Spatial dependence

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Spatial dependence

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Notwithstanding the comments in the previous section about spatial heterogeneity, anyone examining the Earth’s surface in detail would be struck by how conditions tend to persist locally, and how it is possible to divide the surface into regions that exhibit substantial internal similarity. For example, the desert regions are characterized by lack of rainfall, the tropical rainforests by abundant rainfall and dense vegetation, and the Polar Regions by extreme cold. Conditions at nearby points are not sampled independently and randomly from global distributions, but instead show remarkable levels of interdependence. Of course there are exceptions, where conditions change very rapidly over short distances, for example between the plains of India and the high Himalayas, or between coastal plains and adjacent ocean.

The general term for this phenomenon is spatial dependence.  Its pervasiveness was aptly captured by Tobler in what has become known as his First Law of Geography: “All things are related, but nearby things are more related than distant things”. The magnitude of the effect can be measured using a number of statistics of spatial autocorrelation (Cliff and Ord, 1981, Goodchild CATMOG 47). It also underlies the discipline known as geostatistics, which describes spatial variation in terms of a function (known as a correlogram) that shows how spatial autocorrelation decreases with increasing distance (for a general overview of geostatistics see Section 6.7.1, Core concepts in Geostatistics, and the book by Isaaks and Srivastava, 1989). This correlogram reaches zero, or independence, at a distance known as the range. The implications of Tobler’s observation are profound. If it were not true, the full range of conditions anywhere on the Earth’s surface could in principle be found packed within any small area. There would be no regions of approximately homogeneous conditions to be described by giving attributes to area objects. Topographic surfaces would vary chaotically, with slopes that were everywhere infinite, and the contours of such surfaces would be infinitely dense and contorted. Spatial analysis, and indeed life itself, would be impossible.