Beyond Mapping IV

Topic 7 – Spatial Data Mining in Geo-business

 

 

GIS Modeling book

 

 

Twisting the Perspective of Map Surfaces — describes the character of spatial distributions through the generation of a customer density surface

Linking Numeric and Geographic Distributions — investigates the link between numeric and geographic distributions of mapped data 

Myriad Techniques Help to Interpolate Spatial Distributions — discusses the basic concepts underlying spatial interpolation

Use Map Analysis to Characterize Data Groups — describes the use of “data distance” to derive similarity among the data patterns in a set of map layers

Discover the “Miracles” in Mapping Data Clusters — describes the use of “clustering” to identify inherent groupings of similar data patterns 

The Universal Key for Unlocking GIS’s Full Potential — outlines a global referencing system approach compatible with standard DBMS systems

Further Reading — four additional sections

 

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Twisting the Perspective of Map Surfaces

(GeoWorld, April 2008)  

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This section’s theme grabs some earlier concepts, adds an eye of newt and then a twist of perspective to concoct a slightly shaken (not stirred) new perception of map surfaces.  Traditionally one thinks of a map surface in terms of a postcard scene of the Rocky Mountains with peaks and valleys recording uplift and erosion over thousands of years—a down to earth view of a map surface one can stand on. 

 

However, a geomorphological point of view of a digital elevation model (DEM) isn’t the only type of map surface.  For example, a Customer Density Surface can be derived from sales data that depicts the peaks and valleys of customer concentrations throughout a city as discussed in an earlier topic (see author’s note).  Figure 1 summarizes the processing steps involved—1) a customer’s street address is geocoded to identify its Lat/Lon coordinates, 2) vector to raster conversion is used to place and aggregate the number of customers in each grid cell of an analysis frame (discrete mapped data), 3) a rowing window is used to count the total number of customers within a specified radius of each cell (continuous mapped data), and then 4) classified into logical ranges of customer density.

 

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Figure 1.  Geocoding can be used to connect customer addresses to their geographic locations for subsequent  map analysis, such as generating a map surface of customer density.

 

The important thing to note is that the peaks and valleys characterize the Spatial Distribution of the customers, a concept closely akin to a Numerical Distribution that serves as the foundation for traditional statistics.  However in this instance, three dimensions are needed to characterize the data’s dispersion—X and Y coordinates to position the data in geographic space and a Z coordinate to indicate the relative magnitude of the variable (# of customers).  Traditional statistics needs only two dimensions—X to identify the magnitude and Y to identify the number of occurrences.  

 

While both perspectives track the relative frequency of occurrence of the values within a data set, the spatial distribution extends the information to variations in geographic space, as well as in numerical variations in magnitude— from just “what” to “where is what.”  In this case, it describes the geographic pattern of customer density as peaks (lots of customers nearby) and valleys (not many).

 

Within our historical perspective of mapping the ability to plot “where is what” is an end in itself.  Like Inspector Columbo’s crime scene pins poked on a map, the mere visualization of the pattern is thought to be sufficient for solving crimes.  However, the volume of sales transactions and their subtle relationships are far too complex for a visual (visceral?) solution using just a Google Earth mashed-up image.

 

The interaction of numerical and spatial distributions provides fertile turf for a better understanding of the mounds of data we inherently collect every day.  Each credit card swipe identifies a basket of goods and services purchased by a customer that can place on a map for grid-based map analysis to make sense of it all—“why” and “so what.” 

 

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Figure 2.  Merging traditional statistics and map analysis techniques increases understanding of spatial patterns and relationships (variance-focused; continuous) beyond the usual central tendency (mean-focused; scalar) characterization of geo-business data.

 

For example, consider the left side of figure 2 that relates the unusually high response range of customers of greater than 1 standard deviation above the mean (a numerical distribution perspective) to the right side that identifies the location of these pockets of high customer density (a spatial distribution perspective).  As discussed in a previous Beyond Mapping column (May 2002; see author’s note), this simple analysis uses a typical transaction database to 1) map the customer locations, 2) derive a map surface of customer density, 3) identify a statistic for determining unusually high customer concentrations (> Mean + 1SD), and 4) apply the statistic to locate the areas with lots of customers.  The result is a map view of a commonly used technique in traditional statistics—an outcome where the combined result of an integrated spatial and numerical analysis is far greater than their individual contributions.

