One of the challenges of 21st-century science is to model the evolution of complex systems. One example of practical importance is urban structure, for which the dynamics may be described by a series of non-linear first-order ordinary differential equations. Whilst this approach provides a reasonable model of urban retail structure, it is somewhat restrictive owing to uncertainties arising in the modelling process.
We address these shortcomings by developing a statistical model of urban retail structure, based on a system of stochastic differential equations. Our model is ergodic and the invariant distribution encodes our prior knowledge of spatio-temporal interactions. We proceed by performing inference and prediction in a Bayesian setting, and explore the resulting probability distributions with a position-specific metrolpolis-adjusted Langevin algorithm.