GeoWorld March 2013

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To Summarize: Mapping Depends on Where Is What BEYONDMAPPING E arly procedures in spatial statistics largely focused on the characterization of spatial patterns formed by the relative positioning of discrete spatial objects: points, lines and polygons. The area, density, edge, shape, core area, neighbors, diversity and arrangement of map features are summarized by numerous landscape-analysis indices (see "Author's Note 1," page 11). Most of these techniques are direct extensions of manual procedures using paper maps and subsequently coded for digital maps. Grid-based map analysis, however, BY JOSEPH BERRY expands this classical view by the direct application of advanced statistical techniques in analyzing spatial relationships that consider continuous geographic space. Some of the earliest applications (circa 1960) were in climatology and used map surfaces to generate isotherms of temperature and isobars of barometric pressure. In the 1970s, the analysis of remotely sensed data (raster images) began employing traditional statistical techniques that had been used in analyzing nonspatial data for decades. By the 1990s, these classification-oriented procedures, operating on spectral bands, were extended to include the full wealth of statistical operations such as correlation and regression, utilizing diverse sets of georegistered map variables (grid-based map layers). Spatial autocorrelation follows Tobler's first law of geography: "Near things are more alike than distant things." This condition provides the foundation for surface modeling used to identify the continuous spatial distribution implied in a set of discrete point data based on one of four fundamental approaches (see Figure 2 and "Author's Note 2"). The first two approaches, map generalization and geometric facets, consider the entire set of point values in determining the "best fit" of a polynomial equation—a set of 3-D geographic shapes. The lower two approaches in Figure 1, density analysis and spatial interpolation, are based on localized summaries of point data utilizing "roving windows." Density analysis counts the number of data points in the window (e.g., number of crime incidents within half a kilometer) or computes the sum of the values (e.g., total loan value within half a kilometer). However, the most-frequently used surface-modeling approach is spatial interpolation, which "weightaverages" data values within a roving window based on some function of distance. For example, inverse distance-weighting interpolation uses the geometric equation 1/DPower to diminish the influence of distant data values in computing the weighted average. The bottom portion of Figure 2 encapsulates the basis for Kriging, which derives the weighting equation from the point-data values themselves, instead of assuming a fixed geometric equation. A variogram plot of the joint variation among the data values (blue curve) shows the differences in the values as a function of distance. The inverse of this derived equation (red curve) is used to calculate the distance-affected weights used in weight-averaging the data values. Branching Out Joseph Berry is a principal in Berry & Associates, consultants in GIS technology. He can be reached via e-mail at 10 It's the historical distinction between "spatial-pattern characterization of discrete objects" and "spatial-relationship analysis of continuous map surfaces" that identifies the primary conceptual branches in spatial statistics. The spatial-relationship branch can be further grouped by two types of spatial dependency driving the relationships: 1) spatial autocorrelation, involving spatial relationships within a single map layer, and 2) spatial correlation, involving spatial Figure 1. Spatial dependency involves relationships within a relationships among multiple map laysingle map layer (spatial autocorrelation) or among multiple map ers (see Figure 1). layers (spatial correlation). G E O W O R L D / M A R C H 2 O 1 3

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