 

However, another step is needed to complete the process and fully illustrate the geo-business framework.  The left side of figure 3 depicts the translation of a non-spatial customer database into a mapped representation using vector-based processing that determines Lat/Lon coordinates for each record—the mapping component.  Translation from vector to raster data structure establishes the analytic frame needed for map analysis, such as generating a customer density surface and classifying it into statistically defined pockets of high customer density—then map analysis component. 

 

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Figure 3.  The column, row index of the grid cells in the analytic frame serves as a primary key for linking spatial analysis results with individual records in a database.

 

The importance of the analytic frame is paramount in the process as it provides the link back to the original customer database—the spatial data mining component.  The column, row index of the matrix for each customer record is appended to the database and serves as a primary key for “walking” between the GIS and the database.  For example, the areas of unusually high customer density can be appended to the customer database using the column, row index in both data sets, and then used to identify individual customers within these areas. 

 

Similarly, maps of demographics, sales by product, travel-time to our store, and the like can be used in customer segmentation and propensity modeling to identify maps of future sales probabilities.  Areas of high probability can be cross-walked to an existing customer database (or zip +4 or other generic databases) to identify new sales leads, product mix, stocking levels, inventory management, and competition analysis.  At the core of this vast potential for geo-business application set is the analytic frame and its continuous map surfaces that underwrite a spatially aware database.    

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Author’s Note: Related discussion on Density Surface Analysis is presented in Beyond Mapping Compilation Series, book III, Topic 6, section 2 “Milking Spatial Context Information” posted at www.innovativegis.com.

 

 

Linking Numeric and Geographic Distributions

(GeoWorld, June 2008) 

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The previous section set the stage for discussion of the similarities and differences of geo-business applications to more conventional GIS solutions in municipalities, infrastructure, natural resources, and other areas with deep roots in the mapping sciences.  Underlying the discussion was the concept of a Grid-based Analysis Frame that serves as a primary key for “walking” between a continuous geographic space representation and the individual records in a customer database.

 

It illustrated converting a point map of customer locations into a Customer Density Surface that depicts the “peaks and valleys” of customer concentrations throughout a city.  While the process is analogous to Inspector Columbo’s crime scene pins poked on a map, the mere visualization of a point pattern is rarely sufficient for solving crimes.  Nor is it adequate for a detailed understanding of customer distribution and spatial relationships, such as determining areas of statistically high concentrations—Customer Pockets—that can be used in targeted marketing, locating ATMs or sales force allocation. 

 

Another grid-based technique for investigating the customer pattern involves Point Territories assignment.  This procedure looks for inherent spatial groups in the point data and assigns customers to contiguous areas.  In the example in figure 1, you might want to divide the customer locations into ten groups for door-to-door contact on separate days.

 

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Figure 1.  Clustering on the latitude and longitude coordinates of point locations can be used to identify geographically balanced customer territories.

 

The two small inserts on the left show the general pattern of customers, and then the partitioning of the pattern into spatially balanced groups.  This initial step was achieved by applying a K-means clustering algorithm to the latitude and longitude coordinates of the customer locations.  In effect this procedure maximizes the differences between the groups while minimizing the differences within each group.  There are several alternative approaches that could be applied, but K-means is an often-used procedure that is available in all statistical packages and a growing number of GIS systems.

 

The final step to assign territories uses a nearest neighbor interpolation algorithm to assign all non-customer locations to the nearest customer group.  The result is the customer territories map shown on the right.  The partitioning based on customer locations is geographically balanced (approximately the same area in each cluster), however it doesn’t consider the number of customers within each group—that varies from 69 in the lower right (Territory #8) to 252 (Territory #5) near the upper right… that twist of map analysis will be tackled in a future beyond mapping column.

 

However it does bring up an opportunity to discuss the close relationship between spatial and non-spatial statistics.  Most of us are familiar with the old “bell-curve” for school grades.  You know, with lots of C’s, fewer B’s and D’s, and a truly select set of A’s and F’s.  Its shape is a perfect bell, symmetrical about the center with the tails smoothly falling off toward less frequent conditions.   

However the normal distribution (bell-shaped) isn’t as normal (typical) as you might think.  For example, Newsweek noted that the average grade at a major ivy-league university isn’t a solid C with a few A’s and F’s sprinkled about as you might imagine, but an A- with a lot of A’s trailing off to lesser amounts of B’s, C’s and (heaven forbid) the very rare D or F.  

