Beyond Mapping III
|
Map
Analysis book with companion CD-ROM
for hands-on exercises and further reading |
Moving Mapping to Analysis of Mapped Data — describes
Spatial Analysis and Spatial Statistics as extensions to traditional mapping
and statistics
Bending Our Understanding of Distance — uses
effective distance in establishing erosion setback to demonstrate spatial
analysis
Simultaneously Trivializing and
Complicating GIS — describes
a mathematical structure for spatial analysis operations
Use Spatial Statistics to Map Abnormal
Averages — discusses
surface modeling to characterize the spatial distribution inherent in a data
set
Making Space for Mapped Data — investigates
the link between geographic space and data space for mapping data patterns
Infusing Spatial Character into Statistics — describes
a statistical structure for spatial statistics operations
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.
<Click here>
right-click to download a printer-friendly version of this topic (.pdf).
(Back to the Table of Contents)
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Moving Mapping to Analysis
of Mapped Data
(GeoWorld, December 2004, pg. 18-19)
The evolution (or is it revolution?) of
Figure 1 identifies two key trends in the movement from
mapping to map analysis. Traditional
Figure 1. Spatial Analysis and Spatial Statistics are extensions of
traditional ways of analyzing mapped data.
Spatial Analysis extends the basic set of discrete map features of points, lines and polygons to surfaces that represent continuous geographic space as a set of contiguous grid cells. The consistency of this grid-based structuring provides a wealth of new analytical tools for characterizing “contextual spatial relationships”, such as effective distance, optimal paths, visual connectivity and micro-terrain analysis.
In addition, it provides a mathematical/statistical framework by numerically representing geographic space. Traditional Statistics is inherently non-spatial as it seeks to represent a data set by its typical response regardless of spatial patterns. The mean, standard deviation and other statistics are computed to describe the central tendency of the data in abstract numerical space without regard to the relative positioning of the data in real-world geographic space.
Spatial Statistics, on the other hand, extends traditional statistics on two fronts. First, it seeks to map the variation in a data set to show where unusual responses occur, instead of focusing on a single typical response. Secondly, it can uncover “numerical spatial relationships” within and among mapped data layers, such as generating a prediction map identifying where likely customers are within a city based on existing sales and demographic information.
The next few columns will investigate important aspects and procedures in Spatial Analysis and Spatial Statistics through a couple of example applications in natural resources and geo-business. Figure 2 shows the processing logic for generating a map of potential erosion for part of a large watershed. The model assumes that erosion potential is primarily a function of terrain steepness and water flow. Admittedly the model is simplistic but serves as a good starting point for a spatial analysis example in natural resources.
Figure 2. Erosion potential is in large
part dependent on the spatial combination of slope and flow.
The first step calculates a slope map that is interpreted into three classes of relative steepness—gentle (green), moderate (yellow) and steep (red) terrain. Similarly, an accumulation flow map is generated and then interpreted into three classes of surface confluence—light (green), moderate (yellow) and heavy (red) overland flows. The slope and flow maps are shown draped over the terrain surface to help visually verify the results. How the slope and flow maps were derived has been discussed in previous columns focusing on procedures and algorithms involved (see Author’s Notes). What is important for this discussion is the realization that realistic spatial considerations beyond our paper-map legacy can be derived and incorporated into map processing logic.
The third step combines the two maps into nine possible coincidence combinations and then interprets the combinations into relative erosion potential. For example, areas with heavy flows and steep slopes have the greatest potential for erosion while areas that have light flows and gentle slopes have the least. At this point, the calibration identifying relative erosion potential should raise some concern, but discussion of this critical step is reserved for later. What is important at this point is a basic understanding of the spatial reasoning supporting the model’s logic.
The Erosion Potential map in figure 2 clearly shows that not all locations have the same potential to get dirt balls rolling downhill. Given this information, a series of simple buffers based on planimetric distance would be ludicrous for protecting against sediment loading to streams. It is common sense that a fixed distance likely would be an insufficient setback in areas of high erosion potential with heavy flows and steep intervening conditions. Similarly the buffer would reach too far in conditions of light flow and gentle slopes.
Figure 3. Effective distance from streams
considering erosion potential generate realistic protective buffers.
