Topic 9 – A
Math/Stat Framework for Map Analysis |
GIS
Modeling book |
SpatialSTEM
Has Deep Mathematical Roots — provides a conceptual
framework for a map-ematical treatment of mapped data
Simultaneously
Trivializing and Complicating GIS — describes a mathematical structure
for spatial analysis operations
Infusing
Spatial Character into Statistics — describes a statistical structure
for spatial statistics operations
To
Boldly Go Where No Map Has Gone Before — identifies Lat/Lon as a
Universal Spatial Key for joining database tables
Depending
on Where is What — develops an organizational
structure for spatial statistics
Laying
the Foundation for SpatialSTEM: Spatial Mathematics, Map Algebra and Map
Analysis — discusses the conceptual foundation and intellectual shifts
needed for SpatialSTEM
Further Reading
— four additional sections
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______________________________
SpatialSTEM Has Deep Mathematical Roots
(GeoWorld, January 2012)
Recently my interest has
been captured by a new arena and expression for the contention that “maps are
data”—spatialSTEM (or sSTEM for short)—as a means for redirecting
education in general, and GIS education in particular. I suspect you have heard of STEM (Science,
Technology, Engineering and Mathematics) and the educational crisis that puts
U.S. students well behind many other nations in these quantitatively-based
disciplines.
While Googling
around the globe makes for great homework in cultural geography, it doesn’t
advance quantitative proficiency, nor does it stimulate the spatial reasoning
skills needed for problem solving. Lots
of folks from Freed Zakaria to Bill Gates to
President Obama are looking for ways that we can recapture our leadership in
the quantitative fields. That’s the
premise of spatialSTEM– that “maps are numbers first, pictures later”
and we do mathematical things to mapped data for insight and better
understanding of spatial patterns and relationships within decision-making
contexts.
This contention suggests
that there is a map-ematics that can be
employed to solve problems that go beyond mapping, geo-query, visualization and
GPS navigation. This column’s discussion
about the quantitative nature of maps is the first part of a three-part series
that sets the stage to fully develop this thesis— that grid-based Spatial
Analysis Operations are extensions of traditional mathematics
(Part 2 investigating map math, algebra, calculus, plane and solid geometry,
etc.) and that grid-based Spatial Statistics Operations
are extensions of traditional statistics (Part 3 looking at map descriptive
statistics, normalization, comparison, classification, surface modeling,
predictive statistics, etc.).
Figure 1. Conceptual overview of the
SpatialSTEM framework.
Figure 1 outlines the
important components of map analysis and modeling within a mathematical
structure that has been in play since the 1980s (see author’s note). Of the three disciplines forming
Geotechnology (Remote Sensing, Geographic Information Systems and Global
Positioning System), GIS is at the heart of converting mapped data into spatial
information. There are two primary
approaches used in generating this information—Mapping/Geo-query and Map
Analysis/Modeling.
The major difference
between the two approaches lies in the structuring of mapped data and their
intended use. Mapping and geo-query
utilizes a data structure akin to manual mapping in which discrete spatial
objects (points, lines and polygons) form a collection
of independent, irregular features to characterize geographic space. For example, a Water map might contain
categories of Spring (points), Stream (lines) and Lake
(polygons) with the features scattered throughout a landscape.
Map analysis and modeling
procedures, on the other hand, operate on continuous map variables
(termed map surfaces) composed of thousands upon thousands of map values
stored in geo-registered matrices.
Within this context, a Water map no longer contains separate and
distinct features but is a collection of adjoining grid cells with a map value
indicating the characteristic at each location (e.g., Spring=1, Stream= 2 and Lake= 3).
Figure 2. Basic data structure for Vector and Raster map
types.
Figure 2 illustrates two
broad types of digital maps, formally termed Vector for storing discrete
spatial objects and Raster for storing continuous map surfaces. In vector format, spatial data is stored as
two linked data tables. A “spatial
table” contains all of the X,Y coordinates defining a
set of spatial objects that are grouped by object identification numbers. For example, the location of the Forest
polygon identified on the left side of the figure is stored as ID#32 followed
by an ordered series of X,Y coordinate pairs
delineating its border (connect-the-dots).