 

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Figure 2.  Mapped data are characterized by their geographic distribution (maps on the left) and their numeric distribution (descriptive statistics and histogram on the right).

 

The frequency distributions of mapped data also tend toward the ab-normal (formally termed asymmetrical).  For example, consider the customer density data shown in the figure 2 that was derived by counting the total number of customers within a specified radius of each cell (roving window).  The geographic distribution of the data is characterized in the Map View by the 2D contour map and 3D surface on the left.  Note the distinct pattern of the terrain with bigger bumps (higher customer density) in the central portion of the project area.  As is normally the case with mapped data, the map values are neither uniformly nor randomly distributed in geographic space.  The unique pattern is the result of complex spatial processes determining where people live that are driven by a host of factors—not spurious, arbitrary, constant or even “normal” events.

 

Now turn your attention to the numeric distribution of the data depicted in the right side of the figure.  The Data View was generated by simply transferring the grid values in the analysis frame to Excel, then applying the Histogram and Descriptive Statistics options of the Data Analysis add-in tools.  The map organizes the data as 100 rows by 100 columns (X,Y) while the non-spatial view simply summarizes the 10,000 values into a set of statistical indices characterizing the overall central tendency of the data.  The mechanics used to plot the histogram and generate the statistics are a piece-of-cake, but the real challenge is to make sense of it all. 

 

Note that the data aren’t distributed as a normal bell-curve, but appear shifted (termed skewed) to the left.  The tallest spike and the intervals to its left, match the large expanse of grey values in the map view—frequently occurring low customer density values.  If the surface contained a disproportionably set of high value locations, there would be a spike at the high end of the histogram.  The red line in the histogram locates the mean (average) value for the numeric distribution.  The red line in the 3D map surface shows the same thing, except its located in the geographic distribution. 

The mental exercise linking geographic space with data space is a good one, and some general points ought to be noted.  First, there isn’t a fixed relationship between the two views of the data’s distribution (geographic and numeric).  A myriad of geographic patterns can result in the same histogram.  That’s because spatial data contains additional information—where, as well as what—and the same data summary of the “what’s” can reflect a multitude of spatial arrangements (“where’s”).   

But is the reverse true?  Can a given geographic arrangement result in different data views?  Nope, and it’s this relationship that catapults mapping and geo-query into the arena of mapped data analysis.  Traditional analysis techniques assume a functional form for the frequency distribution (histogram shape), with the standard normal (bell-shaped) being the most prevalent. 

 

Spatial statistics, the foundation of geo-business applications, doesn’t predispose any geographic or numeric functional forms—it simply responds to the inherent patterns and relationships in a data set.  The next several sections describe some of the surface modeling and spatial data mining techniques available to the venturesome few who are willing to work “outside the lines” of traditional mapping and statistics.

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Author’s Note: a more detailed treatise and hands-on exercises in GIS concepts, procedures and applications in business is in the workbook Analyzing Geo-Business Data (Berry, 2003) available for download from www.innovativegis.com.

 

 

Myriad Techniques Help to Interpolate Spatial Distributions

(GeoWorld, July 2008) 

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Statistical sampling has long been at the core of business research and practice.  Traditionally data analysis used non-spatial statistics to identify the “typical” level of sales, housing prices, customer income, etc. throughout an entire neighborhood, city or region.  Considerable effort was expended to determine the best single estimate and assess just how good the “average” estimate was in typifying the extended geographic area.

 

However non-spatial techniques fail to make use of the geographic patterns inherent in the data to refine the estimate—the typical level is assumed everywhere the same throughout a project area.  The computed variance (or standard deviation) indicates just how good this assumption is—the larger the standard deviation, the less valid is the assumption “everywhere the same.”  But no information is provided as to where values might be more or less than the computed typical value (average). 

 

Spatial Interpolation, on the other hand, utilizes spatial patterns in a data set to generate localized estimates throughout the sampled area.  Conceptually it “maps the variance” by using geographic position to help explain the differences in the sample values.  In practice, it simply fits a continuous surface (kind of like a blanket) to the point data spikes (figure 1).

 

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Figure 1.  Spatial interpolation involves fitting a continuous surface to sample points.

 

While the extension from non-spatial to spatial statistics is a theoretical leap, the practical steps are relatively easy.  The left side of figure 1 shows 2D and 3D “point maps” of data samples depicting the percentage of home equity loan to market value.  Note that the samples are geo-referenced and that the sampling pattern and intensity are different than those generally used in traditional non-spatial statistics and tend to be more regularly spaced and numerous.