Figure 3 shows an extension of the model that reaches farther from streams under adverse erosion conditions and not as far in favorable conditions. The result is a buffer that constricts and expands with erosion potential around the streams (see Author’s Notes). While this “rubber-ruler” approach to establishing protective buffers isn’t part of our traditional map paradigm, it is part of real-world experience that recognizes “all buffer-feet are not the same.”
This simple example of a Spatial Analysis application
illustrates how the evolution of
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Author’s Note: See www.innovativegis.com/basis,
select the online book Map Analysis and Topic 11, Characterizing
Micro-Terrain Features for discussion of slope and flow calculation; select
Topic 13, Creating Variable-Width Buffers for discussion on calculating
effective distance.
Bending
Our Understanding of Distance
(GeoWorld, January 2005, pg. 18-19)
The last section introduced two dominant forces that are moving mapping toward map analysis. Spatial Statistics is used to characterize the geographic distribution in a set of data and uncover the “numerical spatial relationships” among data layers. Spatial Analysis extends traditional discrete map objects of point, lines and polygons to continuous surfaces that support a wealth of new analytical operations for characterizing “contextual spatial relationships.”
In describing spatial analysis, an example of generating an effective erosion buffer was used. A comparison of simple distance (as the crow flies) and effective distance (as the crow walks) showed how a more realistic protective buffer around a stream expands and contracts depending on the erosion potential conditions surrounding the stream. What was left unexplained was how the computer calculates the effective distance short of using a rubber ruler.
The left side of figure 1 diagrams how we traditionally measure distance between two points—manually and mathematically. We can place a ruler alongside the points and note the number of tic-marks separating them, and then multiply the map distance times the scale factor to determine the geographic distance. Or we can calculate the X and Y differences between the points’ coordinates and use the Pythagorean Theorem to solve for the shortest straight line distance between two points (hypotenuse of a right triangle).
Figure 1. Distance measures the space between two points while proximity
identifies the spacing throughout a continuous geographic area.
Spatial analysis takes the concept of distance a bit farther to that of proximity—the set of shortest straight lines among all points within a geographic area. It is calculated as a series of distance waves that move out from one of the points. In effect this process is analogous to nailing a ruler at the point and spinning it with the tic-marks scribing concentric circles of distance. The distance to any point within a project area is simply its stored value indicating the number of rings away from the starting location.
If more than one feature is considered (sets of starting points, lines or polygons) the computer successively calculates proximity a starting location at a time, and keeps track of the shortest distance that is calculated for each map location. The result is a proximity surface that identifies the distance to the closest starting point. In figure 1 the three maps on the right identify proximity surfaces for a set of housing locations (points), the entire road network (lines) and critical habitat areas (polygons).
Figure 2. Simple proximity considers uniform impedance as a Proximity Wave front propagates away from a starting location—“as the crow flies.”
Figure 2 illustrates the calculations for a portion of the proximity surface shown in figure 1. The algorithm first identifies the adjoining grid cell locations that a starting location could move into (eight adjacent light grey cells). Then it determines the geographic distance of the step as orthogonal (up/down or across = 1.000 grid space) or diagonal (slanting = 1.414) and multiples this distance times its relative impedance factor. In the case of simple proximity the factor is constant (1.0) throughout the project area and the result is reduced to simply the geographic distance of the step.
The process is repeated for all cells within successive distance rings with the computer keeping track of the smallest distance value for each location as the waves advance (like tossing a rock into a pond). Note that the values stored in the table increment by one in the orthogonal directions (1.0, 2.0, etc) and are adjusted for longer steps in the diagonal directions (1.414, 2.828, etc.) and off-diagonal directions (2.414, etc.) as combinations of orthogonal and diagonal steps.
Effective proximity relaxes the assumption that the relative ease of movement (impedance) is the same throughout a project area. In the example shown in figure 3, the Proximity Wave for the first step is 1.0 (orthogonal step) times its difficulty of 8.0 (impedance factor) results in an effective distance of 8.0. The second step has an effective distance of 3.0 (1.0 * 3.0) and combines for a total movement of 11.0 (8.0 + 3.0) from the starting location. Note that the movement in the opposite direction is effectively a little further away ((1.0 * 7.0) + (1.0 * 5.0) = 12.0)). In practice, half steps are used for more precise calculations (see author’s notes).
In the practical example discussed last month, a map of erosion potential was used to identify variable impedance based on slope and flow characteristics of the surrounding terrain. Areas with high erosion potential were assigned low friction values and therefore the effective buffers reached farther than in areas of minimal erosion potential having higher friction values. The result was a map of effective protection buffers around streams that expanded and contracted in response to localized conditions.
