Topic 7 – Overlaying Maps and Summarizing the Results Beyond Mapping book

Characterizing Spatial Coincidence the Computer’s Way  describes point-by-point overlay techniques

discusses region-wide summary and map coincidence techniques

If I Hadn’t of Believed It, I Wouldn’t Have Seen It  discusses map-wide overlay techniques and the spatial evaluation of algebraic equations, such as regression

Note: The processing and figures discussed in this topic were derived using MapCalcTM software.  See www.innovativegis.com to download a free MapCalc Learner version with tutorial materials for classroom and self-learning map analysis concepts and procedures.

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Characterizing Spatial Coincidence the Computer’s Way

(GIS World, Jan/Feb 1992)

...that’s the Beauty of the Pseudo-Sciences, since they don't depend on empirical verification, anything can be explained (Doonesbury).

As noted in many previous sections, GIS maps are numbers and a rigorous, quantitative approach to map analysis should be maintained.  However, most of our prior experience with maps is non-quantitative, using map sheets composed of inked lines, shadings, symbols and zip-a-tone.  We rarely think of map uncertainty and error propagation.  And we certainly wouldn't think of demanding such capabilities in our GIS software.  That is, not as of yet.

Everybody knows the 'bread and butter' of a GIS is its ability to overlay maps.  Why it's one of the first things we think of asking a vendor (right after viewing the 3-D plot that knocks your socks off).  Most often the question and the answer are framed in our common understanding of "light-table gymnastics."  We conceptualize peering through a stack of acetate sheets and interpreting the subtle hues of resulting colors.  To a GIS you're asking the computer to identify the condition from each map layer for every location in a project area.  From the computer's perspective, however, this is simply one of a host of ways to characterize the spatial coincidence.

Let's compare how you and your computer approach the task of identifying coincidence.  Your eye moves randomly about the stack, pausing for a nanosecond at each location and mentally establishing the conditions by interpreting the color.  Your summary might conclude that the northeastern portion of the area is unfavorable as it has "kind of a magenta tone."  This is the result of visually combining steep slopes portrayed as bright red with unstable soils portrayed as bright blue with minimal vegetation portrayed as dark green.  If you want to express the result in map form, you would tape a clear acetate sheet on top and delineate globs of color differences and label each parcel with your interpretation.  Whew!  No wonder you want a GIS.

The GIS goes about the task in a very similar manner.  In a vector system, line segments defining polygon boundaries are tested to determine if they cross.  When a line on one map crosses a line on another map, a new combinatorial polygonal is indicated.  Trigonometry is employed, and the X,Y coordinate of the intersection of the lines is computed.  The two line segments are split into four and values identifying the combined map conditions are assigned.  The result of all this crossing and splitting is the set of polygonal prodigy you so laboriously delineated by hand.

A raster system has things a bit easier.  As all locations are predefined as a consistent set of cells within a matrix, the computer merely 'goes' to a location, retrieves the information stored for each map layer and assigns a value indicating the combined map conditions.  The result is a new set of values for the matrix identifying the coincidence of the maps.

The big difference between ocular and computer approaches to map overlay isn't so much in technique, as it is in the treatment of the data.  If you have several maps to overlay you quickly run out of distinct colors and the whole stack of maps goes to an indistinguishable dark, purplish hue.  One remedy is to classify each map layer into just two categories, such as suitable and unsuitable.  Keep one as clear acetate (good) and shade the other as light grey (bad).  The resulting stack avoids the ambiguities of color combinations, and depicts the best areas as lighter tones.  However, in making the technique operable you have severely limited the content of the data-- just good and bad.