 

The surface map on the right side of figure 1 translates pattern of the “spikes” into the peaks and valleys of the surface map representing the data’s spatial distribution.  The traditional, non-spatial approach when mapped is a flat plane (average everywhere) aligned within the yellow zone.  Its “everywhere the same” assumption fails to recognize the patterns of larger levels (reds) and smaller levels (greens).  A decision based on the average level (42.88%) would be ideal for the yellow zone but would likely be inappropriate for most of the project area as the data vary from 16.8 to 72.4 percent.

 

The process of converting point-sampled data into continuous map surfaces representing a spatial distribution is termed Surface Modeling involving density analysis and map generalization (discussed last month), as well as spatial interpolation techniques.  All spatial interpolation techniques establish a "roving window" that—

 

-        moves to a grid location in a project area (analysis frame),

-        calculates an estimate based on the point samples around it (roving window),

-        assigns the estimate to the center cell of the window, and then

-        moves to the next grid location.

 

The extent of the window (both size and shape) affects the result, regardless of the summary technique.  In general, a large window capturing a larger number of values tends to "smooth" the data.  A smaller window tends to result in a "rougher" surface with more abrupt transitions.

 

Three factors affect the window's extent: its reach, the number of samples, balancing.   The reach, or search radius, sets a limit on how far the computer will go in collecting data values.  The number of samples establishes how many data values should be used.  If there is more than the specified number of values within a specified reach, the computer uses just the closest ones.  If there are not enough values, it uses all that it can find within the reach.  Balancing of the data attempts to eliminate directional bias by ensuring that the values are selected in all directions around window's center. 

 

Once a window is established, one of several summary techniques comes into play.  Inverse distance weighted (IDW) is an easy spatial interpolation technique to conceptualize (see figure 2).  It estimates the value for a location as an average of the data values within its vicinity.  The average is weighted in a manner that decreases the influence of the surrounding sample values as the distance increases.  In the figure, the estimate of 53.35 is the "inverse distance-squared (1/D2) weighted average" of the four samples in the window.  Sample #15 (the closest) influences the average a great deal more than sample #14 (farthest away).  

 

The right portion of figure 2 contains three-dimensional (3-D) plots of the point sample data and the inverse distance-squared surface generated.  The estimated value in the example can be conceptualized as "sitting on the surface," 53.35 units above the base (zero).  

 

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Figure 2.  Inverse distance weighted interpolation weight-averages sample values within a roving window.

 

 

 

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Figure 3.  Example Calculations for Inverse Distance Squared Interpolation.

 

 

Figure 3 shows the weighted average calculations for spatially interpolating the example location in figure 2.  The Pythagorean Theorem is used to calculate the Distance from the Grid Location to each of the data Samples within the summary window.  The distances are converted to Weights that are inversely proportional (1/D2; see example calculation in the figure).  The sample Values are multiplied by their computed Weights and the “sum of the products” is divided by the “sum of the weights” to calculate the weighted average value (53.35) for the location on the interpolated surface.

 

Because the inverse distance procedure is a fixed, geometric-based method, the estimated values can never exceed the range of values in the original field data.  Also, IDW tends to "pull-down peaks and pull-up valleys" in the data, as well as generate “bull’s-eyes” around sampled locations.  The technique is best suited for data sets with samples that exhibit minimal regional trends.

 

 

However, there are numerous other spatial interpolation techniques that map the spatial distribution inherent in a data set.  Critical to surface modeling is an ability to benchmark interpolation results from different techniques and describe a procedure for assessing which is best (see Author’s Note) …certainly a “must read” for techie-types. 

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Author’s Note:  The first section of the Further Online Reading listing at the end of this topic describes the application of residual analysis for “Interpreting Interpolation Results (and why it is important).”

 

 

Use Map Analysis to Characterize Data Groups

(GeoWorld, September 2008) 

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One of the most fundamental techniques in map analysis is the comparison of a set of maps.  This usually involves staring at some side-by-side map displays and formulating an impression about how the colorful patterns do and don’t appear to align.   

 

But just how similar is one location to another?  Really similar, or just a little bit similar?  And just how dissimilar are all of the other areas?  While visual (visceral?) analysis can identify broad relationships, it takes quantitative map analysis to generate the detailed scrutiny demanded by most Geo-business applications.

