High-dimensional problems for spatial point processes

如今，新技术一方面使我们能够获取越来越多的数据，另一方面使我们能够观察到越来越复杂的现象。统计，特别是空间统计的分支并没有回避这些问题。本研究计划拟考虑一类特定空间模型的高维问题，即（空间）点过程。******上下文***点处理模型在交互中的点或事件的随机集。点模式出现在广泛的领域。当观测域S对应于Rd的一个子集（维度d=2,3）时，这样的过程可以建模，例如天体物理学中的星系、林业中的数百种树种、流行病学中疾病爆发的来源、不同个体在视觉上观看图像或视频时的眼睛注视，等等。经典的问题是关于点模式之间依赖关系的建模（两种树种是独立的吗？）和/或将点的分布与额外的信息（如高程图、土壤性质）联系起来，用于林业应用。文献中存在许多统计方法，但是当同时观察到许多点模式和/或当额外信息的数量很重要时，很少有东西是已知的。如何提取信息，有效地选择协变量是隐含的问题。最近，当S的维度很大时（想象一个d=50的单位立方体[0,1]d），点过程已经出现在计算机实验中以构建随机设计，当S是一个离散空间时，它们已经出现在机器学习中，压缩感知作为对可能的高维数据集进行子采样的有效工具。为了说明其中一个问题，从随机模型中获得的点样本的预期特征是“很好地”覆盖单位立方体，如果模式显示出某种规律性，则可以实现这一点。但是，当相同的点样本投射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，这仍然是一个悬而未决的问题，这是拉丁超立方体等经典实验设计能够处理的性质。******目的***本研究计划的目标是将由高维特征引起的现代问题引入相对较新的一类点过程模型，这些模型在越来越多的应用中出现。从这两个研究领域的性质来看，这个研究项目是现代的和创新的。问题将从理论的角度进行调查，为统计社区提供新的方法和结果，以了解其局限性，并从实践/计算的角度向从业者提供在（免费）R软件中开发的方法的系统实现。

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.