Figure 3. Effective proximity considers varying impedance that changes with intervening conditions—“as the crow walks.”
Effective buffers are but one example of a multitude of new spatial analysis procedures that are altering our traditional view mapped data analysis. Next month’s column will investigate some of the new procedures in spatial statistics that uncover numerical relationships within and among map layers.
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Author’s Note: The more exacting half-step calculation for
the example is (0.5 * 5.0) + (0.5 * 8.0) + (0.5 * 8.0) + (0.5 * 3.0) + (.5 *
3.0) + (.5 * 3.0) = 15.0. See www.innovativegis.com/basis, and
select the online book Map Analysis, Topic 25, Understanding Effective
Distance and Connectivity, for an in-depth discussion of how effective
proximity is calculated.
Simultaneously
Trivializing and Complicating GIS
(GeoWorld, April 2012)
Several things seem to be coalescing in my mind (or maybe colliding is
a better word). GIS has moved up the
technology adoption curve from Innovators in the 1970s to Early
Adopters in the 80s, to Early Majority in the 90s, to Late
Majority in the 00s and is poised to capture the Laggards this
decade. Somewhere along this
progression, however, the field seems to have bifurcated along technical and
analytical lines.
The lion’s share of this growth has been GIS’s ever expanding
capabilities as a “technical tool” for corralling vast amounts of
spatial data and providing near instantaneous access to remote sensing images,
GPS navigation, interactive maps, asset management records, geo-queries and
awesome displays. In just forty years
GIS has morphed from boxes of cards passed through a window to a megabuck
mainframe that generated page-printer maps, to today’s sizzle of a 3D
fly-through rendering of terrain anywhere in the world with back-dropped
imagery and semi-transparent map layers draped on top—all pushed from the cloud
to a GPS enabled tablet or smart phone.
What a ride!
However, GIS as an “analytical tool” hasn’t experienced the same
meteoric rise—in fact it might be argued that the analytic side of GIS has
somewhat stalled over the last decade. I
suspect that in large part this is due to the interests, backgrounds, education
and excitement of the ever enlarging GIS tent.
Several years ago (see figure 1 and author’s note 1) I described the
changes in breadth and depth of the community as flattening from the 1970s
through the 2000s. By sheer numbers, the
balance point has been shifting to the right toward general and public users
with commercial systems responding to market demand for more technological
advancements.
Figure 1. Changes in breadth and depth of the community.
The 2010s will likely see billions of general and public users with the
average depth of science and technology knowledge supporting GIS nearly
“flatlining.” Success stories in
quantitative map analysis and modeling applications have been all but lost in
the glitz n' flash of the technological whirlwind. The vast potential of GIS to change how
society perceives maps, mapped data and their use in spatial reasoning and
problem solving seems relatively derailed.
In a recent editorial in Science entitled Trivializing Science
Education, Editor-in-Chief Bruce Alberts laments
that “Tragically, we have managed to simultaneously trivialize and complicate
science education” (author’s note 2). A
similar assessment might be made for GIS education. For most students and faculty on campus, GIS
technology is simply a set of highly useful apps on their smart phone that can
direct them to the cheapest gas for tomorrow’s ski trip and locate the nearest
pizza pub when they arrive. Or it is a
Google fly-by of the beaches around Cancun.
Or a means to screen grab a map for a paper on community-based
conservation of howler monkeys in Belize.
To a smaller contingent on campus, it is career path that requires
mastery of the mechanics, procedures and buttons of extremely complex
commercial software systems for acquiring, storage, processing, and display
spatial information. Both perspectives are
valid. However neither fully grasps the
radical nature of the digital map and how it can drastically change how we
perceive and infuse spatial information and reasoning into science, policy
formation and decision-making—in essence, how we can “think with maps.”
A large part of missing the mark on GIS’s full potential is our lack of
“reaching” out to the larger science, technology, engineering and math (STEM)
communities on campus by insisting 1) that non-GIS students interested in
understanding map analysis and modeling must be tracked into general GIS
courses that are designed for GIS specialists, and 2) that the material
presented primarily focuses on commercial GIS software mechanics that
GIS-specialists need to know to function in the workplace.
Figure 2. Alternative frameworks for quantitative map analysis.
