Collaborative Research: Approximating Eigensystems of Matrices Used in Spatial Analysis

合作研究：空间分析中矩阵特征系统的近似。摘要：该项目研究关系结构的理论性质及其使用，关系结构的使用在几个学科中有着悠久的传统。在许多其他问题中，社会科学和行为科学关注社会网络或空间联系结构，而更抽象的科学处理图论问题或大矩阵的代数。在这些关系结构的不同用法和相关问题中，一个共同的主题是检索它们的特征系统，这允许对它们的底层数据生成过程做出显式的陈述。特别是，该项目侧重于稀疏的性质，二进制链接矩阵封装空间对象之间的成对对称关系。根据其底层空间结构、规范和编码，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后续影响。随着地理信息系统的扩散，这些空间结构的规模也在不断扩大。本项目为选定的大型特征系统寻求有效和准确的近似方法，以及与这些空间结构特征系统相关的新定理和猜想。更一般地说，地理上分布和位置上与地球表面相关的现象和活动倾向于显示地图模式，因为附近的现象有相互吸引或排斥的倾向。这同样适用于地理领域之外的许多其他现象的安排，例如组织结构图中的等级制度，或传染病的潜在传播途径。在空间域中，由于局部相对位置，这种距离限制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》这样描述这种情况：[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的数据，以及越来越小的地理编码观测单元，这种尴尬的平均变得更加复杂和困难，计算能力不断超过计算机技术的进步。这个项目的目的是通过进一步发展数学理论来帮助缩小这一差距，这一理论是在正确的统计计算中涉及的尴尬的平均。

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.