A general mathematical framework for the analysis of spatiotemporal point processes

Ovaskainen et al. motivate their study with the difficulty of mathematical analysis of spatial stochastic models. A central problem is that each spatial moment in such models depends on the next order moment resulting in an infinite hierarchy of moments. We’ve discussed moment closure as one option for addressing this problem by making basic assumptions about the relationship between lower and higher order moments for the sake of mathematical expediency. The authors, however, highlight the concept that most such methods available rely primarily on heuristic assumptions, making the choice of moment closure “more of an art than a science.” They propose the application of the mathematical field studying “Markov evolutions in locally finite configurations” as an as yet overlooked direct analogue to the ecological study of spatiotemporal point processes.

They begin by defining an infinite domain where the number of points/individuals within a finite space is finite and offering mu sub t as the probability that a system is in a given state at time t given its initial conditions. Rather than formulating an equation for the very complex mu sub t in order to study its evolution they employ functions on the system which take real values (known as observables). A given model is then defined by the way that individual-level events change the observable (here they provide a helpful example of a spatial reproduction-dispersal individual-based ecological model).

We are then introduced to the formulation of spatial moments and spatial cumulants.  The first order spatial moment corresponds to the expected population density at a given location, the second-order tot the density of pairs of individuals, and the vector of all spatial moments contains the same statistical info as the above discussed measure mu. Spatial cumulants map 1-1 to moments and often provide simpler analytical solutions. The authors then employ moment genenerating functionals (the spatial equivalent of typical m.g.f.s) to derive the time evolution of a system in the form of the time evolution of its spatial moments. This formulation still suffers from the infinite hierarchy of moments problem. The conversion of this formulation into the terms of spatial cumulants is used as a mathematically reasonable remedy. Though spatial cumulants are also infinitely hierarchical spatial cumulants of higher orders can be reasonably expected to be small. Finally the authors use these tools to demonstrate that the spatial stochastic model can be considered a perturbation expansion around the mean-field model (in that its moments converge to the mean-field model as dispersal and competition kernels become increasingly expansive). They prove that the second order spatial cumulant tends to dominate the spatial patterning, making higher order cumulants less important producing a mathematically sound alternative to heurisitcally derived moment-closures.

Ovaskainen, Otso, et al. “A general mathematical framework for the analysis of spatiotemporal point processes.” Theoretical ecology 7.1 (2014): 101-113.