Much of the earlier efforts in structuring a framework for quantitative
map analysis has focused on how the analytical operations work within the
context of Focal, Local and Zonal classification by
Tomlin, or even my own the Reclassify, Overlay, Distance
and Neighbors classification scheme (see top portion of figure 2 and author’s
note 3). The problem with these structuring approaches is that most STEM
folks just want to understand and use the analytical operations properly—not
appreciate the theoretical geographic-related elegance, or code the
algorithm.
The bottom portion of figure 2 outlines restructuring of the basic
spatial analysis operations to align with traditional mathematical concepts and
operations (author’s note 4). This
provides a means for the STEM community to jump right into map analysis without
learning a whole new lexicon or an alternative GIS-centric mindset. For example, the GIS concept/operation of Slope=
spatial “derivative”, Zonal functions= spatial “integral”, Eucdistance= extension of “planimetric distance” and
the Pythagorean Theorem to proximity, Costdistance=
extension of distance to effective proximity considering absolute and relative
barriers that is not possible in non-spatial mathematics, and Viewshed=
“solid geometry connectivity”.
Figure 3. Conceptual extension of derivative, trigonometric functions and
integral to mapped data and map analysis operations.
Figure 3 outlines the conceptual development of three of these
operations. The top set of graphics
identifies the Calculus Derivative as a measure of how a mathematical
function changes as its input changes by assessing the slope along a curve in
2-dimensional abstract space—calculated as the “slope of the tangent line” at
any location along the curve. In an
equivalent manner the Spatial Derivative creates a slope map depicting
the rate of change of a continuous map variable in 3-dimensional geographic
space—calculated as the slope of the “best fitted plane” at any location along
the map surface.
Advanced Grid
Math
includes most of the buttons on a scientific calculator to include
trigonometric functions. For example,
calculating the “cosine of the slope values” along a terrain surface and then
multiplying times the planimetric surface area of a grid cell will solve for
the increased real-world surface area of the “inclined plane” at each grid
location.
The Calculus Integral is identified as the “area of a region
under a curve” expressing a mathematical function. The Spatial Integral counterpart
“summarizes map surface values within specified geographic regions.” The data summaries are not limited to just a
total but can be extended to most statistical metrics. For example, the average map surface value
can be calculated for each district in a project area. Similarly, the coefficient of variation ((Stdev / Average) * 100) can be calculated to assess data
dispersion about the average for each of the regions.
By recasting GIS concepts and operations of map analysis within the
general scientific language of math/stat we can more easily educate tomorrow’s
movers and shakers in other fields in “spatial reasoning”—to think of maps as
“mapped data” and express the wealth of quantitative analysis thinking they
already understand on spatial variables.
Innovation and creativity in spatial problem solving is being held
hostage to a trivial mindset of maps as pictures and a non-spatial mathematics
that presuppose mapped data can be collapsed to a single central tendency value
that ignores the spatial variability inherent in the data.
Simultaneously, the “build it (GIS) and they will come (and take our existing
courses)” educational paradigm is not working as it requires potential users to
become a GIS’perts in complicated software
systems.
GIS must take an active leadership role in “leading” the STEM community
to the similarities/differences and advantages/disadvantages in the
quantitative analysis of mapped data—there is little hope that the STEM folks
will make the move on their own. Next
month we’ll consider recasting spatial statistics concepts and operations into
a traditional statistics framework.
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Author’s Notes: 1) see
“A Multifaceted GIS Community” in Topic 27, GIS Evolution
and Future Trends in the online book Beyond Mapping III, posted at www.innovativegis.com. 2) Bruce Alberts in
Science, 20 January 2012:Vol. 335 no. 6066 p. 263. 3) see “An Analytical Framework for GIS Modeling” posted at www.innovativegis.com/basis/Papers/Other/GISmodelingFramework/. 4) see “SpatialSTEM: Extending Traditional Mathematics and
Statistics to Grid-based Map Analysis and Modeling” posted at www.innovativegis.com/basis/Papers/Other/SpatialSTEM/.
Use
Spatial Statistics to Map Abnormal Averages
(GeoWorld, February, 2005, pg. 18-19)
The last couple of sections identified two dominant forces that are moving mapping toward map analysis—Spatial Analysis and Spatial Statistics. The discussion focused on Spatial Analysis that supports a wealth of new analytical operations for characterizing “contextual spatial relationships.” Now we can turn our attention to Spatial Statistics and how it characterizes the geographic distribution of mapped data to uncover the “numerical spatial relationships.”
