Wednesday, December 1, 2021

A Gentle, Original Approach to Stochastic Point Processes

 In this article, original stochastic processes are introduced. They may constitute one of the simplest examples and definitions of point processes. A limiting case is the standard Poisson process - the mother of all point processes - used in so many spatial statistics applications. We first start with the one-dimensional case, before moving to 2-D. Little probability theory  knowledge is required to understand the content. In particular, we avoid discussing measure theory, Palm distributions, and other hard-to-understand concepts that would deter many users. Nevertheless, we dive pretty deep into the details, using simple English rather than arcane abstractions, to show the potential. A spreadsheet with simulations is provided, and model-free statistical inference techniques are discussed, including model-fitting and test for radial distribution. It is shown that two realizations of very different processes can look almost identical to the naked eye, while they actually have fundamental differences that can only be detected with machine learning techniques. Cluster processes are also presented in a very intuitive way.  Various probability distributions, including logistic, Poisson-binomial and Erlang, are attached to these processes, for instance to model the distribution and/or number of points in some area, or distances to nearest neighbors. 

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