The computer can mimic this technique by using binary maps.  A "0" is assigned to good conditions and a "1" is assigned to bad conditions.  The sum of the maps has the same information as the brightness scale you observe-- the smaller the value the better.  The two basic forms of logical combination can be computed.  "Find those locations which have good slopes .AND. good soils .AND. good vegetative cover."  Your eye sees this as the perfectly clear locations.  The computer sees this as the numeric pattern 0-0-0.  "Find those locations which have good slopes .OR. good soils .OR. good vegetative cover."  To you this is could be any location that is not the darkest shading; to the computer it is any numeric pattern that has at least one 0.  But how would you handle, "Find those locations which have good slopes .OR. good soils .AND. good vegetative cover"?  You can't find them by simply viewing the stack of maps.  You would have to spent a lot of time flipping through the stack.  To the computer, this is simply the patterns 0-1-0, 1-0-0 and 0-0-0.  It's a piece of cake from the digital perspective.

In fact any combination is easy to identify.  Let's say we expand our informational scale and redefine each map from just good and bad to not suitable (0), poor (1), marginal (2), good (3) and excellent (4).  We could ask the computer to INTERSECT SLOPES WITH SOILS WITH COVER COMPLETELY FOR ALL-COMBINATIONS.  The result is a map indicating all combinations that actually occur among the three maps.  Likely this map would be too complex for human viewing enjoyment, but it contains the detailed information basic to many application models.  A more direct approach is a geographic search for the best areas invoked by asking to INTERSECT SLOPES WITH SOILS WITH COVER FOR EXCELLENT-AREAS ASSIGNING 1 TO 4 AND 4 AND 4.  Any combination not assigned a value drops to 0, leaving a map with 1's indicating the excellent areas.

Let's try another way of combining these maps by asking to COMPUTE SLOPES MINIMIZE SOILS MINIMIZE COVER FOR WEAK-LINK.  The resulting map's values indicate the minimal coincidence rating for each location.  Low values indicate areas of concern and a 0 indicate areas to dismiss as not suitable from at least one map's information.  There is a host of other computational operations you could invoke, such as plus, minus, times, divided, and exponentiation.  Just look at the functional keys on your hand calculator.  But you may wonder, "why would someone want to raise one map to the power of another"?  Spatial modelers who have gone beyond mapping, that's who.

What would happen if, for each location (be it a polygon or a cell), we computed the sum of the three maps, then divided by the number of maps?  That would yield the average rating for each location.  Those with the higher averages are better.  Right?  You might want to take it a few steps further.  First, in a particular application, some maps may be more important than others in determining the best areas.  Ask the computer to AVERAGE SLOPES TIMES 5 WITH SOILS TIMES 3 WITH COVER TIMES 1 FOR WEIGHTED-AVERAGE.  The result is a map whose average ratings are more heavily influenced by slope and soil conditions.

Just to get a handle on the variability of ratings at each location, you can determine the standard deviation-- either simple or weighted.  Or for even more information, determine the coefficient of variation, which is the ratio of the standard deviation to the average, expressed as a percent.  What will that tell you?  It hints at the degree of confidence you should put into the average rating.  A high COFFVAR indicates wildly fluctuating ratings among the maps and you might want to look at the actual combinations before making a decision.

How about one final consideration?  Combine the information on minimal rating (WEAKEST-LINK) with that of the average rating (WEIGHTED-AVERAGE).  A prudent decision-maker would be interested in those areas with the highest average rating, but score at least 2 (marginal) in any of the map layers.  This level of detail should be running through your head while viewing a stack of acetate sheets, or a simple GIS product depicting map coincidence.  Is it?  If not, you might consider stepping beyond mapping.

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As with all Beyond Mapping articles, allow me to apologize in advance for the "poetic license" invoked in this terse treatment of a technical subject.  Readers interested in more information should read a "classic" paper in map overlay by Charles J. Robinove entitled "Principles of Logic and the Use of Digital Geographic Information Systems," published by U.S. Geological Survey, 1977.

Map Overlay Techniques— there’s more than one

(GIS World, March 1992)

…I have the feeling we aren’t in Kansas anymore (Dorothy to Todo).

Last section's discussion of map overlay procedures may have felt like a scene from the Wizard of Oz.  The simple concept of throwing a couple of maps on a light-table was blown all out of proportion into the techy terms of combinatorial, computational and statistical summaries of map coincidence.  An uncomfortable, unfriendly and unfathomable way of thinking.  But that's the reality of GIS-- the surrealistic world of map-ematics.