 

Consider the three maps shown in figure 1— what areas identify similar data patterns?  If you focus your attention on a location in the southeastern portion, how similar are all of the other locations?  Or how about a northeastern section?  The answers to these questions are far too complex for visual analysis and certainly beyond the geo-query and display procedures of standard desktop mapping packages. 

 

The mapped data in the example show the geographic patterns of housing density, value and age for a project area.  In visual analysis you move your focus among the maps to summarize the color assignments (2D) or relative surface height (3D) at different locations.  In the southeastern portion the general pattern appears to be low Density, high Value and low Age— low, high, low.  The northeastern portion appears just the opposite—high, low, high.

 

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Figure 1.  Map surfaces identifying the spatial distribution of housing density, value and age.

 

The difficulty in visual analysis is two-fold— remembering the color patterns and calculating the difference.  Quantitative map analysis does the same thing except it uses the actual map values in place of discrete color bands.  In addition, the computer doesn’t tire as easily as you and completes the comparison throughout an entire map window in a second or two (10,000 grid cells in this example).

 

The upper-left portion of figure 2 illustrates capturing the data patterns for comparing two map locations.  The “data spear” at Point #1 identifies the housing Density as 2.4 units/ac, Value as $407,000 and Age as 18.3 years.  This step is analogous to your eye noting a color pattern of green, red, and green.  The other speared location at Point #2 locates the least similar data pattern with housing Density of 4.8 units/ac, Value of $190,000 and Age of 51.2 years— or as your eye sees it, a color pattern of red, green, red. 

 

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Figure 2.  Conceptually linking geographic space and data space.

 

The right side of the figure schematically depicts how the computer determines similarity in the data patterns by analyzing them in three-dimensional “data space.”  Similar data patterns plot close to one another with increasing distance indicating decreasing similarity.  The realization that mapped data can be expressed in both geographic space and data space is paramount to understanding how a computer quantitatively analyses numerical relationships among mapped data. 

 

Geographic space uses earth coordinates, such as latitude and longitude, to locate things in the real world.  The geographic expression of the complete set of measurements depicts their spatial distribution in familiar map form.  Data space, on the other hand, is a bit less familiar.  While you can’t stroll through data space you can conceptualize it as a box with a bunch of balls floating within it. 

 

In the example, the three axes defining the extent of the box correspond to housing Density (D), Value (V) and Age (A).  The floating balls represent data patterns of the grid cells defining the geographic space—one “floating ball” (data point) for each grid cell.  The data values locating the balls extend from the data axes—2.4, 407.0 and 18.3 for the comparison point identified in figure 2.  The other point has considerably higher values in D and A with a much lower V values so it plots at a different location in data space (4.8, 51.2 and 190.0 respectively). 

 

The bottom line for data space analysis is that the position of a point in data space identifies its numerical pattern—low, low, low in the back-left corner, and high, high, high in the upper-right corner of the box.  Points that plot in data space close to each other are similar; those that plot farther away are less similar.  Data distance is the way computers “see” what you see in the map displays.  The real difference in the graphical and quantitative approaches is in the details—the tireless computer “sees” extremely subtle differences between all of the data points and can generate a detailed map of similarity.

 

In the example in figure 2, the floating ball closest to you is least similar—greatest “data distance” from the comparison point.  This distance becomes the reference for “most different” and sets the bottom value of the similarity scale (0% similar).  A point with an identical data pattern plots at exactly the same position in data space resulting in a data distance of 0 equating to the highest similarity value (100% similar).

 

The similarity map shown in figure 3 applies a consistent scale to the data distances calculated between the comparison point and all of the other points.  The green tones indicate locations having fairly similar D, V and A levels to the comparison location—with the darkest green identifying locations with an identical data pattern (100% similar).  It is interesting to note that most of the very similar locations are in the southern portion of the project area.  The light-green to red tones indicate increasingly dissimilar areas occurring in the northern portion of the project area. 

 

A similarity map can be an extremely valuable tool for investigating spatial patterns in a complex set of mapped data.  The similarity calculations can handle any number of input maps, yet humans are unable to even conceptualize more than three variables (data space box).  Also, the different map layers can be weighted to reflect relative importance in determining overall similarity.  For example, housing Value could be specified as ten times more important in assessing similarity.  The result would be a different map than the one shown in figure 3— how different depends on the unique coincidence and weighting of the data patterns themselves.

