Dependent radius marks of Laguerre tessellations: a case study

IF 0.8 4区数学 Q3 STATISTICS & PROBABILITY

Australian & New Zealand Journal of Statistics Pub Date : 2021-07-21 DOI:10.1111/anzs.12314

Dietrich Stoyan, Viktor Beneš, Filip Seitl

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引用次数: 2

Abstract

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.

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拉盖尔镶嵌的依赖半径标记:一个案例研究

我们研究了一个特殊的标记三维点过程样本，它代表了拉盖尔镶嵌。它来自于铝合金材料的多晶样品。“点”是单元生成器，而“标记”是半径标记，控制镶嵌单元的大小和形状。我们的统计标记相关性分析表明，样品的标记具有清晰而合理的空间相关性:靠近在一起的发电机的标记往往小而相似，相关函数的形式不能证明地质统计标记。我们表明，通过具有独立半径标记的拉盖尔镶嵌来简化镶嵌建模可能会导致错误的结果。当我们从铝合金数据出发，通过随机排列产生随机标记时，我们得到了与原始特征截然不同的镶嵌。我们在模拟拉盖尔镶嵌中观察到类似的行为。对于给定的数据类型来说，这似乎是很自然的事实，但这使得模型拟合到经验拉盖尔镶嵌非常困难:生成器点和半径标记必须同时建模。这可能意味着，如果只需要类似拉盖尔镶嵌的样本，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配方。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Australian & New Zealand Journal of Statistics 数学-统计学与概率论

CiteScore

1.30

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Australian & New Zealand Journal of Statistics is an international journal managed jointly by the Statistical Society of Australia and the New Zealand Statistical Association. Its purpose is to report significant and novel contributions in statistics, ranging across articles on statistical theory, methodology, applications and computing. The journal has a particular focus on statistical techniques that can be readily applied to real-world problems, and on application papers with an Australasian emphasis. Outstanding articles submitted to the journal may be selected as Discussion Papers, to be read at a meeting of either the Statistical Society of Australia or the New Zealand Statistical Association. The main body of the journal is divided into three sections. The Theory and Methods Section publishes papers containing original contributions to the theory and methodology of statistics, econometrics and probability, and seeks papers motivated by a real problem and which demonstrate the proposed theory or methodology in that situation. There is a strong preference for papers motivated by, and illustrated with, real data. The Applications Section publishes papers demonstrating applications of statistical techniques to problems faced by users of statistics in the sciences, government and industry. A particular focus is the application of newly developed statistical methodology to real data and the demonstration of better use of established statistical methodology in an area of application. It seeks to aid teachers of statistics by placing statistical methods in context. The Statistical Computing Section publishes papers containing new algorithms, code snippets, or software descriptions (for open source software only) which enhance the field through the application of computing. Preference is given to papers featuring publically available code and/or data, and to those motivated by statistical methods for practical problems.