Quantitative Methods for Analyzing Map Similarity and Zoning


Joseph K. Berry1

President, Berry & Associates // Spatial Information Systems

2000 South College Avenue, Suite 300

Fort Collins, Colorado, USA 80525

Phone: 970-215-0825

Email: jberry@innovativegis.com



In the past thirty years GIS technology has progressed from computer mapping to spatial database management, and more recently, to quantitative map analysis and modeling.  However, most applications still rely on visceral/visual analysis for determining similarity within and among maps.  Various quantitative techniques for analysis and comparison of mapped data are becoming available in inexpensive and easy-to-use packages that seamlessly interface with existing desktop mapping systems.  This paper investigates the conceptual basis and practical considerations in evaluating data patterns within a single map and relationships among sets of map layers.  Procedures discussed include comparison of discrete/continuous maps, similarity index, and data zoning/clustering.  Considerable attention is given to the nature of mapped data, assumptions inherent in the techniques, and mechanisms for evaluating results.  Examples emphasize environmental and natural resource applications yet apply to any discipline seeking to quantify similarities within and among maps.




I bet you will see it over and over at this and other conferences¾a speaker waves a laser pointer at a couple of maps and says something like "see how similar the maps are."  But what determines similarity… a few similarly shaped gobs appearing in the same general area?  Or do all of the globs have to kind of align?  Do display factors, such as color selection, number of classes and thematic break points, affect perceived similarity?  What about the areas that misalign?  How dissimilar are they?


The ability to objectively compare maps is fundamental to map analysis yet is often neglected.  Visual comparison is far too limited and, in most cases, must generalize mapped data before human consumption.  The result is a subjective analysis of generalized data with minimal spatial specificity.  The problem is magnified for visual assessment of spatial relationships among a set of maps.  This paper presents several techniques for comparing maps and quantifying spatial relationships implied in their coincident data patterns.


Comparing Discrete Maps

Consider the three maps on the right side of figure 1-1. The maps generalize the same map surface into discrete zones (white= low, light gray= medium and dark gray= high).  Are the maps similar?  Is the top map more similar to the middle one, than the map on the bottom?  If so, how much more similar? 

Figure 1-1.  Coincidence Summary and Proximal Alignment can be used to assess the similarity between maps.  Note: the tables identified in the figure are discussed in the referenced articles. 

One way to find out for certain is to overlay the two maps and note where the classifications are the same and where they are different.  At one extreme, the maps could perfectly coincide with the same conditions everywhere (identical).  At the other extreme, the conditions might be different everywhere.  Somewhere in between these extremes, the high and low areas could be swapped and the pattern inverted¾similar but opposite. 


A “coincidence summary generates a cross-tabular listing of the intersection of two maps (Berry, 1999a).  The percentage of the map area in agreement indicates the overall similarity.  Of more interest, however, are the areas of disagreement.  The summary table indicates for each location the specific nature of the discrepancy.  For example areas that are dark gray in the top map in the figure that align with areas that are light gray in the bottom map are a different type disagreement than a misalignment with white areas.  Summary percentages for the different types of misalignments provide insight into which classes are more often confused than others.


In vector analysis maps are intersected and aggregating the areas of the son-and-daughter polygons that are derived summarizes the type of disagreement.  In grid-based analysis the process simply involves noting the number of grid cells falling into each category combination.  All of the remaining techniques for assessing similarity and data pattern relationships require grid-based processing and are not available in vector systems.

A more powerful technique for comparing discrete maps, termed “proximal alignment,” isolates one of the map categories (e.g., the dark-toned areas on Map3 in the figure) then generates its proximity map (Berry, 1999a).  The proximity values are "masked" for the corresponding feature on the other map (e.g., enlarged dark-toned area on Map1).  The result is proximity values appearing within areas of misalignment between the two maps.  These values provide further information about map disagreement as larger values indicate locations where the two maps are drastically misaligned.  Small proximity values indicate areas that are different but not too different.  These proximity data on the geographic extent of disagreement can be summarized for a statistic of “how far” off the two maps are—not just “how often” (percentage of area) they are off.