 

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Figure 3.  A similarity map identifies how related locations are to a given point.

 

In effect, a similarity map replaces a lot of general impressions and subjective suggestions for comparing maps with an objective similarity measure assigned to each map location.  The technique moves map analysis well beyond traditional visual/visceral map interpretation by transforming digital map values into to a quantitative/consistent index of percent similarity.  Just click on a location and up pops a map that shows how similar every other location is to the data pattern at the comparison point— an unbiased appraisal of similarity.

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Author’s Note: a more detailed treatise and hands-on exercises in GIS concepts, procedures and applications in business is in the workbook Analyzing Geo-Business Data (Berry, 2003) available for download from www.innovativegis.com.).

 

 

Discover the “Miracles” in Mapping Data Clusters

(GeoWorld, November 2008) 

(return to top of Topic)

 

The last couple of sections have focused on analyzing data similarities within a stack of maps.   The first technique, termed Map Similarity, generates a map showing how similar all other areas are to a selected location.  A user simply clicks on an area and all of the other map locations are assigned a value from zero (0% similar—as different as you can get) to one hundred (100% similar—exactly the same data pattern).

 

The other technique, Level Slicing, enables a user to specify a data range of interest for each map layer in the stack then generate a map identifying the locations meeting the criteria.  Level Slice output identifies combinations of the criteria met—from only one criterion (and which one it is), to those locations where all of the criteria are met.

 

While both of these techniques are useful in examining spatial relationships, they require the user to specify data analysis parameters.  But what if you don’t know which locations in a project area warrant Map Similarity investigation or what Level Slice intervals to use?  Can the computer on its own identify groups of similar data?  How would such a classification work?  How well would it work?

 

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Figure 1.  Example output from map clustering.

 

Figure 1 shows some example spatial patterns derived from Map Clustering.  The “floating” map layers on the left show the input map stack used for the cluster analysis.  The maps are the same ones used in previous examples and identify the geographic and numeric distributions of housing Density, home Value and home Age levels throughout the example project area. 

 

The map in the center of the figure shows the results of classifying the D, V and A 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 mapped data into two groups would be worse at mathematically balancing the two criteria.  

 

The two smaller maps on 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 cluster with relatively high Density, low Value and high Age responses (less wealthy) and green to the one with the most opposite conditions (wealthy areas).   Note the encroachment at the margin on these basic groups by the added clusters that are formed by reassigning data patterns at the classification boundaries.  The procedure is effectively dividing the project area into “data neighborhoods” based on relative D, V and A values throughout the map area.  Whereas traditional neighborhoods usually are established by historical legacy, cluster partitions respond to similarity of mapped data values and can be useful in establishing insurance zones, sales areas and marketing clusters.

 

The mechanics of generating cluster maps are quite simple.  Just specify the input maps and the number of clusters you want then miraculously a map appears with discrete data groupings.  So how is this miracle performed?  What happens inside clustering’s black box?

 

The schematic in figure 2 depicts the process.  The floating balls identify the data pattern for each map location (in Geographic Space) plotted against the P, K and N axes (in Data Space).  For example, the tiny green ball in the upper-right corner corresponds to a map location in the wealthiest part of town (low D, high V and low A).  The large red ball appearing closest depicts a location in a less wealthy part (high D, low V and high A).  It seems sensible that these two extreme responses would belong to different data groupings (clusters 1 and 2, respectively).

 

While the specific algorithm used in clustering is beyond the scope of this discussion (see author’s note), it suffices to recognize that data distances between the floating balls are used to identify cluster membership— groups of balls that are relatively far from other groups (different between groups) and relatively close to each other (similar within a group) form separate data clusters.  In this example, the red balls identify relatively less wealthy locations while green ones identify wealthier locations.  The geographic pattern of the classification (wealthier in the south) is shown in the 2D maps in the lower right portion of the figure.

 

Identifying groups of neighboring data points to form clusters can be tricky business.  Ideally, the clusters will form distinct “clouds” in data space.  But that rarely happens and the clustering technique has to enforce decision rules that slice a boundary between nearly identical responses.  Also, extended techniques can be used to impose weighted boundaries based on data trends or expert knowledge.  Treatment of categorical data and leveraging spatial autocorrelation are additional considerations.

 

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Figure 2.  Data patterns for map locations are depicted as floating balls in data space with groups of nearby patterns identifying data clusters.

