Collaborative Research: Approximating Eigensystems of Matrices Used in Spatial Analysis

合作研究：空间分析中使用的矩阵特征系统的近似

• 批准号: 0611882
• 负责人: Michael Tiefelsdorf
• 金额: --
• 依托单位: University of Texas at Dallas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-30 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0611882&HistoricalAwards=false
• 关键词: Collaborative; Research; Approximating; Eigensystems; Matrices

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.

合作研究：空间分析中使用的矩阵的近似特征系。摘要：该项目研究关系结构的理论属性及其使用，其使用在多个学科中有着悠久的传统。社会科学和行为科学关注的是社交网络或空间联系结构，而更抽象的科学则处理图论问题或大型矩阵的代数。这些关系结构的不同用法和相关问题的一个共同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进行明确的陈述。特别是，这个项目的重点是稀疏，二进制链接矩阵封装空间对象之间的成对对称关系的属性。根据基础的空间结构，其规格和编码，其相关的本征系统的属性的变化，与数据分析和解释的后续后果。伴随着地理信息系统的扩散，这些空间结构的规模不断增加。该项目寻求有效和精确的近似方法，为选定的大型本征系统，和新的定理和与这些本征系统的空间结构相关联的图解。在更一般的术语，现象和活动，地理分布和位置参照地球表面往往显示地图模式，因为附近的现象有一种倾向，相互吸引或排斥。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。在空间领域，由于当地的相对位置，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》将这种情况描述如下：[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为越来越多以及越来越小的地理编码观测单元收集数据，这种笨拙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。该项目旨在通过进一步发展数学理论来帮助缩小这一差距，这些数学理论是正确统计计算中所涉及的笨拙平均值的基础。