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Paint by Numbers outside the Traditional Statistics Box BEYONDMAPPING T BY JOSEPH BERRY surface involve normalization techniques. For example, a Standard Normal Variable map can be generated to identify "how unusual" (above or below) each map loca- tion is compared to the typical value in a project area. Direct comparisons among continuous map surfaces include appropriate statistical tests (e.g., F-test), difference maps and surface-configuration differences based on variations in surface slope and orientation at each grid location. Map correlations provide a foothold for advanced he last two "Beyond Mapping" columns described a general framework and approach for teaching spatial analysis within a mathematical context that resonates with science, technology, engineering and math/stat communities (spatialSTEM). The following discussion focuses on extending tradi- tional statistics to spatial statistics for understanding geographic-based patterns and relationships. Whereas spatial analysis focuses on "contextual relationships" in geographic space (such as effective proximity and visual exposure), spatial statistics focuses on "numerical relationships" within and among mapped data (see Figure 1). From a spatial-statistics perspective, there are three primary analytical arenas: Summaries, Comparisons and Correlations. Enter the Arena Statistical summaries provide generalizations of the grid values comprising a single map layer (within) or set of map layers (among). Most common is a tabular summary included in a discrete map's legend that identifies the area and proportion of occurrence for each map category, such as extremely steep terrain comprising 286 acres (19 percent) of a project area. Or for a continuous map surface of slope values, the generalization might identify the data range as from 0-65 percent and note that the average slope is 24.4 with a standard deviation of 16.7. Summaries among two or more discrete Joseph Berry is a principal in Berry & Associates, consultants in GIS technology. He can be reached via e-mail at jkberry@du.edu. 10 maps generate cross-tabular tables that "count" the joint occurrence of all categori- cal combinations of the map layers. For example, the coincidence of steepness and cover maps might identify that there are 242 acres of forest cover on extremely steep slopes (16 percent), a particularly hazardous wildfire condition. Map comparison and correlation techniques only apply to continuous mapped data. Comparisons within a single map GEO W ORLD / MA R CH 2O12 Figure 1. Spatial statistics use numerical analysis to uncover spatial relationships and patterns. inferential spatial statistics. Spatial autocorrelation within a single map surface identifies the similarity among nearby values for each grid location. Spatial correlation, however, identifies the degree of geographic dependence among two or more map layers and is the foundation of spatial data mining. For example, a map surface of a bank's existing concentra- tion of home-equity loans within a city can be regressed against a map surface of home values. If a high level of spatial dependence exists, the derived regression equation can be used on home-value data for another city. The resulting map surface of estimated loan concentration proves useful in locating branch offices. Lost in Space Figure 2 describes an example of basic surface mod- eling and the link between numeric-space and geo- graphic-space representations using environmentally oriented mapped data. Soil samples are collected and analyzed, ensuring that geographic coordinates accompany the field samples. The resulting discrete point map of the soil-chemistry data are spatially interpolated into a continuous map surface charac- terizing the dataset's geographic distribution. The bottom portion of Figure 2 depicts the link among data-space and geographic-space representa- tions of mapped data. In data space, a standard