Beyond Mapping II

Topic 10 – A Futuristic GIS



Spatial Reasoning book



The Unique Character of Spatial Analysis  discusses spatial analysis as deriving new spatial information based on geographic dependence within and among map variables

Analyzing Spatial Dependency within a Map  investigates univariate analysis involving spatial relationships within a single map layer

Analyzing Spatial Dependency between Maps  Analyzing Spatial Dependency Between Maps — investigates multivariate analysis involving the coincidence of two or more map layers


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The Unique Character of Spatial Analysis

(GeoWorld, April 1996)  

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GIS mapping and management capabilities are becoming common in the workplace.  The mapping revolution will be complete when most office automation packages offer GIS at the touch of a button.  So can the GIS technocrats declare a victory and fade into a well-deserved (and well appointed) retirement?  Is that all there is to GIS (as Peggy Lee sings)?  Or have we merely attained another milestone along GIS's evolutionary path? 


GIS is often described as a "decision-support tool."  Currently most of that support comes from its data mapping and management (inventory) capabilities.  The abilities to geo-query datasets and generate tabular and graphical renderings of the results are already recognized; they are invaluable tools but are still basically data handling operations.  It is the ability of GIS to analyze the data that will eventually revolutionize the way we deal with spatial information. 


The heart of GIS analysis is the spatial/relational model, which expresses complex spatial relationships among map entities.  The relationships can be stored in the data structure (topology) or derived through analysis.  For example, the cascading relationships among river tributaries can be encapsulated within a dataset.  The actual path a rain drop takes to the sea and the time it takes to get there, however, must be derived from the data by optimal path analysis.  Derivation techniques like these form the spatial analysis toolbox. 


The term "spatial analysis" has assumed various definitions over time and discipline. To some, the geoquery for all locations of dense, old Douglas fir stands from the set of all forest stands is spatial analysis.  But to the GIS purist, the inquiry is a nonspatial database management operation.  It involves manipulating the attribute database and producing a map as its graphical expression, but it doesn't involve spatial analysis.  Spatial analysis, strictly defined, involves operations in which results depend on data locations-move the data, and the results change.  For example, if you move a bunch of elk in a park their population center moves, but the average weight for an elk doesn't.  That distinction identifies the two basic types of geo-referenced measures: spatially dependent or independent.  The population center calculation is a spatially dependent measurement, and the average weight considering the entire population is independent.  Note that the term "measurement" is a derived relationship, not a dataset characteristic.  Spatial analysis involves deriving new spatial information, not repackaging existing data. 


With that definition diatribe under your belt, you have one more distinction to complete the conceptual framework for spatial analysis: derivation mechanics, whether the data are spatially aggregated or disaggregated.  In the elk example, the average weight for the entire population in the park is spatially independent and aggregated.  Spatial patterns can be inferred, however, if disaggregated analysis is employed by partitioning space into subunits and calculating independent measures.  Such analysis might reveal that the average weight for an elk is higher in one portion of the park than it is in another. 



Figure 1.  Field data of animal activity.


These concepts are illustrated in figure 1.  There are l6 field samples (samples #l-16), their coordinates (X,Y) and the animal activity values for two 24-hour periods (P1 in June and P2 in August).  Note the varying levels of activity: 1 to 42 for Period I and 0 to 87 for Period 2— sample location #15 has the highest activity in both periods.  Also note that the average animal activity increased (19 to 23), as well as its variability (12 to 26).  These traditional statistics tell us things have changed, but they fail to tie the changes to the ground. 


A simple spatial summary of the data's geographic distribution is its centroid, calculated as the weighted average of the X and Y coordinates.  That's done by multiplying each of the sample coordinates (Xi,Yi) by the number of animals for a period at that location (Pi), then dividing the sum of the products by the total animal activity (SUMXiPi / SUMPi and SUMYiPi / SUMPi where i = 1 to 16).  Whew! 


The calculations show the centroid for Period 1 as X = 1,979 and y = 1,728, and it shifts to X = 2,218 and Y = 1,893 for Period 2.  Because the measure moved, the centroid must involve spatial analysis.  Both periods show a "displaced centroid" from the geographic center of the project area.  If the data were uniformly or randomly distributed, the centroid and the geographic center would align.  The magnitude of the difference indicates the degree of the displacement and its direction indicates the orientation of the shift.  Comparing the centroids for the two periods shows a shift toward the northeast (X = 2,218 - 1,979 = 239 meters to the east and Y = 1,893 - 728 = 165 meters to the north). 