Comparing Continuous Map Surfaces

Comparing map surfaces involves similar approaches, but employs different techniques taking advantage of the more robust nature of continuous grid-based data.  Consider the two map surfaces shown on the left side of figure 1-2.  Are they similar, or different?  Where are they more similar or different?  Where's the greatest difference?  How would you know?

Figure 1-2.  Map surfaces can be compared by statistically testing for significant differences in data sets, differences in spatial coincidence, or surface configuration alignment.  Note: the tables identified in the figure are discussed in the referenced articles. 

In visual comparison, your eye looks back-and-forth between the two surfaces attempting to compare the relative “heights" at corresponding locations on the surfaces.  In the computer, the relative heights are stored as individual map “values” (in this case, 1380 numbers in an analysis grid of 46 rows by 30 columns). 

One approach that quantitatively compares the surfaces involves “statistical tests whether the data sets are significantly different (Berry, 1999b).  In this approach, GIS is used to "package" the data and pass it to standard statistical program for analysis of differences within and between data groups.  The packaging can be done in a variety of ways including systematic/random sampling, specified administration/management zones, or inferred spatial groupings.  Statistical tests, such as t- or F-tests can be used to determine if the mapped data are different.

The non-spatial statistical tests compare entire areas yet they fail to provide any insight into the spatial distribution of the differences.  Comparison using “percent difference,” on the other hand, capitalizes on the juxtaposition of the differences by simply evaluating the percent change equation ([[map1_value – map2_value] / map1_value] *100) for each grid cell (Berry, 1999b).  This measure is the most direct and easy to interpret comparison between two maps as it uses the full data range and geographically depicts the differences—e.g., “…more different over here than it is over there.”  The area statistics in the legend of a difference map can be summarized to identify the "thirds rule of thumb" for comparison¾if two-thirds of the map area is within one-third (33 percent) difference, the surfaces are fairly similar; if less than one-third of the area is within one-third difference, the surfaces are fairly different¾generally speaking that is. 

Yet another approach to compare map surfaces, termed surface configuration, focuses on the differences in the localized trends between two map surfaces instead of the individual values (Berry, 1999b).  Like you, the computer can "see" the bumps in the surfaces, but it does it with a couple of derived maps.  A “slope map indicates the relative steepness while an “aspect” map denotes the orientation of locations along the surface. 

The computer “sees” differences based on the slope and aspect by evaluating some fairly complex trigonometry equations that are beyond the scope of this paper (see the appended discussion at the end of Topic 10 of the online Map Analysis book identified in the reference section for detailed information).  Conceptually speaking, the immediate neighborhood around each grid location identifies a small plane with steepness and orientation defined by the slope and aspect maps.  The equations simply solve for the normalized difference in slope and aspect angles between the two planes.  Locations with flat/vertical differences in inclination (slope difference = 90o) and diametrically opposed orientations (Aspect difference = 180o) are as different as different can get.  Zero differences for both, on the other hand, are as similar as things can get (exactly the same slope and aspect).  All other slope/aspect differences fall somewhere in between on a scale of 0-100.

Identifying Unusual Data Zones

In assessing similarity among maps, the link between “Geographic Space” and “Data Space” is key.  As shown in figure 1-3, spatial data can be viewed as a map or a histogram.  While a map shows us “where is what,” a histogram summarizes “how often” measurements occur (regardless where they occur).


Figure 1-3.  Identifying areas of unusually high measurements based on the numerical distribution of the data.


The top-left portion of the figure shows a 2D/3D map display of the relative amount of phosphorous (P) throughout a farmer’s field.  Note the spikes of high measurements along the edge of the field, with a particularly big spike in the north portion.