Now that maps are digital, all GIS processing is the mathematical or statistical summary of map values.  What characterized last issue's discussion was that the values to be summarized were obtained from a set of spatially registered maps at a particular location, termed point-by-point map overlay.  Like the movie TRON, imagine you shrank small enough to crawl into your computer and found yourself standing atop a stack of maps.  You look down and see numbers aligned beneath you.  You grab a spear and thrust it straight down into the stack.  As you pull it up, the impaled values form a shish kabob of numbers.  You run with the kabob to the CPU and mathematically or statistically summarize the numbers as they are pealed off.  Then you run back to the stack, place the summary value where you previously stood, and then move over to next cell in a raster system.  Or, if your using a vector system, you would move over to the next 'polygonal prodigy' (see last issue).

What filled the pages last issue, were some of ways to summarize the values.  Let's continue with the smorgasbord of possibilities.  Consider a 'coincidence summary' identifying the frequency of joint occurrence.  If you CROSSTAB FORESTS WITH SOILS a table results identifying how often each forest type jointly occurs with each soil type.  In a vector system, this is the total area of all the polygonal prodigy for each of the forest/soil combinations.  In a raster system, this is simply a count of all the cell locations for each forest/soil combination.

 TABLE 1.  Coincidence Table                                                  For Map1 = FORESTS With Map2 = SOILS Map1 Forests Number of Cells Map1 Soils Number of Cells Number of Crosses Percent of 625 Total % of Map 1 % of Map 2 1 Deciduous 303 1 Lowland 427 299 47.84 98.68 70.02 1 Deciduous 303 2 Upland 198 4 0.64 1.32 2.02 2 Conifer 322 1 Lowland 427 128 20.48 39.75 29.98 2 Conifer 322 2 Upland 198 194 31.04 60.25 97.98

For example, reading across the first row of Table 1 notes that Forest category 1 (Deciduous) contains 303 cells distributed throughout the map.  The Soils category 1 (Lowland) totals 427 cells.  The next section of the table notes that the joint condition of Deciduous/Lowland occurs 299 times for 47.84 percent of the total map area.  Contrast this result with that of Deciduous/Upland occurrence on the row below indicating only four 'crosses' for less than one percent of the map.  The coincidence statistics for the Conifer category is more balanced with 128 cells (20.48%) occurring with the Lowland soil type and 194 cells (31.04%) occurring with the Upland soil type.

These data may cause you to jump to some conclusions, but you had better consider the right side of the table before you do.  These columns normalize the coincidence count to the total number of cells in each category.  For example, the 299 Deciduous/Lowland coincidence accounts for 98.68 percent of all occurrences of Deciduous trees ((299/303)*100).  That's a very strong relationship.  However, from Lowland soil occurrence the 299 Deciduous/Lowland coincidence is a bit weaker as it accounts for only 70.02 percent of all occurrences of Lowland soils ((299/427)*100).  In a similar vein, the Conifer/Upland coincidence is very strong as it accounts for 97.98 percent of the occurrence of all Upland soil occurrences.  Both columns of coincidence percentages must be considered as a single high percent might be merely the result of the other category occurring just about everywhere.

Whew!  What a bunch of droning gibberish.  Maybe you had better read that paragraph again (and again...).  It's important, as it is the basis of spatial statistic's concept of "correlation"-- the direct relationship between two variables.  For the non-techy types seeking just 'the big picture,' the upshot is that a coincidence table provides insight into the relationships among map categories.  A search of the table for unusually high percent overlap of map categories uncovers strong positive relationships.  Relatively low percent overlap indicates negative relationships.

The one and two percent overlaps for Deciduous/Upland suggests the trees are avoiding these soils.  I wonder what spatial relationship exists for Spotted Owl activity and forest type?  For Owl activity and human activity?  For convenience store locations and housing density?  For incidence of respiratory disease and proximity to highways?

