High-dimensional problems for spatial point processes

空间点过程的高维问题

• 批准号: RGPIN-2017-05257
• 负责人: Coeurjolly, JeanFrançois
• 金额: $ 1.15万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711555
• 关键词: dimensional; problems; spatial; point; processes

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.

如今，新技术一方面可以获取越来越多的数据，另一方面可以观察到越来越复杂的现象。统计，特别是空间统计的分支机构，并不回避这些问题。目前的研究计划打算考虑一类特定的空间模型的高维问题，这类空间模型是(空间)点过程。 语境 点过程对交互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，比如S，对应于RD的一个子集(维度d=2，3)时，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，流行病学中的疾病暴发源，视觉中观看图像或视频的不同个人的眼睛注视，等等。经典问题是关于点模式之间的依赖性的建模(两个树木物种独立吗？)和/或将点的分布与额外的信息相关联，例如用于林业应用的海拔地图、土壤性质。文献中有许多统计方法，但当同时观察到许多点模式和/或当额外信息量很重要时，很少有人知道这一点。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(想想d=50的单位立方体[0，1]d)，点过程出现在计算机实验中来构造随机设计，当S是离散空间时，它们出现在机器学习中，压缩感知作为对可能高维数据集进行二次采样的有效工具。为了说明其中一个问题，从随机模型派生的点样本的预期特征是“很好地”覆盖单位立方体，如果图案显示出某种规律性，就可以实现这一点。但是，当相同的点样本被投影到单位立方体的任意子空间上时，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，这是像拉丁超立方体这样的经典实验设计能够处理的特性。 客观化 这项研究的目标是将高维特征引发的现代问题引入到相对较新的一类点过程模型中，这类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，本研究方案具有时代性和创新性。将从理论角度调查问题，向统计界提供新的方法和结果以了解其局限性，并从实际/计算角度调查问题，方法是向从业人员提供在(免费)R软件中系统地实施已开发的方法。