4. Semantics of Point Spaces Through the Topological Weighted Centroid and Other Mathematical Quantities: Theory and Applications

Given a scattering of observations on a map it is natural for one to want to determine the most likely origin of those points, and the origin is typically hidden within data. Using an example to illustrate the point, suppose the police authorities have a map on which is noted the actual distribution of home break-ins. It is natural for the police to want to know the point of origin of those crimes so that they might be able to quickly apprehend the criminals. If the points were those of an outbreak of an epidemic the public health officials would want to know the location of the source of the disease. A new methodology is introduced to solve just this kind of set of problems. It is based on the notion of the Topological Weighted Centroid that permits one to draw powerful inferences about these kinds of center points, even in cases containing very few observations or in which the points are based on a poorly understood underlying data system. Two kinds of problems, based on the degree of spatialization, are addressed: the first kind of problem possesses an inherent spatial semantic in which all of the relevant characteristics of the observed entities are of a spatial nature; the other kind of problem involves those possessing full semantics in which some of the characteristics have a non-spatial nature and must therefore be properly spatialized. The theory is backed up by case studies involving criminal network detection, tracking down the course of an epidemic, and the reconstruction of terrorist attacks relationships on the basis of a small-dimensional qualitative dataset.