High-dimensional problems for spatial point processes

如今，新技术一方面使人们能够获得越来越多的数据，另一方面也使人们能够观察越来越复杂的现象。统计学，特别是空间统计学的分支并没有回避这些问题。本研究计划打算考虑高维问题的一类特定的空间模型，这是类（空间）点过程。 上下文 点过程对相互作用中的点或事件的随机集合进行建模。点模式出现在广泛的领域。当观察域S对应于Rd的子集时，（维度d= 2，3），这样的过程可以模拟例如天体物理学中的星系、林业中的数百种树木、流行病学中疾病爆发的来源、来自观看视觉中的图像或视频的不同个体的眼睛注视，经典的问题是关于点模式之间的依赖关系的建模（两个树种独立吗？）和/或将点的分布与额外的信息（如海拔图、土壤性质）相关联，以用于林业应用。许多统计方法存在于文献中，但是当同时观察到许多点模式和/或当额外信息的量很重要时，知道的东西很少。如何提取信息，有效地选择协变量是隐含的问题。最近，当S的维数很大时（考虑一个单位立方体[0，1]d，d=50），点过程已经出现在计算机实验中来构建随机设计，当S是一个离散空间时，它们已经出现在机器学习中，压缩感知作为一种有效的工具，用于对可能的高维数据集进行子采样。为了说明其中一个问题，从随机模型导出的点样本的预期特征是“很好地”覆盖单位立方体，如果图案表现出某种规律性，则可以实现这一点。但是，当相同的点样本被投影到单位立方体的任何子空间上时，是否有一个简单的模型也满足这种规律性仍然是一个悬而未决的问题，这是一个经典的实验设计（如拉丁超立方体）能够处理的性质。 目的 该研究计划的目标是将由高维特征引起的现代问题带到相对较新的点过程模型中，这些模型在越来越多的应用中出现。根据这两个研究领域的性质，该研究计划是现代和创新的。问题将从理论的角度进行调查，提供统计界的新方法和结果，以了解其局限性，并从实际/计算的角度，提供从业人员与（免费）R软件内开发的方法的系统实施。

Nowadays, new technologies allow, on the one hand, the acquisition of an increasing mass of data and on the other hand the observation of more and more complex phenomena. Statistics and in particular the sub-branch of spatial statistics does not avoid these questions. The present research program intends to consider high-dimensional problems for one specific class of spatial models which is the class of (spatial) point processes. Context Point processes model random sets of points or events in interaction. Point patterns arise in a broad range of fields. When the observation domain, say S, corresponds to a subset of Rd (with the dimension d=2,3), such processes can model for instance galaxies in astrophysics, hundreds of trees species in forestry, sources of outbreak of a disease in epidemiology, ocular fixations from different individuals watching images or videos in vision, etc. Classical questions are about the modelling of the dependency between point patterns (are two trees species independent?) and/or to relate the distribution of points to extra information like the altitude map, soil nature, for forestry applications. Many statistical methodologies exist in the literature, however very few things are known when many point patterns are simultaneously observed and/or when the amount of extra information is important. How to extract information, to efficiently select covariates are the implicit questions. Very recently, when the dimension of S is large (think of a unit cube [0,1]d with d=50), point processes have appeared in computer experiments to construct random designs and when S is a discrete space they have emerged in machine learning, compressed sensing as an efficient tool for subsampling a possibly high-dimensional dataset. To illustrate one of of the questions an expected feature for a sample of points derived from a stochastic model is to "nicely" cover the unit cube, which can be achieved if the pattern exhibits some kind of regularity. But it is still an open question to have a simple model which satisfies also this kind of regularity when the same sample of points is projected on any subspace of the unit cube, a property that classical experimental designs like Latin hypercubes are able to handle. Objective The goal of this research program is to bring modern questions induced by the high-dimension feature to the relatively recent class of point processes models, which arises in an increasing number of applications. By the nature of these two research areas, this research program is modern and innovative. Problems will be investigated both from a theoretical point of view by providing the statistics community new methodologies and results to understand their limitations and from a practical/computational point of view by providing practitioners with a systematic implementation of the developed methodologies within the (free) R software.