Tobler's Law in a Multivariate World

Tobler's first law of geography is widely recognized as reflecting broad empirical realities in geography. Its key concepts of "near" and "related" are intuitive in a univariate setting. However, when moving to the joint consideration of spatial patterns among multiple variables, the combination of attribute similarity and geographical similarity that underlies the concept of spatial autocorrelation is much harder to deal with. This article uses the notion of distance in multiattribute space to explore and visualize the connection between "near" and "related" in a multivariate context. We approach this from a global, local, and regional perspective. We outline a number of ways to combine different visualization techniques and introduce a new local neighbor match test for multivariate local clusters. We illustrate the methods by means of Guerry's classic data set on moral statistics in 1833 France.