过随机排列生成随机标记时，我们获得了与原始曲面有很大不同的特征的曲面细分。我们观察到模拟 Laguerre 镶嵌的类似行为。这一事实对于给定的数据类型来说似乎是自然的，这使得将模型拟合到经验 Laguerre 细分非常困难：必须同时对生成点和半径标记进行建模。这可能意味着如果只需要类似 Laguerre 镶嵌的样本，则重建方法比点过程建模更有效。我们还发现应该谨慎使用用于估计相关函数的带宽选择的文献方法。这一事实对于给定的数据类型来说似乎是很自然的，这使得将模型拟合到经验 Laguerre 细分非常困难：必须同时对生成点和半径标记进行建模。这可能意味着如果只需要类似 Laguerre 镶嵌的样本，则重建方法比点过程建模更有效。我们还发现应该谨慎使用用于估计相关函数的带宽选择的文献方法。这一事实对于给定的数据类型来说似乎是很自然的，这使得将模型拟合到经验 Laguerre 细分非常困难：必须同时对生成点和半径标记进行建模。这可能意味着如果只需要类似 Laguerre 镶嵌的样本，则重建方法比点过程建模更有效。我们还发现应该谨慎使用用于估计相关函数的带宽选择的文献方法。