The centroid identifies the data's "balance point," or centrality.  Another technique to characterize the data's geographic distribution is a table of the spatially disaggregated means.  First the study area is partitioned into quarter-sections, or "quads."  The average for the data within each quad is computed, and then compared to the average of the entire area.  The calculations show the following:



Whew!  So what does that tell you (other than "being digital" with maps is a pain)?  It appears the southeast and northeast quads have consistently high populations (always above the period averages of 19 and 23), which squares with the centroid's northeast displacement.  Also, the northeast quad consistently has the greatest overage (33 - 19 = 14 for Pl and 58 - 23 = 35 for P2), and the southwest quad has the greatest percentage decrease ([(17 - 19) - (4 -23) / (17 - 19)] * 100 = -850%) from the averages for the two time periods. 


The analysis would be even more spatially disaggregated if you were to "quad the quads," and compute their means.  With this example, however, there would be only one sample point in each of the 16 subdivisions, and their mean would be meaningless.  What if you "quaded the quaded quads" (8 x 8 = 64 cells)?  Most of the partitions wouldn't have a sample value, so what would you do?  That's where the previous discussions of spatial interpolation (Topic 2) come in  to fill the holes. 


The next section will build on the spatial interpolation surface and fill in a few of the conceptual holes as well, such as assumptions about spatial dependency, autocorrelation, and cross-correlation.  Heck, by the time this topic is over, at least you'll have a useful bunch of intimidating techno-science terms to throw around.  If you're really into this stuff, consider the following calculations for the centroid and disaggregated means. 





Analyzing Spatial Dependency within a Map

(GeoWorld, May 1996) 

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The previous section identified two measurements that characterize the geographic distribution of field data: centroid and spatially disaggregated means.  Both techniques reduce findings to discrete, numeric summaries.  The centroid's X,Y coordinates identify the data's balance point, or centrality.  The spatially disaggregated means are expressed in a table of localized averages for an area's successive quarter-sections.  Both techniques reveal the geographic bias in a dataset, but fail to map the data's continuous distribution.  That's where spatial interpolation comes in to estimate the characteristics of unsampled locations from nearby sampled ones. 


Consider the 3-D plot in the center of figure 1.  It identifies a weighted nearest-neighbors interpolated surface of the geographic distribution for Period 2 animal activity data.  Note that the peak in the northeast and the dip in the northwest are consistent with the centroid and disaggregated means characterizations discussed in the previous section.  With the graphical rendering, however, you can "see" the subtle fluctuations in animal activity within the landscape.  High activity appears as a mountain in the northeast and a smaller hill to the south— sort of a two-bumper distribution. 



Figure 1.  Calculation of Standard Normal Variable (SNV) map (univariate spatial analysis).


The surface looks cool and is generally consistent with the bias reported by the centroid and aggregated means, but is it really a good picture of the distribution?  What are the assumptions ingrained in spatial interpolation?  How well do they hold in this case?  That brings us to the concept of spatial dependence— what occurs at one location depends on the characteristics of nearby locations, and near things tend to be more related than distant things.  Spatial dependence can be negative (near things are less alike) or positive (near things are more similar).  But common sense and most interpolation techniques are based on positive spatial dependence. 


That implies a measure of spatial dependence within a dataset should provide insight into how well spatial interpolation might perform.  Such a measure is termed "spatial autocorrelation."  For the techy-types, autocorrelation (in a nonspatial statistics context) means residuals tend to occur as clumps of adjacent deviations on the same side of a regression line— a bunch above, then a bunch below (high autocorrelation), which is a radically different situation than every other residual alternating above then below (low autocorrelation).  For the rest of us, it simply means how good one sample is at predicting a similar sample (or a near sample in GIS's case). 


It shouldn't take a rocket scientist to figure out that high spatial autocorrelation in a set of sample data should yield good interpolated results.  Low autocorrelation should lower your faith in the results.  Two measures often are used: the Geary index and the Moran index.  The Geary index compares the squared differences in value between neighboring samples with the overall variance in values among all samples.  The Moran index is calculated similarly, except it's based on the product of values. 


