Fundamentals of Spatial Prediction

In the process of creating a map, geographers often have to engage in the activity of spatial prediction. Although there are many tools we use to accomplish this task, they generally boil down to the use of one or two fundamental concepts.

Waldo Tobler is credited for identifying the ‘first law of geography’, stating “Everything is related to everything else, but near things are more related than distant things.” This concept is essentially synonymous with spatial dependence and spatial autocorrelation. Spatial interpolation methods rely on this principle to make predictions about the attributes of areas between sampled locations. For example, kriging utilizes an observed spatial lag relationship to determine the range of autocorrelation. In other words, it quantifies the degree to which locations are similar with respect to their distance from each other (i.e. semivariogram). Kriging then uses that information to optimize predictions for the unobserved locations.

This colorized version of a diagram from the USDA-NRCS illustrates spatial association by identifying similar soils on similar landscape positions. It is also true that adjacent soils are more likely to be more similar than non-adjacent soils, but with the exception of distant soils that have the same processes of soil formation.

If spatial autocorrelation is the first law of geography, then spatial association should be the second. Actually, spatial association has arguably been in use longer, so maybe it should be the first law. In any case, spatial association describes how phenomena are similarly distributed. To put it in a phrase parallel to Tobler’s law, “Everything is related to everything else, but things sharing similar conditions are more related than things under dissimilar conditions.” A quantitative form of spatial association is spatial correlation or regression. A more focused use of spatial association is often called environmental correlation, which uses environmental covariates as predictors. Soil science has heavily relied on this concept. Although vegetation was recognized as an indicator of soil quality by the ancient Greeks, E.W. Hilgard formally described the relationship of soil properties with the more readily observable characteristics of vegetation in 1860. V.V. Dokuchaev went further in 1883 by recognizing that soil characteristics could be predicted by considering the factors of climate, organisms, relief, parent material, and time. The guiding principle that where these five factors are the same, similar soil will be found remains the primary strategy for mapping soils today.

The concept of spatial association has also been referred to as regionalization, but that term is easily confused with different forms of spatial analysis. The term ‘regionalization’ has also been used to describe the process of identifying regions based on clusters and it has been used to describe spatial interpolation methods, such as kriging. To add to the confusion, spatial association has also been used to describe statistics that evaluate the existence of clusters. To make things worse, spatial autocorrelation has been sometimes described as a type of spatial association. For these reasons, I think it is important that we establish spatial autocorrelation and spatial association as independent, fundamental concepts in spatial prediction.

Recognizing these two separate concepts in geography makes it easier to explain the variety of spatial prediction methods that attempt to utilize some blend of both. For example, co-kriging adds information from covariates with similar spatial distributions to improve upon interpolations based on spatial autocorrelation. Conversely, geographically weighted regression identifies spatial association relationships within different spatial units, which can be based on cluster analysis or some other form of analysis that recognizes similarity by spatial autocorrelation.

The importance and utility of spatial autocorrelation and spatial association, as defined here, is clear. However, the consistent use of these terms, especially for spatial association, has clouded the recognition of their widespread use. Regardless, these are fundamental and unifying concepts in geography.

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