The histogram to the right of the map view forms a different perspective of the same data.  Rather than positioning the measurements in geographic space it summarizes their relative frequency of occurrence in data space.  The X-axis of the graph corresponds to the Z-axis of the map—amount of phosphorous.  In this case, the spikes in the graph indicate measurements that occur more frequently.  Note the high occurrence of phosphorous around 11ppm. 


Now to put the geographic-data space link to use.  The shaded area in the histogram view identifies measurements that are unusually high—more than one standard deviation above the mean.  This statistical cutoff is used to isolate locations of high measurements as shown in the map on the right.  The level slicing procedure (Berry, 2001b) is repeated for the potassium (K) and the nitrogen (N) surfaces to identify their locations of unusually high concentrations.


Figure 1-4.  Level-slice classification is used to identify data zones of specified levels.


The box in figure 1-4 depicts how the computer “sees” data patterns.  In this case “data space” is composed of three axes defining the extent of the box that corresponds to the data ranges of P, K and N.  The floating balls represent grid cells—one ball for each grid cell in “geographic space”—and the position of the balls identify their data pattern. 


The balls plotting in the shaded area of the diagram identify field locations that have unusually high P, K and N concentrations.  The non-shaded portions of the box identify conditions in which at least on of the nutrient levels is not unusually high.  The map in the lower right identifies the eight possible combinations from no unusually high concentrations (near the origin of the scatter plot) to unusually high in all three nutrients (shaded area of high, high, high response).  The level slicing in essence carved data space into eight boxes with the map identifying the geographic distribution of the assignments—most of the field does not have unusually high concentration of any of the nutrients (light gray).

Calculating Map Similarity

Extending the geographic-data space link using “data distance” measures develop a map similarity index (Berry, 2001a).  Consider the same three maps shown in the upper-left portion of figure 1-5.  If you focus your attention on the location in the southeastern portion, how similar are all of the other locations in the field?  What about the location in the extreme northern portion… how similar is its data pattern? 


Figure 1-5.  Conceptually linking geographic space and data space for the spatial distribution of P,K and N throughout an agricultural field.


The “data spear” at map location column 45, row 18 identifies that the P-level as 11.0ppm, the K-level as 177.0 and N-level as 32.9.  This step is analogous to your eye noting a color pattern of burnt-red, dark-orange and light green.  The other location for comparison (32c, 62r) has a data pattern of P= 53.2, K= 412.0 and N= 27.9—or as your eye sees it, a color pattern of dark-green, dark-green and yellow. 


As previously noted, the position of any point in data space identifies its numerical pattern—low, low, low is in the back-left corner, while high, high, high is in the upper-right corner.  Points that plot in data space close to each other are similar and those that plot farther away are less similar.  In the example, the floating ball closest to you is the farthest one from the comparison point in the southeastern portion of the field.  This distance becomes the reference for “most different” data pattern and sets the bottom value of the similarity scale (0%).  A point with an identical data pattern plots at exactly the same position in data space resulting in a data distance of 0 that equates to the highest similarity value (100%).


In practice, a user clicks on a location and its data pattern for a selected set of maps becomes the “comparison shishkebab” and the data distance for all other locations are calculated.  An index is normalized to the most different pattern and a similarity map is generated with from 0 (most different) to 100 (identical).  The result is a map that quantitatively shows how similar all of the other locations are to the selected location.


Mapping Data Clusters


While level slicing and map similarity are useful in examining spatial relationships and data patterns, they require a user to specify data analysis parameters.  But what if you don’t know what level slice intervals to use or which locations in the field warrant map similarity investigation?  Can the computer on its own identify groups of similar data?  How would such a classification work?  How well would it work?


Figure 1-6.  Relative data distance is used to divide a map area into clusters of similar data patterns.


Figure 1 shows some examples derived from map clustering (Berry, 2001c) the nutrient data.  The map in the center of the figure shows the results of classifying the P, K and N map stack into two clusters.  The data pattern for each cell location is used to partition the field into two groups that are 1) as different as possible between groups and 2) as similar as possible within a group.  If all went well, any other division of the field into two groups would be not as good at balancing the two criteria.  