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
...in the classroom or in
Figure 1. Mapped data are characterized by their
geographic distribution (maps on the left) and their numeric distribution
(descriptive statistics and histogram on the right).
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 geographic pattern of the surface 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 defining the map surface to Excel, and then applying the Histogram and Descriptive Statistics options of the Data Analysis add-in tools. The mechanics used to plot the histogram and generate the statistics were a piece-of-cake, but the intellectual challenge is to make some sense of it all.
Note that the data aren’t distributed as a normal bell-curve,
but appear shifted 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 values. If the surface contained disproportionably
higher 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 2D and 3D maps shows the
same thing, except it’s identified in the geographic distribution.
The mental exercise linking geographic space with data space is fundamental to
spatial statistics and leads to important points about the nature of mapped
data. First, there isn’t a fixed
relationship between the two views of the data’s distributions (geographic and
data). For example, 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 be reflected in 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.
Figure 2 offers yet another perspective of the link between
numeric and geographic distributions.
The upper-left inset identifies the spatial pattern formed by 16 samples
of the “percent of home equity loan limit” for a small project area— ranging
from 16.8 to 72.4. The table reports the
numerical pattern of the data— mean= 42.88 and standard deviation= 19.57. The coefficient of variation is 45.6%
((19.57/42.88) * 100= 45.6%) suggesting a fairly large unexplained variation
among the data values.
In a geographic context, the mean represents a horizontal plane hovering over the project area. However, the point map suggests a geographic trend in the data from values lower than the mean in the west toward higher values in the east. The inset in the upper-right portion of figure 2 shows a “nearest neighbor” surface generated by assigning the closest sample value to all of the other non-sampled grid locations in the project area. While the distribution is blocky it serves as a first-order estimate of the geographic distribution you see in the point map—lower in the east and higher in the west.
Figure 2.
The spatial distribution implied by a set of discrete sample points can
be estimated by iterative smoothing of the point values.
The series of plots in the lower portion of figure 2 shows the results of iteratively smoothing the blocky data. This process repeatedly passes a “roving window” over the area that calculates the average value within quarter-mile. The process is analogous to chipping away at the stair steps with the rubble filling in the bottom. The first smoothing still retains most of the sharp peak and much of the angular pattern in the blocky surface. As the smoothing progresses the surface takes on the general geographic trend of the data (Smooth10).
Eventually the surface is eroded to a flat plane— the arithmetic mean of the data. The progressive series of plots illustrate a very important concept in surface modeling— the geographic distribution maps the variance. Or, in other words, a map surface uses the geographic pattern in a data set to help explain the variation among the sample points.
Keep this in mind the next time you reduce a bunch of sample data to their arithmetic average and then assign that value to an entire polygon, such as a sales district, county or other administrative boundary. In essence, this simple mapping procedure often strips out the inherent spatial information in a set of data—sort of like throwing the baby out with the bath water.
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Author’s Note: See www.innovativegis.com/basis, and
select the online book Map Analysis, Topic 2, Assessing Interpolation Results
and Topic 8, Investigating Spatial Dependency.
(GeoWorld, March 2005, pg. 20-21)
The past three sections have investigated two dominant
forces that are driving traditional
The other force, Spatial
Statistics, is used to derive “numerical spatial relationships” that map
the variation in a set of data (surface modeling) and investigate the
similarity among map layers to classify data patterns and develop predictive
models (spatial data mining). This final
installment in the series focuses on the underlying concepts supporting data
mining capabilities of analytical
Figure 1. Conceptually linking geographic
space and data space.
The most fundamental notion is that geographic space and data space are interconnected. The three maps on the left side of figure 1 depict surfaces of housing Density, Value and Age for a portion of a city. It is important that “maps are numbers first, pictures later” and that the peaks and valleys on the surfaces are simply the graphic portrayal of actual differences in the values stored at each map location (grid cell).
The “data spear” at Point
#1 identifies the housing Density as 2.1 units/ac, Value as $407,000 and Age as
18.3 years and is analogous to your eye noting a color pattern of green, red,
and green. The other speared location
locates a very dissimilar data pattern with Density= 4.8 units/ac, Value= $190,000 and Age= 51.2
years—or as your eye sees it, a color pattern of red, green and red.
