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Map-ematically Messing with Mapped Data BEYONDMAPPING L BY JOSEPH BERRY forming sharp, abrupt boundaries (choropleth), such as a cover-type map. Discrete maps generally provide limited footholds for quantitative map analysis. However, continuous maps contain a range of values (quantitative only) that form spatial gradients (isopleth), such as an elevation surface. They provide a wealth of analytics from basic grid math to map algebra, calculus and geometry. ast month's "Beyond Mapping" column intro- duced the idea of spatialSTEM for teaching map analysis and modeling fundamentals within a mathematical context that resonates with science, technology, engineering and math/stat communities (see "Spatial- STEM Has Deep Mathematical Roots," GeoWorld, January 2012, page 10). This column focuses on the nature of mapped data, an example of a grid-math/algebra application and discussion of extended spatial-analysis operations. Numeric vs. Geographic Figure 1 identifies the two primary perspectives of spa- tial data: 1) numeric, which indicates how numbers are distributed in "number space" ("what" condition) and 2) geographic, which indicates how numbers are distributed in "geographic space" ("where" condition). The numeric perspective can be grouped into categories of qualitative numbers that deal with general descriptions based on perceived "quality" as well as quantitative numbers that deal with measured characteristics or "quantity." Further classification identifies the familiar numeric data types of nominal, ordinal, interval, ratio and binary. It's generally well known that few math/stat operations can be performed using qualitative data (nominal, ordinal), whereas a wealth of operations can be used with quantitative data (interval, ratio). Only a specialized few operations utilize binary data. Less familiar are the two geographic Joseph Berry is a principal in Berry & Associates, consultants in GIS technology. He can be reached via e-mail at jkberry@du.edu. 10 data types. Choropleth numbers form sharp and unpredictable boundaries in space, such as the values assigned to the discrete map features on a road or cover-type map. Isopleth numbers, however, form continuous and often predictable gradients in geographic space, such as the values on an elevation or temperature surface. Putting the where and what perspectives of spatial data together, discrete maps identify mapped data with spatially indepen- dent numbers (qualitative or quantitative), GEO W ORLD / FEB R UA R Y 2O12 Figure 1. Spatial data perspectives indicate "where is what." There Would be Math Site-specific farming provides a good example of basic grid math and map algebra using continuous maps (see Figure 2). Yield mapping involves simultaneously record- ing yield flow and GPS position as a combine harvests a crop, resulting in a grid map of thousands of georeg- istered numbers that track crop yield throughout a field. Grid math can be used to calculate the mathematical difference in yield at each location between two years by simply subtracting the respective yield maps. Map algebra extends the processing by spatially evaluating the full algebraic percent-change equation. The paradigm shift in this map-ematical approach is that map variables, comprised of thousands of georegistered numbers, are substituted for traditional variables defined by a single value. Map algebra's con- tinuous map solution shows localized variation, rather than a single "typical" value being calculated (i.e., 37.3-percent increase in the example) and assumed everywhere the same in nonspatial analysis. Getting Context Figure 3 expands basic grid math and map algebra into other mathematical arenas. Advanced grid math includes most of the buttons on a scientific calculator to include trigonometric functions. For example, taking the cosine of a slope map expressed in degrees and multiplying it times the planimetric surface area of a grid