 

So how do know if the clustering results are acceptable?  Most statisticians would respond, “…you can’t tell for sure.”  While there are some elaborate procedures focusing on the cluster assignments at the boundaries, the most frequently used benchmarks use standard statistical indices, such as T- and F-statistics used in comparing sample populations.

 

Figure 3 shows the performance table and box-and-whisker plots for the map containing two data clusters.  The average, standard deviation, minimum and maximum values within each cluster are calculated.  Ideally the averages between the two clusters would be radically different and the standard deviations small—large difference between groups and small differences within groups.

 

Box-and-whisker plots enable us to visualize these differences.  The box is aligned on the Average (center of the box) and extends above and below one Standard Deviation (height of the box) with the whiskers drawn to the minimum and maximum values to provide a visual sense of the data range.  If the plots tend to overlap a great deal, it suggests that the clusters are not very distinct and indicates significant overlapping of data patterns between clusters. 

 

 

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Figure 3.  Clustering results can be evaluated using “box and whisker” diagrams and basic statistics.

 

The separation between the boxes in all three of the data layers of the example suggests good distinction between the two clusters with the Home Value grouping the best with even the Min/Max whiskers not overlapping.  Given the results, it appears that the clustering classification in the example is acceptable… and hopefully the statisticians among us will accept in advance my apologies for such an introductory and visual treatment of a complex topic.

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Author’s Note: The clustering algorithm used in the examples was described in Beyond Mapping Compilation Series, book III, Topic 10, Further Reading section 4, “Identify Data Patterns.”  Related discussion and hands-on exercises on Clustering are in Topic 7, “Spatial Data Mining” in the workbook Analyzing Geo-Business Data (Berry, 2003) available for download from www.innovativegis.com.

 

 

The Universal Key for Unlocking GIS’s Full Potential

(GeoWorld, October 2011) 

(return to top of Topic)

 

Geotechnology is rapidly changing how we perceive, process and provide spatial information.  It is generally expected that one can click anywhere in the world and instantly access images and basic information about a location.  However, accessing your own specialized and proprietary data is much more difficult—often requiring wholesale changes to your corporate database and staffing.

 

The emergence of Geo-Web Applications involving the integration/interaction of GIS, visualization and social media has set the stage for entirely new perspectives on corporate DBMS.  For example, one can upload sales figures for individual customers into Google Earth and view as clusters of pins draped on an aerial image of a city, or receive GPS-tagged photos of potholes on county roads, or track shipments and field crews, or even locate vacant parking spaces at the mall—all this and more from your somewhat overly-smart cell phone.

 

Generally speaking there are three information processing modes…   

 

-        Visualizeto recall or form mental images or pictures involving map display (Charting capabilities),

-        Synthesizeto form a material or abstract entity by combining parts or elements involving the re-packaging of existing information (Geo-query capabilities), and   

-        Analyzeto separate a material or abstract entity into constituent parts or elements to determine their relationship involving deviation of new spatial information identifying key factors, connections and associations (Map Analysis capabilities).

 

Map Analysis is by far the least developed of the three.  Visualization and query of mapped data are direct extensions of our paper map and filing systems legacy.  Analysis of mapped data, on the other hand, involves somewhat unfamiliar territory for most organizations.  Like “the chicken and the egg” quandary, the demand for map analysis hasn’t been there because prior experience with map analysis hasn’t been there.  But even more basic is the lack of mapped data in a form amenable for analysis.  

 

Your grade school exposure to geography and mapping can change all that.  Recall that Latitude (north/south) and Longitude (east/west) lines can be drawn on the globe to identify a location anywhere in the world.  Using typical single precision floating point storage of Lat/Long coordinates in a standard database enables grid cell referencing of about half a foot or less anywhere in the continental U.S. (365214 ft/degree * 0.000001= .365214 ft *12 = 4.38257 inch grid precision along the equator).  That means appending Lat/Long fields to any database record locates that record with more than enough precision for most map analysis applications. 

 

As review, recall that the Lat/Long coordinate system uses solid angles measured from the center of the earth (figure 1).  A line passing through the Royal Observatory in Greenwich near London (termed the Prime Meridian) serves as the international zero-longitude reference.  Locations to the east are in the eastern hemisphere, and places to the west are in the western hemisphere with each half divided into 180 degrees. 

