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Coverage and connectivity in stochastic geometry

随机几何中的覆盖范围和连通性

基本信息

批准号：
EP/T028653/1
负责人：
Mathew Penrose
金额：
$ 58.73万
依托单位：
University of Bath
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT028653%2F1
关键词：
Coverage connectivity stochastic geometry

项目摘要

Geometrical patterns requiring statistical analysis arise in many branches of science and technology, and spatial probabilistic modelling is important in diverse areas such as materials science and telecommunications. Stochastic geometry is the mathematical analysis of these geometrical probabilistic models.A fundamental model in stochastic geometry is the Boolean model, a version of which goes as follows. Consider a collection of small (possibly overlapping) `droplets' centred on the points of a large random `point cloud' in a smoothly bounded region of space (either Euclidean space, or more generally a Riemannian manifold). Under appropriate mathematical assumptions, we may ask questions such as the following.What is the probability that the random set, given by the union of the droplets, fully covers the region into which they are placed? What is the probability that this random set is connected? If not connected, how many components does it split into? Given two fixed points in space, what is the probability that they are connected by a path through this random set? What is the probability that the complement of this random set is connected?Exact formulae to answer these kinds of question are not available. Moreover, in applications the sample size is often very large. Thus we propose to investigate these types of question, in appropriate limiting regimes where the point cloud is large and the droplets are small.Much of the difficulty in addressing these problems arises from having to handle the boundary of the region in which the points are placed. Essentially this is because it is harder for any location near the boundary to be covered, but the volume of such locations is less than the volume of interior locations so one has to estimate the trade-off between these two effects. We shall develop new methods to deal with boundary effects, at least when the boundary is smooth, thereby deriving complete answers to many of the questions above; previously, such answers have been mainly available only in simpler cases where there is no boundary (for example, in a torus). Our methods should be relevant to various other models of stochastic geometry, and also to regions with polyhedral boundariesThese kinds of problem are relevant to wireless communications; a random point cloud in a planar region may represent a collection of wireless transmitters. They are also relevant to statistical set estimation and to topological data analysis (TDA), where the random points in a region of (possibly high-dimensional) Euclidean space or manifold may represent multivariate statistical data. In TDA one aims to learn about the topology of the underlying space from the point cloud, often through a discrete structure such as a graph on the points with edges between nearby points.

需要统计分析的几何图案出现在科学和技术的许多分支中，空间概率建模在材料科学和电信等不同领域都很重要。随机几何是这些几何概率模型的数学分析。随机几何中的一个基本模型是布尔模型，其版本如下。考虑一个小的（可能重叠的）“液滴”的集合，它们集中在一个光滑有界的空间区域（欧几里得空间，或者更一般的黎曼流形）中的一个大的随机“点云”的点上。在适当的数学假设下，我们可能会问这样的问题：由液滴的并集给出的随机集完全覆盖它们所处区域的概率是多少？这个随机集合连通的概率是多少？如果没有连接，它分为多少个组件？给定空间中的两个不动点，它们通过这个随机集的路径连接的概率是多少？这个随机集的补集连通的概率是多少？回答这类问题的精确公式是没有的。此外，在应用中，样本量往往非常大。因此，我们建议调查这些类型的问题，在适当的限制制度，点云是大的，液滴是small. Most的困难，在解决这些问题的产生，从必须处理的区域中的点被放置的边界。基本上，这是因为边界附近的任何位置都很难被覆盖，但这些位置的体积小于内部位置的体积，因此必须估计这两种效果之间的权衡。我们将开发新的方法来处理边界效应，至少当边界是光滑的，从而得出上述许多问题的完整答案;以前，这样的答案主要是在没有边界的简单情况下（例如，在环面中）。我们的方法应该是相关的各种其他模型的随机几何，也与多面体边界的区域这些类型的问题是相关的无线通信，一个随机点云在一个平面区域可以代表一个集合的无线发射机。它们也与统计集估计和拓扑数据分析（TDA）有关，其中（可能是高维的）欧氏空间或流形区域中的随机点可以表示多变量统计数据。在TDA中，人们的目标是从点云中了解底层空间的拓扑结构，通常是通过离散结构，例如点上的图，相邻点之间有边。