The two smaller maps at the right show the division of the data set into three and four clusters.  In all three of the cluster maps red is assigned to the data group with relatively low responses and green to the one with relatively high responses.   Note the encroachment on these marginal groups by the added clusters that are formed by data patterns at the boundaries.


Figure 1-7.  Data patterns for map locations are depicted as floating balls in data space.


The schematic in figure 1-7 depicts the process.  The floating balls identify the data patterns for each map location (geographic space) plotted against the P, K and N axes (data space).  For example, the large ball appearing closest to you depicts a location with high values on all three input maps.  The tiny ball in the opposite corner (near the plot origin) depicts a map location with small map values.  It seems sensible that these two extreme responses would belong to different data groupings.


Discussion of the algorithm used in clustering is beyond the scope of this paper but it suffices to note that “data distances” between the floating balls are used to identify cluster membership—groups of balls that are relatively far from other groups and relatively close to each other form separate data clusters.  In this example, the red balls identify relatively low responses while green ones have relatively high responses.  The geographic pattern of the classification is shown in the map at the lower right of the figure.




While the discussion has focused on agricultural data, keep in mind that the input maps could be crime, pollution or sales data—most sets of application related data.  The fundamental nature of map comparison, data zoning, map similarity and clustering cuts across disciplines, data types and applications.  The techniques address a basic need for evaluating spatial relationships and data patterns beyond simply viewing a set of side-by-side maps.  The capabilities discussed have been known for years and are part of many GIS systems yet they are rarely employed.  As spatial technology gains a larger foothold beyond mapping and geo-query, the quantitative nature of digital mapped data will increasingly become a focus.  In the future, laser pointer dancing about a set of map displays will be replaced by objective measures that evaluate relationships within and among maps.  Coincidence summary, proximal alignment, statistical tests, percent difference, surface configuration, level-slicing, map similarity, clustering and a host of other quantitative measures will replace subjective “visceral visions” of relationships thought to exist when viewing graphic displays.




Note:  The topics discussed in this paper are treated in greater detail in the Beyond Mapping columns appearing in GeoWorld cited below.  These articles have been compiled into an online text at www.innovativegis.com/basis select the “Map Analysis” book and refer to Topics 10 and 12.


Berry, 1999a.  “Comparing Maps by the Numbers,” Beyond Mapping column, GEOWorld, September issue, 1999, pages 23-24.

Berry, 1999b.  “Use Statistics to Compare Map Surfaces,” Beyond Mapping column, GEOWorld, October issue, 1999, pages 23-24.

Berry, 2001a.  “Geographic Software Removes Guesswork from Map Comparison,” Beyond Mapping column, GEOWorld, October, 2001, pages 23-24.

Berry, 2001b.  “Use Similarity to Identify Data Zones,” Beyond Mapping column, GEOWorld, November, 2001, pages 23-24.

Berry, 2001c.  “Use Statistics to Map Data Clusters,” Beyond Mapping column, GEOWorld, December issue, 2001, pages 23-24.




1Joseph K. Berry is a leading consultant and educator in the application of Geographic Information Systems (GIS) technology.  He is the president of BASIS, consultants and software developers in GIS and the author of the “Beyond Mapping” column for GEOWorld magazine.  He has written over two hundred papers on the theory and application of map analysis, and is the author of the popular books Beyond Mapping and Spatial Reasoning.  Since 1976, he has presented workshops on GIS to thousands of individuals from a wide variety of disciplines.  Dr. Berry conducted basic research and taught courses in GIS for twelve years at Yale University's Graduate School of Forestry and Environmental Studies, and is currently a Special Faculty member at Colorado State University and the W. M. Keck Visiting Scholar in Geography at the University of Denver.  He holds a B.S. degree in forestry, an M.B.A. in business management and a Ph.D. emphasizing remote sensing and land use planning.