 

Geographic latitude measures the angle from the equator to the poles that trace circles on the Earth’s surface called parallels, as they are parallel to the equator and to each other.  The equator divides the globe into Northern and Southern Hemispheres— 0 to 90° North and 0 to 90° South.  Figure 1 shows a ten degree Lat/Long gridding steps representing approximately 692 mile movements along the equator. 

   

T28_10a

 

Figure 1. The Latitude/Longitude coordinate system forms a comprehensive grid covering the entire world with cells of about half a foot or less over the continental U.S.

 

So much for a conceptual review of Lat/Long, keeping in mind that there are lot of geographic and mathematical considerations in implementing the coordinate system.  Thankfully they were hammered-out years ago resulting in a set universal standards that form the foundation for contemporary GPS use. 

 

The next conceptual step involves extending the paper map paradigm to grid-based data layers.  Traditional mapping holds that there are three fundamental map features— discrete points, lines and polygons.  With the advent of the digital map, a fourth feature type emerges—continuous surfaces.  The Lat/Long grid forms a surface for geographic referencing that is analogous to a digital image with a “dot” (pixel) for every location in viewed area.  In the case of a grid map layer, a map value that identifies the characteristic/condition at a location replaces the pixel value denoting color.

 

Like the image on a computer screen, a point feature is represented by a single dot/cell containing that feature; a line feature by a series of connected cells; and a polygon feature by all of the cells defining it, both its interior and boundary.  A surface is represented by the entire gridded continuum and like an elevation surface it depicts continuous changes in a map variable.  Whereas points, lines and polygons have sharp abrupt boundaries, surfaces form gradients of change.

 

T28_10b

 

Figure 2. The Lat/Long grids are used to construct a map stack of geo-registered map layers that are pre-conditioned for map analysis and modeling.

 

The final conceptual leap is shown in figure 2.  Like Google Earth’s registration of satellite images to Lat/Long referencing, mapped data of all types can be stored, retrieved and analyzed based on the stored coordinates for the records—thereby forming a “Universal DBMS Key” that can link seemingly disparate database files.  The process is similar to a date/time stamp, except the “where information” provides a spatial context for joining data sets.  Demographic records can be linked to resource records that in turn can be linked to business records, etc.    

 

In practice, extensive data is stored in “Virtual Tiles” for efficient storage, access and processing.  A user identifies a boundary extent, then an “Analysis Frame resolution (cell size) and pertinent data layers that are automatically extracted into a “Map Stack” for grid-based map analysis and modeling.  Solution map(s) resulting from analysis can be exported in either vector, raster or DBMS form and the map stack retained for subsequent processing or deleted and reconstructed as needed. 

 

So what is holding back this seemingly utopian mapping world?  The short answer is “some practical and legacy considerations.”  On the practical front, the geographic stretching and pinching of the grid cells with increasing latitude confounds map analysis at global scales and lacks the precision necessary for detailed cadastral/surveying applications.  On the legacy front, the approach relies more on DBMS and image processing mindsets than on traditional mapping paradigms and geographic principles underlying most flagship GIS packages. 

 

It is like a Rorschach inkblot.  Since the middle ages we have thought of Lat/Long as intersecting lines, whereas the new perspective is flipped to a continuum of grid cells.  Current thinking is more like a worldwide egg crate with the grid spaces as locations for placing map values that indicate the characteristics/conditions of a multitude of map variables anywhere in the world.

 

As the Geo-Web mindset gains acceptance and data storage becomes ubiquitous, more and more maps will take on image characteristics—a raster map where Lat/Long grid cells replace pixels and map values amenable to analysis replace color codes.  While the vector map model will continue, it is the raster model (Lat/Long grids in particular) that takes us well beyond traditional mapping.

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Author’s Notes:  For more information on grid-based mapped data considerations, see Beyond Mapping Compilation Series, book IV, Topic 1 “Extending Grid-based Data Concepts” posted at www.innovativegis.com.

 

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Further Online Reading: (Chronological listing posted at www.innovativegis.com/basis/BeyondMappingSeries/)

 

Interpreting Interpolation Results (and why it is important) — describes the use of “residual analysis” for evaluating spatial interpolation performance (August 2008)

Get “Map-ematical” to Identify Data Zones — describes the use of “level-slicing” for classifying locations with a specified data pattern (October 2008)

Can We Really Map the Future? — describes the use of “linear regression” to develop prediction equations relating dependent and independent map variables (December 2008)

Follow These Steps to Map Potential Sales — describes an extensive geo-business application that combines retail competition analysis and product sales prediction (January 2009)

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