In a similar manner, the
ID#s and X,Y coordinates defining the other cover type
polygons are sequentially listed in the table.
The ID#s link the spatial table (Where) to a corresponding “attribute
table” (What) containing information about each spatial object as a separate
record. For example, polygon ID#31 is
characterized as a mature 60 year old Ponderosa Pine (PP) Forest stand.
The right side of figure
2 depicts raster storage of the same cover type information. Each grid space is assigned a number
corresponding to the dominant cover type present— the “cell position” in the
matrix determines the location (Where) and the “cell value” determines the
characteristic/condition (What). It is
important to note that the raster representation stores information about the
interior of polygons and “pre-conditions geographic space” for analysis by
applying a consistent grid configuration to each grid map. Since each map’s underlying data structure is
the same, the computer simply “hits disk” to get information and does not have
to calculate whether irregular sets of points, lines or polygons on different
maps intersect.
Figure 3 depicts the
fundamental concepts supporting raster data.
As a comparison between vector and raster data structures consider how
the two approaches represent an Elevation surface. In vector, contour lines are used to identify
lines of constant elevation and contour interval polygons are used to identify
specified ranges of elevation. While
contour lines are exacting, they fail to describe the intervening surface
configuration.
Contour intervals
describe the interiors but overly generalize the actual “ups and downs” of the
terrain into broad ranges that form an unrealistic stair-step configuration
(center-left portion of figure 3). As
depicted in the figure, rock climbers would need to summit each of the contour
interval “200-foot cliffs” rising from presumed flat mesas. Similarly, surface water flow presumably
would cascade like waterfalls from each contour interval “lake” like a Spanish
multi-tiered fountain.
The upshot is that within
a mathematical context, vector maps are ineffective representations of
real-world gradients and actual movements and flows over these surfaces— while
contour line/interval maps have formed colorful and comfortable visualizations
for generations, the data structure is too limited for modern map analysis and
modeling.
Figure 3. Organizational considerations and
terminology for grid-based mapped data.
The remainder of figure 3
depicts the basic Raster/Grid organizational structure. Each grid map is termed a Map Layer
and a set of geo-registered layers constitutes a Map Stack. All of the map layers in a project conform to
a common Analysis Frame with a fixed number of rows and columns at a
specified cell size that can be positioned anywhere in geographic space. As in the case of the Elevation surface in
the lower-left portion of figure 3, a continuous gradient is formed with subtle
elevation differences that allow hikers to step from cell to cell while
considering relative steepness. Or
surface water to sequentially stream from a location to its steepest downhill
neighbor thereby identifying a flow-path.
The underlying concept of
this data structure is that grid cells for all of the map layers precisely
coincide, and by simply accessing map values at a row, column location a
computer can “drill” down through the map layers noting their
characteristics. Similarly, noting the
map values of surrounding cells identifies the characteristics within a
location’s vicinity on a given map layer, or set of map layers.
Keep in mind that while
terrain elevation is the most common example of a map surface, it is by no
means the only one. In natural systems,
temperature, barometric pressure, air pollution concentration, soil chemistry
and water turbidity are but a few examples of continuous mapped data
gradients. In human systems, population
density, income level, life style concentration, crime occurrence, disease
incidence rate all form continuous map surfaces. In economic systems, home values, sales
activity and travel-time to/from stores form map variables that that track
spatial patterns.
In fact the preponderance
of spatial data is easily and best represented as grid-based continuous map
surfaces that are preconditioned for use in map analysis and modeling. The computer does the heavy-lifting of the
computation …what is needed is a new generation of creative minds that goes
beyond mapping to “thinking with maps” within this less familiar, quantitative
framework— a SpatialSTEM environment.
_____________________________
Author’s Notes: My
involvement in map analysis/modeling began in the 1970s with doctoral work in
computer-assisted analysis of remotely sensed data a couple of years before we
had civilian satellites. The extension
from digital imagery classification using multivariate statistics and pattern
recognition algorithms in the 70s to a comprehensive grid-based mathematical
structure for all forms of mapped data in the 80s was a natural evolution. See www.innovativegis.com, select “Online Papers” for a link to a 1986
paper on “A Mathematical Structure for Analyzing Maps” that serves as an early
introduction to a comprehensive framework for map analysis/modeling.