There are still a couple of loose ends before we can wrap-up point-by-point overlay summaries.  One is direct map comparison, or 'change detection'.  For example, if you encode a series of land use maps for an area, then subtract each successive pair of maps, the locations that underwent change will appear as non-zero values for each time step.  In GIS-speak, you would enter COMPUTE LANDUSE-T2 MINUS LANDUSE-T1 FOR CHANGE-T2&1 for a map of the land use change between Time 1 and Time 2.

If you are real tricky and think 'digitally,' you will assign a binary progression to the land use categories (1,2,4,8,16, etc.), as the differences will automatically identify the nature of the change.  The only way you can get a 1 is 2-1; a 2 is 4-2; a 3 is 4-1; a 6 is 8-2; etc.  A negative sign indicates the opposite change, and now all bases are covered.  Prime numbers will also work, but they require more brain power to interpret.

Our last point-by-point operation is a weird one-- 'covering'.  This operation is truly spatial and has no traditional math counterpart.  Imagine you prepared two acetate sheets by coloring all of the forested areas an opaque green on one sheet and all of the roads an opaque red on the other sheet.  Now overlay them on a light-table.  If you place the forest sheet down first the red roads will 'cover' the green forests and you will see the roads passing through the forests.  If the roads map goes down first, the red lines will stop abruptly at the green forest globs.

In a GIS, however, the colors become numbers and the clear acetate is assigned zero.  The command COVER FORESTS WITH ROADS causes the computer to go to a location and assess the shish kabob of values it finds.  If the kabob value for roads is 0 (clear), keep the forest value underneath it.  If the road value is non-zero, place that value at the location, regardless of the value underneath.

So What?  What's it good for?  There is a lot of advanced modeling uses, however covering is most frequently used for masking map information.  Say you just computed a slope map for a large area and you want to identify the slope for just your district.  You would create a mask by assigning 0 to your district and some wild number like 32,000 to the area outside your district.  Now cover the slope map with your mask and the slopes will show through for just your district.  This should be a comfortable operation.  It is just like you do on the light-table.

But so much for that comfortable feeling.  Let's extend our thinking to region-wide map overlay.  Imagine you're back inside your computer, but this time you end up sandwiched between two maps.  It's a horrible place and you are up to your ankles in numbers.  You glance up and note there is a pattern in the numbers on the map above.  Why it is the exact shape of your district!  This time you take the spear and attach a rope, like an oversized needle and thread.  You wander around threading the numbers at your feet until you have impaled all of them within the boundary of your district.  Now run to the CPU, calculate their average and assign the average value to your district.  Voila, you now know the average slope for your district provided you were sloshing around in slope values.

Since you're computerized and moving a megahertz speed, you decide to repeat the process for all of the other districts denoted on the template map above you.  You are sort of a digital cookie-cutter summarizing the numbers you find on one map within the boundaries identified on another map.  That's the weird world of region-wide map overlay.  In GIS-speak, you would enter COMPOSITE DISTRICTS WITH SLOPE AVERAGE.

However, average isn't the only summary you can perform with your lace of numbers.  Some other summary statistics you might use include total, maximum, minimum, median, mode or minority value; the standard deviation, variance or diversity of values; and the correlation, deviation or uniqueness of a particular combination of values.  See, math and stat are the cornerstones of GIS.

For example, a map indicating the proportion of undeveloped land within each of several counties could be generated by superimposing a map of county boundaries on a map of land use and computing the ratio of undeveloped land to the total land area for each county.  Or a map of postal codes could be superimposed over maps of demographic data to determine the average income, average age and dominant ethnic group within each zip code.  Or a map of dead timber stands could be used as a template to determine average slope, dominant aspect, average elevation and dominant soil for each stand.  If they tend to be dying at steep, northerly, high elevations on acidic soils this information might help you locate areas of threatened living trees that would benefit from management action.  Sort of a preventative health plan for the woods.

In the next section, point-by-point and region-wide overlaying will be extended to concepts of map-wide overlay.  If all goes well, this will complete our overview of map overlay and we can forge ahead to other interesting (?) topics.

