A Methodology for Neural Spatial Interaction Modelling

This paper attempts to develop a mathematically rigid and unified framework for neural spatial interaction modelling. Families of classical neural network models, but also less classical ones such as product unit neural network ones are considered for the cases of unconstrained and singly constrained spatial interaction flows. Current practice appears to suffer from least squares and normality assumptions that ignore the true integer nature of the flows and approximate a discrete-valued process by an almost certainly misrepresentative continuous distribution. To overcome this deficiency we suggest a more suitable estimation approach, maximum likelihood estimation under more realistic distributional assumptions of Poisson processes, and utilise a global search procedure, called Alopex, to solve the maximum likelihood estimation problem. To identify the transition from underfitting to overfitting we split the data into training, internal validation and test sets. The bootstrapping pairs approach with replacement is adopted to combine the purity of data splitting with the power of a resampling procedure to overcome the generally neglected issue of fixed data splitting and the problem of scarce data. In addition, the approach has power to provide a better statistical picture of the prediction variability. Finally, a benchmark comparison against the classical gravity models illustrates the superiority of both, the unconstrained and the origin-constrained neural network model versions in terms of generalisation performance measured by Kullback and Leibler’s information criterion.