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.
_____________________________
Author’s Notes: 1) See “A Multifaceted
GIS Community” in Beyond mapping Compilation Series book III, Epilog section 2 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/.
Infusing
Spatial Character into Statistics
(GeoWorld, May 2012)
The previous section
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
non-spatial 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.
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 1. Alternative frameworks for quantitative map analysis.
Figure 2 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.”
The top portion of figure
2 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 2 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.
Figure 2. Comparison and linkage between spatial and non-spatial statistics.
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 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.
Figure 3. Conceptual extension of clustering and correlation to mapped data
and analysis.
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/.
To Boldly Go Where No Map Has Gone Before
(GeoWorld, October 2012)
Previous sections have
described a mathematical framework (dare I say a “map-ematical”
framework?) for quantitative analysis of mapped data. Recall that Spatial Analysis
operations investigate the “contextual” relationships within and among
maps, such as variable-width buffers that account for intervening
conditions. Spatial Statistics
operations, on the other hand, examine the “numerical” relationships,
such as map clustering to uncover inherent geographic patterns in the
data.
The cornerstone of these
capabilities lies in the grid-based nature of the data that treats geographic
space as continuous map surfaces composed of thousands upon thousands of cells
with each containing data values that identify the characteristics/conditions
occurring at each location. This simple
matrix structure provides a detailed account of the unique spatial distribution
of each map variable and a geo-registered stack of map layers provides the
foothold to quantitatively explore their spatial patterns and
relationships.
The most fundamental and
ubiquitous grid form is the Latitude/Longitude coordinate system that enables
every location on the Earth to be specified by a pair of numbers. The upper portion of figure
1, depicts a 2.5^{0} Lat/Lon grid forming a matrix of 73 rows by 144
columns= 10,512 cells in total with each cell having an area of about 18,735mi^{2}.
The lower portion of the
figure shows that the data could be stored in Excel with each spreadsheet cell
directly corresponding to a geographic grid cell. In turn, additional map layers could be
stored as separate spreadsheet pages to form a map stack for analysis.
Of course this resolution
is far too coarse for most map analysis applications, but it doesn’t have to
be. Using the standard single precision
floating point storage of Lat/Long coordinates expressed in decimal degrees,
the precision tightens to less than half a foot anywhere in the world (365214
ft/degree * 0.000001= .365214 ft *12 = 4.38257 inches or 0.11132 meters). However, current grid-based technology limits
the practical resolution to about 1m (e.g., Ikonos
satellite images) to 10m (e.g., Google Earth) due to the massive amounts of
data storage required.
For example, to store a
10m grid for the state of Colorado it would take over two and half billion grid
cells (26,960km²= 269,601,000,000m² / 100m² per cell= 2,696,010,000 cells). To store the entire earth surface it would
take nearly a trillion and a half cells (148,300,000km^{2} = 148,000,000,000,000m^{2} / 100m² per
cell= 1,483,000,000,000
cells).
Figure 1. Latitude and Longitude coordinates provide a
universal framework for parsing the earth’s surface into a standardized set of
grid cells.
At first these storage
loads seem outrageous but with distributed cloud computing the massive grid can
be “easily” broken into manageable mouthfuls.
A user selects an area of interest and data for that area is downloaded
and stitched together. For example,
Google Earth responds to your screen interactions to nearly instantaneously
download millions of pixels, allowing you to pan/zoom and turn on/off map
layers that are just a drop in the bucket of the trillions upon trillions of
pixels and grid data available in the cloud.
Figure 2 identifies
another, more practical mechanism for storage using a relational database. In essence, each of the conceptual grid map
spreadsheets can be converted to an interlaced format with a long string of
numbers forming the columns (data fields); the rows (records) identify the
information available each of the individual grid cells that form the reference
grid.
Figure 2. Within a relational database, Lat/Lon forms a
Universal DBMS Key for joining tables.
For fairly small areas of
up to a million or so cells this is an excellent way to store grid maps as
their spatial coincidence is inherent in the organization and the robust
standard set of database queries and processing operations is available. Larger grids use more advanced, specialized
mechanisms of storage to facilitate data compression and virtual paging of
fully configured grid layers.