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As with all Beyond Mapping articles, allow me to apologize in advance for the "poetic license" invoked in this terse treatment of a technical subject.  Readers interested in an in-depth presentation of this material should consult a recent text entitled "Statistics for Spatial Data," by Noel Cressie, Wiley Series in Probability and Mathematical Statistics, 1991.

If I Hadn’t Believed It, I wouldn’t have Seen It

(GIS World, April 1992)

…isn’t that the truth, as prejudgment often determines what we see in a map (as well as a ppsychologist’s Rorschach inkblot test).

For better or worse, much of map analysis is left to human viewing.  In many ways the analytical power of the human mind far exceeds the methodical algorithms of a GIS.  As your eye roams across a map, you immediately assess the relationships among spatial features, and your judgment translates these data into meaningful information.  No bits, bytes or buts, that's the way it is.  Just as you see it.

Recently, I had an opportunity to work with an organization that had acquired a major GIS software package, developed an extensive database over a period of several months and had just begun using the system in decision-making.  From the more than one hundred map layers in the database, three composite maps were generated for each of the eighteen topo sheets covering the project area.  The three maps were aligned on top of each other and a fourth clear acetate sheet was attached to complete the bundle.

The eighteen map bundles, in turn, were edge-matched and taped along the wall of a local gymnasium.  A group of decision-makers strolled down the gallery of maps, stopping and flipping through each bundle as they went.  A profusion of discussion ensued.  Finally, with knitted brows and nodding heads, annotations were sketched onto the clear top sheet designating areas available for logging, for development, for wildlife habitat, for recreation, and a myriad of other land uses.  The set of 'solution' sheets were peeled off the top and given to the stunned GIS specialists to be traced into the GIS for final plotting.

Obviously, map overlay means different things to different people.  To the decision-makers it provided a 'data sandwich' for their visual analysis.  To the GIS specialists it not only organizes and displays map composites, but it provides new analytic tools.  To readers of the last couple of issues it means combinatorial, computational and statistical summaries of map coincidence.  As noted, the coincidence data to be summarized can be obtained by point-by-point or region-wide map overlay techniques.

With those discussions behind us, we move on to a third way of combining maps-- map-wide overlay.  Recall that point-by-point overlay can be conceptualized as vertically "spearing" a shish kabob of numbers for each location on a set of registered maps.  By contrast, region-wide overlay horizontally "laces" a string of numbers within an area identified on another map.  Now are you ready for this, map-wide overlay can be thought of as "plunging" an equation through a set of registered maps.  In this instance each map is considered a variable, each location is considered a case and each value is considered a measurement.  These terms (variable, case and measurement) hold special significance for techy types, and have certain rights, privileges and responsibilities when evaluating equations.  For the rest of us, it means that the entire map area is summarized in accordance of an equation.

For example, map-wide overlay can be used for comparing two maps.  Traditional statistics provides several techniques for assessing the similarity among sets of numbers.  The GIS provides the organization of the number sets-- cells in a raster system and polygonal prodigy in a vector system.  A simple "t" or "F" Test uses the means and variances of two sample populations to determine if you can statistically say "they came from the same mold."  Suppose two sample populations of soil lead concentration cover the same map area for two different time periods.  Did lead concentration "significantly" change?  Map-wide comparison statistics provides considerable insight.

Another comparison technique is similarity.  Suppose you have a stack of maps for the world depicting the Gross National Product, Population Density, Animal Protein Consumption and other variables describing human living conditions.  Which areas are similar and which areas are dissimilar?  In the early 1980's I had a chance to put this approach into action.  The concept of "regionalism" had reached its peak and the World Bank was interested in ways it could partition the world into similar groupings.  A program was developed allowing the user to identify a location (say Northeastern Nigeria) and generate a map of similarity for all other locations of the world.  The similarity map contained values from 0 (totally dissimilar) to 100 (identical).

A remote sensing specialist would say "so what, no big deal."  It is a standard multivariate classification procedure.  Spear the characteristics of the area of interest (referred to as a feature vector) and compare this response pattern to those of every other location in the map area.  They're right, it is no big deal.  All that is needed is scale adjustments to normalize map response ranges.  The rest is standard multivariate analysis.  However, to some it is mind wrenching because we normally do not mix map analysis and multivariate classification in the same breadth.  But that's before the digital map took us beyond mapping.

