Defining Spatial Entropy from Multivariate Distributions of Co-occurrences

Abstract

Finding geographical patterns by analysing the spatial configuration distribution of events, objects or their attributes has a long history in geography, ecology and epidemiology. Measuring the presence of patterns, clusters, or comparing the spatial organisation for different attributes, symbols within the same map or for different maps, is often the basis of analysis. Landscape ecology has provided a long list of interesting indicators, e.g. summaries of patch size distribution. Looking at content information, the Shannon entropy is also a measure of a distribution providing insight into the organisation of data, and has been widely used for example in economical geography. Unfortunately, using the Shannon entropy on the bare distribution of categories within the spatial domain does not describe the spatial organisation itself. Particularly in ecology and geography, some authors have proposed integrating some spatial aspects into the entropy: using adjacency properties or distances between and within categories. This paper goes further with adjacency, emphasising the use of co-occurences of categories at multiple orders, the adjacency being seen as a particular co-occurence of order 2 with a distance of collocation null, and proposes a spatial entropy measure framework. The approach allows multivariate data with covariates to be accounted for, and provides the flexibility to design a wide range of spatial interaction models between the attributes. Generating a multivariate multinomial distribution of collocations describing the spatial organisation, allows the interaction to be assessed via an entropy formula. This spatial entropy is dependent on the distance of collocation used, which can be seen as a scale factor in the spatial organisation to be analysed.