But the move to a
relational database structure is far more important than simply corralling
mega-gulps of map values. It provides a
“Universal DBMS Key” that can link seemingly otherwise disparate database
tables (see Authors Note). 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, health records, etc— all sharing a common
Lat/Lon address.
All that is necessary is
to tag your data with its Lat/Lon coordinates (“where” it was collected) just
as you do with the date/time (“when” it was collected) …not a problem with the
ubiquitous availability and increasing precision of GPS that puts a real-time
tool for handling detailed spatial data right in your pocket. In today’s technology, most GPS-enabled smart
phones are accurate to a few meters and specialized data collection devices
precise to a few centimeters.
Once your data is stamped
with its “spatial key,” it can be linked to any other database table with
spatially tagged records without the explicit storage of a fully expanded grid
layer. All of the spatial relationships
are implicit in the relative positioning of the Lat/Lon coordinates.
For example, a selection
operation might be to identify of all health records jointly occurring within
half a kilometer of locations that have high lead concentrations in the top soil. Or, locate all of the customer records within
five miles of my store; better yet, within a ten-minute drive from a
store.
Geotechnology is truly a
mega-technology that will forever change how we perceive and process spatial
information. Gone are the days of manual
measurements and specialized data formats that have driven our mapping
legacy. Lat/Lon coordinates move from
cross-hairs for precise navigation (intersecting lines) to a continuous matrix
of spaces covering the globe for consistent data storage (grid cells). The recognition of a universal spatial key
coupled with spatial analysis/statistics procedures and GPS/RS technologies
provides a firm foothold “to boldly go where no map
has gone before.”
_____________________________
Author’s Note: See
“The Universal Key for Unlocking GIS’s Full Potential,” book IV, Topic 7,
section 6 in the Beyond Mapping Compilation Series posted at www.innovativegis.com.
Depending
on Where is What
(GeoWorld, March 2013)
Early procedures in
spatial statistics were largely focused on the characterization of spatial
patterns formed by the relative positioning of discrete spatial objects—points,
lines, and polygons. The “area, density,
edge, shape, core-area, neighbors, diversity and arrangement” of map features
are summarized by numerous landscape analysis indices, such as Simpson's
Diversity and Shannon's Evenness diversity metrics; Contagion and
Interspersion/Juxtaposition arrangement metrics; and Convexity
and Edge Contrast shape metrics (see Author’s Note 1). Most of these techniques are direct
extensions of manual procedures using paper maps and subsequently coded for
digital maps.
Grid-based map analysis,
however, expands this classical view by the direct application of advanced
statistical techniques in analyzing spatial relationships that consider
continuous geographic space. Some of the
earliest applications (circa 1960) were in climatology and used map surfaces to
generate isotherms of temperature and isobars of barometric pressure.
In the 1970s, the
analysis of remotely sensed data (raster images) began employing traditional
statistical techniques, such as Maximum Likelihood Classification, Principle
Component Analysis and Clustering that had been used in analyzing
non-spatial data for decades. By the
1990s, these classification-oriented procedures operating on spectral bands
were extended to include the full wealth of statistical operations, such as Correlation
and Regression, utilizing diverse sets of geo-registered map variables
(grid-based map layers).
It is the historical
distinction between “Spatial Pattern characterization of discrete
objects” and “Spatial Relationship analysis of continuous map surfaces”
that identifies the primary conceptual branches in spatial statistics. The spatial relationship analysis branch can
be further grouped by two types of spatial dependency driving the relationships—
Spatial Autocorrelation involving spatial relationships within a
single map layer, and Spatial Correlation involving spatial
relationships among multiple map layers (see figure 1).
Spatial Autocorrelation follows Tobler’s first law of geography— that “…near things are more
alike than distant things.” This
condition provides the foundation for Surface Modeling used to identify
the continuous spatial distribution implied in a set of discrete point data
based on one of four fundamental approaches (see figure 2 and Author’s Note
2). The first two approaches—Map
Generalization and Geometric Facets—consider the entire set of point
values in determining the “best fit” of a polynomial equation, or a set of
3-dimentional geographic shapes.
