Spatial Interaction Processes: Formal Development

Abstract

In this chapter, the basic notations of interaction processes and their associated frequency processes are developed in a formal way. We begin in Section 3.2 below with a development of certain mathematical concepts which will be employed throughout the analysis. This is followed in Section 3.3 by a development of a general probability model of interactions which focuses on the locational and frequency attributes of interaction patterns. In Section 3.4, the class of interaction processes discussed in Section 1.4.2 is then formalized, and the important subclass of independent interaction processes discussed in Section 1.4.3 is developed in detail. In Section 3.5, the corresponding concepts of frequency processes and independent frequency processes are developed, and the fundamental Poisson characterization theorem discussed in Section 3.6.2 is established in terms of these concepts. Finally, the application of this general result to the classes of threshold interaction processes and search processes discussed in Examples 2 and 4 of Chapter 2 are developed formally in Sections 3.7 and 3.8, respectively.