Let's try another application perspective.  A natural resource manager might have a set of maps depicting slope, aspect, soil type, depth to bedrock, etc.  Which areas are similar and which areas are dissimilar?  The procedure is like that described above.  In this instance, however, clustering techniques are used to group locations of similar characteristics.  Techy terms of "intra- and inter-cluster distances in multivariate space" report the similarities.  To the manager, the map shows ecological potential-- a radically different way to carve-up the landscape.

Chances are the current carving into timber management units was derived by aerial photo interpretation.  Some of the parcels are the visible result of cut-out/get-out logging and forest fires which failed to respect ecological partitions.  Comparison of the current management parcels to the ecological groupings might bring management actions more into line with Mother Nature.  The alternative is to manage the woods into perpetuity based on how the landscape exposed itself to aerial film several years back.

In addition to comparison and similarity indices, predictive equations can be evaluated.  For example, consider an old application done in the late 1970's.  A timber company in the Pacific Northwest was concerned in predicting "timber felling breakage."  You see, when you cut a tree there is a chance it will crack when it hits the ground.  If it cracks, the sawmill will produce little chunks of wood instead large, valuable boards.  This could cost you millions.  Where should you send your best teams with specialized equipment to minimize breakage?

Common sense tells you if there are big old rotten trees on steep slopes expect kindling at the mill.  A regression equation specifies this bit more rigorously as

Y= -2.490 + 1.670X1 + 0.424X2 - 0.007X3 - 1.120X4 - 5.090X5

where Y= predicted timber felling breakage and X1= percent slope, X2= tree diameter,

X3= tree height, X4= tree volume and X5= percent defect

Now you go to the woods and collect data at several sample plots, and then calculate the average for each variable.  You substitute the averages into the equation and solve.  There that's it, predicted timber felling breakage for the proposed harvest unit.  If it is high, send in the special timber beasts.  If it is low, send them elsewhere.

But is it really that simple?  What if there are big trees to the north and little trees to the south?  There must be medium-sized trees (average) everywhere is the assumption of your analytic procedure.  And what if it is steep to the north and fairly flat to the south?  Why it must be moderately sloped (average) everywhere is the assumption.  This reasoning leads to medium-sized trees on moderate slopes everywhere.  Right?  But hold it, let's be spatially specific-- there are big trees on steep slopes to the north.  This is a real board and profit busting condition.  Your field data is trying to tell you this, yet your non-spatial analysis blurs the information into typical responses assumed everywhere the same.

Figure 1.Schematic of spatially evaluating prediction equations.

As depicted in figure 1, a spatial analysis first creates continuous surface maps from the field data (see Beyond Mapping column, October, 1990).  These mapped variables, in turn, are multiplied by their respective regression coefficients and then summed.  The result is the spatial evaluation of the regression equation in the form of a map of timber felling breakage.  Not a single value assumed everywhere, but a map showing areas of higher and lower breakage.  That's a lot more guidance from the same set of field data.

In fact further investigation showed the overall average of the map predictions was very different from the non-spatial prediction.  Why?  It was due to our last concept-- spatial autocorrelation.  The underlying assumption of the non-spatial analysis is that all of the variables of an equation are independent.  Particularly in natural resource applications this is a poor assumption.  People, birds, bears and even trees tend to spatially select their surroundings.  It is rare that you find any resource randomly distributed in space.  In this example, there is a compelling and easily explainable reason for big trees on steep slopes.  During the 1930's the area was logged and who in their right mind would hassle with the trees on the steep slopes.  The loggers just moved on to the next valley, leaving the big trees on the steeper slopes-- an obvious relationship, or more precisely termed, spatial autocorrelation.  This condition affects many of our quantitative models based on geographic data.

So where does all this lead?  To at least an appreciation that map overlay is more than just a data sandwich.  The ability to render graphic output from a geographic search is just the beginning of what you can do with the map overlay procedures embedded in your GIS.

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