The equations have lots of subscripts and summation signs and their mathematical details are beyond the scope of this discussion, but both indices relate neighboring responses to typical variations in the dataset.  If neighbors tend to be similar, yet there's a fairly high variability throughout the data, spatial dependency is rampant.  If the neighbors tend to be just as dissimilar as the rest of the data, there isn't much hope for spatial interpolation. 


Yep, this is techy stuff, and I bet you're about to turn the page.  But hold on!  This stuff is important if you intend to go beyond mapping or data-painting by the numbers. 


In a modern GIS, you can click the spatial interpolation button and generate a map from field data in a few milliseconds.  But you could be on thin ice if you simply assume it's correct— ask Geary or Moran if it's worth generating a Standard Normal Variable (SNV) map (the tremendously useful map shown on the right side of figure l) to identify statistically unusual locations.  The procedure calculates a normalized difference from the average for each interpolated location, effectively mapping the standard normal curve in geographic space.  The planimetric plot in the figure identifies areas of unusually high animal activity (shaded blob in the extreme northeast) as locations that are one or more standard deviations greater than the average (depicted as the line balancing the surface's volume). 


So what?  Rather, so where!  If your data show lead concentrations in the soil instead of animal activity, you can identify areas of significantly high lead levels.  Or, if your data indicate lead concentration in blood samples, you can identify pockets of potentially sick people.  If your data are monthly purchases by customers, you can see where the big spenders live.  The SNV map directs your attention to unusual areas in space.  The next step is to relate such unusual areas to other mapped variables. 


But that step takes us into another arena-from univariate to multivariate spatial analysis.  Univariate analysis characterizes the relationships within a single mapped variable, such as an SNV that locates statistically unusual areas.  Multivariate analysis, however, uses the coincidence among maps to build relationships among sets of mapped data. 



Figure 2.  Calculation of percent change map (multivariate spatial analysis).


For example, figure 2 calculates a percent-change map between the two periods.  The planimetric plot in the figure shows areas of increased activity in l0 percent contour steps.  The shaded area identifies locations that increased more than 50 percent.  Now if these were sales data, wouldn't you like to know where the big increases recently occurred?  Even better, statistically relate these areas to other factors, such as advertising coverages, demographics, or whatever else you might try as a driving or correlated factor.  But that discussion is for the next section.



Analyzing Spatial Dependency between Maps

(GeoWorld, June 1996) 

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Most traditional mathematical and statistical procedures extend directly into spatial analysis.  At one extreme, GIS is simply a convenient organizational scheme for tracking important variables.  With geo-referenced data hooked to a spreadsheet or database, drab tabular reports can be displayed as colorful maps.  At another level, geo-referencing serves to guide the map-ematical processing of delineated areas, such as total amount of pesticide applied in each state's watersheds.  Finally, spatial relationships themselves can form the basis for extending traditional math/stat concepts. 


The top portion of figure 1 identifies a unique spatial operation: point pattern analysis.  The random pattern is used as a standard that assumes all points are located independently and are equally likely to occur anywhere.  The average distance between neighboring points under the random condition is based on the density of points per unit area: 1 / ((2 * density)**2), to be exact.  Now suppose you record the locations for a set of objects (e.g., trees), or events (e.g., robberies) to determine if they form a random pattern.  The GIS computes the actual distance between each point and its nearest neighbor, then averages the distances.  If the computed average is close to the random statistic, randomness is indicated.  If the computed average is smaller, a clumped pattern is evident.  And if it approaches the maximum average distance possible for a given density, the pattern occurs uniformly. 


An alternate approach involves a roving window, or filter.  It uses disaggregated spatial analysis as it moves about the map calculating the number of points at each position.  Because the window's size is constant, the number of points about each location indicates the relative frequency of point occurrence.  A slight change in the algorithm generates the average distance between neighboring points and compares it to the expected distance for a random pattern of an equal number of points within the window.  Whew!  The result is a surface with values indicating the relative level of randomness throughout the mapped area. 



Figure 1.  Pattern and cross-correlation analysis.


