Studying critical values for global Moran’s I under inhomogeneous Poisson point processes

Abstract. Spatial autocorrelation is a fundamental statistical property of geographical data. A number of estimators have been introduced, with Moran’s I being one of the most commonly used methods. The characterisation of spatial autocorrelation is useful for a number of applications, including finding clusters, testing model assumptions, investigating spatial outliers, and others. Most estimators of spatial autocorrelation are based on assessing the degree of correspondence between structures in an attribute and structures among spatial units, both of which are operationalised in matrix form. Associated inference procedures then rely on holding the spatial configuration fixed, but varying the attribute values over the geometries. Although fixing the geometries is useful in many scenarios, there are cases where it would be more appropriate to allow the geometries to vary as well, such as in the analysis of social media feeds or mobile sensor observations. In this short paper, the case is considered where geometries are the result of inhomogeneous spatial Poisson processes. Using diagonal and circular types of spatial structuring, it is investigated how random geometries affect critical values used to assess the significance of global Moran’s I scores. It is shown that the critical values resulting from an established inference framework often underestimate the bounds that would result if geometric randomness were taken into account. This leads to type-I errors and thus potential false positive patterns.