From urban segregation to spatial structure detection

We develop a ‘multifocal’ approach to reveal spatial dissimilarities in cities, from the most local scale to the metropolitan one. Think, for instance, of a statistical variable that may be measured at different scales, e.g. ethnic group proportions, social housing rate, income distribution, or public transportation network density. Then, to any point in the city there corresponds a sequence of values for the variable, as one zooms out around the starting point, all the way up to the whole city – as if with a varifocal camera lens. The sequences thus produced encode spatial dissimilarities in a precise manner: how much they differ from perfectly random sequences is indeed a signature of the underlying spatial structure. We introduce here a mathematical framework that allows to analyse this signature, and we provide a number of illustrative examples.