The right side of figure 1
depicts how the computer “sees” the data patterns for the two locations in
three-dimensional data space. 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—2.41, 407.0 and 18.3 for Point #1. The other point has considerably higher
values in D and A with a much lower V values (4.83, 51.2 and 190.0
respectively) so it plots at a very different location in data space.
The fact that similar data
patterns plot close to one another in data space with increasing distance
indicating less similarity provides a foothold for mapping numerical
relationships. For example, a cluster
map divides the data into groups of similar data patterns as schematically
depicted in figure 2.
Figure 2. Data patterns for map locations
are depicted as floating balls in data space that can be grouped into clusters
of similar patterns based on their data distances.
As noted before the
floating balls identify the data patterns for each map location (geographic
space) plotted against the P, K and N axes (data space). For example, the tiny green ball in the
upper-left corner depicts a map location in the fairly wealthy part of town
(low D, high V and low A). The large red
ball appearing closest to you depicts a location in the less affluent 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) with the red area locating less wealthy locations while the green area
identifies generally wealthier sections.
In a similar fashion, the project area can be sub-divided into three and
four clusters identifying more detailed data pattern groupings.
While the specific
algorithm used in clustering is beyond the scope of this discussion, it
suffices to note that data distances between the floating balls are used to
identify cluster membership—groups of balls that are relatively far from other
groups and relatively close to each other form separate data clusters. Other techniques, such as map similarity and
regression, use the link between geographic and data space to characterize
spatial relationships and develop prediction maps.
The realization that mapped
data can be expressed in both geographic space and data space is paramount to a
basic understanding of how a computer analyses numerical interrelationships
among mapped data. Geographic space
uses 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 and write algorithms that analyze it … as well as imagine
plenty of potential of applications from geo-business to natural
resources. Coupling the power of Spatial
Statistics with that of Spatial Analysis takes traditional statistics and
____________________________
Author’s Note: See www.innovativegis.com/basis, and
select the online book Map Analysis, Topic 10, Analyzing Map Similarity
and Zoning and Topic 16, Characterizing Patterns and Relationships.
Infusing
Spatial Character into Statistics
(GeoWorld, May 2012)
A previous section (Simultaneously Trivializing and Complicating GIS) discussed the assertion that we might be simultaneously trivializing and complicating GIS. At the root of the argument was the contention that “innovation and creativity in spatial problem solving is being held hostage to a trivial mindset of maps as pictures and a nonspatial mathematics that presuppose mapped data can be collapsed into a single central-tendency value that ignores the spatial variability inherent in data.”
The discussion described a mathematical framework that organizes the spatial analysis toolbox into commonly understood mathematical concepts and procedures. For example, the GIS concept/operation of Slope= spatial “derivative,” Zonal functions= spatial “integral,” Eucdistance= extension of “planimetric distance” and the Pythagorean Theorem to proximity, Costdistance= extension of distance to effective proximity considering absolute and relative barriers that is not possible in non-spatial mathematics, and Viewshed= “solid geometry connectivity.”
This section does a similar translation to describe a statistical framework for organizing the spatial statistics toolbox into commonly understood statistical concepts and procedures. But first we need to clarify the differences between spatial analysis and spatial statistics. Spatial analysis can be thought of as an extension of traditional mathematics involving the “contextual” relationships within and among mapped data layers. It focuses on geographic associations and connections, such as relative positioning, configurations and patterns among map locations.
Spatial statistics,
on the other hand, can be thought of as an extension of traditional statistics
involving the “numerical” relationships within and among mapped data
layers. It focuses on mapping the
variation inherent in a data set rather than characterizing its central
tendency (e.g., average, standard deviation) and then summarizing the
coincidence and correlation of the spatial distributions.
Figure 1. Alternative frameworks for
quantitative map analysis.
The top portion of figure 1 identifies the two dominant GIS
perspectives of spatial statistics— Surface Modeling that derives a
continuous spatial distribution of a map variable from point sampled data and Spatial
Data Mining that investigates numerical relationships of map
variables.
The bottom portion of the figure outlines restructuring of the basic
spatial statistic operations to align with traditional non-spatial statistical
concepts and operations (see author’s note).
The first three groupings are associated with general descriptive
statistics, the middle two involve unique spatial statistics operations and the
final two identify classification and predictive statistics.