The distinction between the randomness statistic for the entire area and the randomness surface is important.  It highlights two dominant perspectives in statistical analysis of mapped data: spatial statistics and geostatistics.  Generally, spatial statistics involves discrete space and a set of predefined objects, or entities.  In contrast, geostatistics involves continuous space and a gradient of relative responses, or fields.  Although the distinction isn't sharp and the terms are frequently interchanged, it generally reflects data structure preferences-vector for entities and raster for fields. 


The center and lower portions of figure 1 demonstrate a different aspect of spatial analysis: multivariate analysis.  The point pattern analysis is univariate, because it investigates spatial relationships within a single variable (map).  Multivariate analysis, however, characterizes the relationships between variables.  For example, if you overlay a couple of maps on a light-table, two features (A and B in the figure) might not align (0 percent overlap).  Or the two features could be totally coincident (100 percent overlap).  In either case, spatial dependency might be at the root of the alignment.  If two conditions never occur jointly in space (e.g., open water and Douglas fir trees), a strong negative spatial correlation is implied.  If they always occur together (spruce budworm infestation and spruce trees), a strong positive relationship is indicated.


What commonly occurs is some intermediate coincidence, as there would be some natural overlap even from random placement: if there are only two features and they each occupy half of the mapped area, you would expect a 50 percent random overlap.  Deviation from the expected overlap indicates spatial cross-correlation, or dependency between maps, but two conditions keep the concept of spatial cross-correlation from being that simple.  First, instances of individual map features are discrete and rarely occur often enough for smooth distribution.  Also, the multitude of features on most maps results in an overwhelmingly complex table of statistics. 


Recall the plot of "big increase" in animal activity (>50 percent change between Period 1 and Period 2) previously discussed.  It was a large shaded glob in the northeastern corner of the study area.  I wonder if the occurrence of the big increase relates to another mapped variable, such as cover type.  The crosstab table at the bottom of figure 1 summarizes the joint occurrences in the area of Big Increase and cover type classes of Lake, Meadow, and Forest.  The last column in the table reports the number of grid cells containing both conditions identified on the table's rows.  Note that Big Increase jointly occurs with Meadows only four times, and it occurs 162 times with Forests.  It never occurs with Lakes. (That's fortuitous, because the animal can't swim.)  A gut interpretation is that they are "dancing in the woods," as the Big Increase (I) is concentrated in the Forest (F). 


But can you jump to that conclusion?  The cell count has to be adjusted for the overall frequency of occurrence.  The diagram to the left of the table illustrates this concept.  From I's point of view, 98 percent of its occurrence is associated with F.  But from F's perspective, only 50 percent of its area coincides with I.  Aaaaahhhh!  All this statistical mumble jumble seems to cloud the obvious.


That's the trouble with being digital with maps.  The traditional mapping community tends to see maps as sets of colorful objects, while the statistics community tends to ignore (or assume away) spatial dependency.  GIS is posed to shatter that barrier.  As a catalyst for communication, consider figure 2.  It identifies several examples of statistical techniques grouped by the Descriptive/Predictive and Univariate/Multivariate dimensions of the statistician and the Discrete/continuous dimension of the GIS’er.  At a minimum, the figure should generate thought, discussion, and constructive dialogue about the spatial analysis revolution. 



Figure 2. Dimensions of spatial statistics.


One thing is certain: GIS is more different from than it is similar to traditional mapping and data analysis.  Many of the map-ematical tools are direct conversions of existing scalar procedures.  Sure you can take the second derivative of an elevation surface, but why would you want to?  Who in their right mind would raise one map to the power of another?  Is map regression valid?  What about spatial cluster analysis?  How would you use such techniques?  What new insights do they provide?  What are the restrictive and enabling conditions? 


Another thing is certain: GIS raises as many questions as it answers.  As we move beyond mapping toward spatial reasoning, the linkage between the spatial and quantitative communities will strengthen.  Both perspectives will benefit from realistic physical and conceptual renderings of geographic space.  But the linkage must be extended to include the user.  The increasing complexity of GIS results from its realistic depiction of spatial relationships.  Its descriptions of space are vivid and intuitive.  Its analyses, however, can be confusing and foreign to new users.  The outcome of the pending spatial analysis revolution hinges as much on users' acceptance as on technological development.



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