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.