Figure 3 depicts the non-spatial and spatial approaches for
characterizing the distribution of mapped data and the direct link between the
two representations. The left side of
the figure illustrates non-spatial statistics analysis of an example set of data
as fitting a standard normal curve in “data space” to assess the central
tendency of the data as its average and standard deviation. In processing, the geographic coordinates are
ignored and the typical value and its dispersion are assumed to be uniformly
(or randomly) distributed in “geographic space.”
Figure 2. Comparison and linkage between spatial and
non-spatial statistics
The top portion of figure 3 illustrates the derivation of a continuous
map surface from geo-registered point data involving spatial
autocorrelation. The discrete point map
locates each sample point on the XY coordinate plane and extends these points
to their relative values (higher values in the NE; lowest in the NW). A roving window is moved throughout the area
that weight-averages the point data as an inverse function of distance—closer
samples are more influential than distant samples. The effect is to fit a surface that
represents the geographic distribution of the data in a manner that is
analogous to fitting a SNV curve to characterize the data’s numeric
distribution. Underlying this process is
the nature of the sampled data which must be numerically quantitative
(measurable as continuous numbers) and geographically isopleth (numbers form
continuous gradients in space).
The lower-right portion of figure 3 shows the direct linkage between
the numerical distribution and the geographic distribution views of the
data. In geographic space, the “typical
value” (average) forms a horizontal plane implying that the average is
everywhere. In reality, the average is
hardly anywhere and the geographic distribution denotes where values tend to be
higher or lower than the average.
In data space, a histogram represents the relative occurrence of each
map value. By clicking anywhere on the
map, the corresponding histogram interval is highlighted; conversely, clicking
anywhere on the histogram highlights all of the corresponding map values within
the interval. By selecting all locations
with values greater than + 1SD, areas of unusually high values are located—a
technique requiring the direct linkage of both numerical and geographic
distributions.
Figure 3. Conceptual extension of
clustering and correlation to mapped data and analysis.
Figure 3 outlines two of the advance spatial statistics operations involving spatial correlation among two or more map layers. The top portion of the figure uses map clustering to identify the location of inherent groupings of elevation and slope data by assigning pairs of values into groups (called clusters) so that the value pairs in the same cluster are more similar to each other than to those in other clusters.
The bottom portion of the figure assesses map correlation by
calculating the degree of dependency among the same maps of elevation and
slope. Spatially “aggregated”
correlation involves solving the standard correlation equation for the entire
set of paired values to represent the overall relationship as a single
metric. Like the statistical average,
this value is assumed
to be uniformly (or randomly) distributed in “geographic space” forming a
horizontal plane.
“Localized” correlation, on the other hand, maps the degree of dependency between the two map variables by successively solving the standard correlation equation within a roving window to generate a continuous map surface. The result is a map representing the geographic distribution of the spatial dependency throughout a project area indicating where the two map variables are highly correlated (both positively, red tones; and negatively, green tones) and where they have minimal correlation (yellow tones).
With the exception of unique Map Descriptive Statistics and Surface Modeling classes of operations, the grid-based map analysis/modeling system simply acts as a mechanism to spatially organize the data. The alignment of the geo-registered grid cells is used to partition and arrange the map values into a format amenable for executing commonly used statistical equations. The critical difference is that the answer often is in map form indicating where the statistical relationship is more or less than typical.
While the technological applications of GIS have soared over the last decade, the analytical applications seem to have flat-lined. The seduction of near instantaneous geo-queries and awesome graphics seem to be masking the underlying character of mapped data— that maps are numbers first, pictures later. However, grid-based map analysis and modeling involving Spatial Analysis and Spatial Statistics is, for the larger part, simply extensions of traditional mathematics and statistics. The recognition by the GIS community that quantitative analysis of maps is a reality and the recognition by the STEM community that spatial relationships exist and are quantifiable should be the glue that binds the two perspectives. That reminds me of a very wise observation about technology evolution—
“Once a new technology rolls over you, if you're not part of the
steamroller, you're part of the road.” ~Stewart Brand, editor
of the Whole Earth Catalog
_____________________________
Author’s Notes: for a more detailed discussion,
see “SpatialSTEM: Extending Traditional Mathematics and Statistics to
Grid-based Map Analysis and Modeling” posted at www.innovativegis.com/basis/Papers/Other/SpatialSTEM/.
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