Collaborative Research: Approximating Eigensystems of Matrices Used in Spatial Analysis

协作研究：空间分析中使用的矩阵的近似特征系统。摘要：该项目调查关系结构的理论属性及其使用，它的使用在几个学科中有很长的传统。在许多问题中，社会和行为科学关注社会网络或空间链接结构，而更抽象的科学处理图论问题或相当大的矩阵的代数。在这些关系结构的不同用法和与之相关的问题上，一个共同的主题是检索它们的特征系统，这允许对它们的基本数据生成过程做出明确的声明。该项目特别关注稀疏的二进制链接矩阵的属性，这些链接矩阵封装了空间对象之间的成对对称关系。根据基本空间结构、其规范和编码的不同，其相关特征系统的属性会发生变化，从而对数据分析和解释产生影响。伴随着地理信息系统的激增，这些空间结构的规模也在不断增加。这个项目寻求对选定的大特征系统的有效和准确的近似方法，以及与这些空间结构的特征系统相关的新的定理和猜想。在更一般的术语中，地理分布和位置参考地球表面的现象和活动倾向于显示地图模式，因为附近的现象具有相互吸引或排斥的倾向。这同样适用于地理领域以外的许多其他现象的安排，如组织结构图中的等级制度，或传染病的潜在传播途径。在空间领域，由于局部相对位置的原因，这种距离受限的相互关系网络使传统的数据分析变得复杂。1992年，《经济学人》将这种情况描述为：[使用地理编码数据进行正确的统计计算]需要在许多不同的地方进行测量，然后进行一些尴尬的平均。随着计算机和数据捕获技术允许收集越来越多以及越来越小的地理编码观测单位的数据，这种尴尬的平均变得更加复杂和困难，计算它的能力不断超过计算机技术的进步。这个项目旨在通过进一步发展在正确的统计计算中涉及的笨拙的平均的数学理论来帮助缩小这一差距。

Collaborative research: approximating eigensystems of matrices used in spatial analysis.ABSTRACT: The project investigates theoretical properties of relationship structures, as well as their usage, whose use has a long-standing tradition in several disciplines. The social and behavioral sciences focus on, among many other issues, social networks or spatial linkage structures, whereas more abstract sciences deal with graph theoretical problems or the algebra of sizeable matrices. A common theme across the different usages of, and problems associated with, these relationship structures is the retrieval of their eigensystems, which allows explicit statements to be made about their underlying data generating processes. In particular, this project focuses on the properties of sparse, binary link matrices that encapsulate pairwise symmetric relationships among spatial objects. Depending upon the underlying spatial structure, its specification and encoding, the properties of their associated eigensystems change, with subsequent consequences for data analysis and interpretation. Accompanying the proliferation of geographic information systems is a constantly increasing size of these spatial structures. This project seeks efficient and accurate approximation methods for selected large eigensystems, and new theorems and conjectures that are associated with these eigensystems of spatial structures.In more general terms, phenomena and activities that are geographically distributed and locationally referenced to the Earth's surface tend to display map patterns because nearby phenomena have a propensity to attract or repel each other. The same holds for arrangements of many other phenomena outside the geographic domain, such as hierarchies in organizational charts, or the potential transmission pathways of infectious diseases. In the spatial domain this web of distance-restricted interrelationships, due to local relative location, complicates conventional data analyses. In 1992 The Economist described this situation as follows: [correct statistical calculation with geocoded data] requires measurements in many different places, followed by some awkward averaging. As computer and data capturing technology allow data to be collected for more and more as well as smaller and smaller geocoded observational units, this awkward averaging becomes more complex and difficult, with an ability to compute it continually outpacing advances in computer technology. This project aims to help close this gap by further developing the mathematical theory underlying the awkward averaging involved in correct statistical calculations.