For example, a 1^{st}
order polynomial (tilted plane) fitted to a set of data points indicates its
spatial trend with decreasing values aligning with the direction cosines of the
plane. Or, a complex set of abutting
tilted triangular planes can be fitted to the data values to capture significant
changes in surface form (triangular tessellation).
Figure 1. Spatial Dependency involves relationships within a single
map layer (Spatial Autocorrelation) or among multiple map layers (Spatial
Correlation).
The lower two approaches—Density
Analysis and Spatial Interpolation—are based on localized summaries
of the point data utilizing “roving windows.” Density Analysis counts the
number of data points in the window (e.g., number of crimes incidents within
half a kilometer) or computes the sum of the values (e.g., total loan value
within half a kilometer).
However, the most
frequently used surface modeling approach is Spatial Interpolation that
“weight-averages” data values within a roving window based on some function of
distance. For example, Inverse Distance
Weighting (IDW) interpolation uses the geometric equation 1/D^{ Power}
to greatly diminish the influence of distant data values in computing the
weighted-average.
Figure 2. Surface Modeling involves generating map surfaces
that portray the continuous spatial distribution implied in a set of discrete
point data.
The bottom portion of
figure 2 encapsulates the basis for Kriging which derives the weighting
equation from the point data values themselves, instead of assuming a fixed
geometric equation. A variogram plot of
the joint variation among the data values (blue curve) shows the differences in
the values as a function of distance.
The inverse of this derived equation (red curve) is used to calculate
the distance affected weights used in weight-averaging the data values.
The other type of spatial
dependency—Spatial Correlation—provides the foundation for analyzing
spatial relationships among map layers.
It involves spatially evaluating traditional statistical procedures
using one of four ways to access the geo-registered data— Local, Focal,
Zonal and Global (see figure3 and Author’s Notes 3 and 4). Once the spatially coincident data is
collected and compatibly formatted, it can be directly passed to standard
multivariate statistics packages or to more advanced statistical engines (CART,
Induction or Neural Net). Also, a
growing number of GIS systems have incorporated many of the most frequently
used statistical operations.
Figure 3. Statistical Analysis of mapped data involves
repackaging mapped data for processing by standard multivariate statistics or
more advanced statistical operations.
The majority of the Statistical
Analysis operations simply “repackage” the map values for processing by
traditional statistics procedures. For
example, “Local” processing of map layers is analogous to what you see when two
maps are overlaid on a light-table. As
your eye moves around, you note the spatial coincidence at each spot. In grid-based map analysis, the cell-by-cell
collection of data for two or more grid layers accomplishes the same thing by
“spearing” the map values at a location, creating a summary (e.g., simple or
weighted-average), storing the new value and repeating the process for the next
location.
“Focal” processing, on
the other hand, “funnels” the map layer data surrounding a location (roving
window), creates a summary (e.g., correlation coefficient), stores the new
value and then repeats the process.
Note that both local and focal procedures store the results on a
cell-by-cell basis.
The other two techniques
(right side of figure 3) generate entirely different summary results. “Zonal” processing uses a predefined template
(termed a map region) to “lace” together the map values for a region-wide
summary. For example, a wildlife habitat
unit might serve as a template map to retrieve slope values from a data map of
terrain steepness, compute the average of the values, and then store the result
for all of the locations defining the region.
Or maps of animal activity for two time periods could be accessed and a
paired t-test performed to determine if a significant difference exists within
the habitat unit. The interpretation of
the resultant map value assigned to all of the template locations is that each
cell is an “element of a spatial entity having that overall summary statistic.”
“Global” processing isn’t
much different from the other techniques in terms of mechanics, but is
radically different in terms of the numerical rigor implied. In map-wide statistical analysis, the entire
map is considered a variable, each cell a case and each value a measurement
(or instance) in mathematical/statistical modeling terminology. Within this context, the processing has “all
of the rights, privileges and responsibilities” afforded non-spatial
quantitative analysis. For example, a
regression could be spatially evaluated by “plunging” the equation through a
set of independent map variables to generate a dependent variable map on
cell-by-cell basis, or reported as an overall map-wide value.
So what’s the take-home
from all this discussion? It is that
maps are “numbers first, pictures later” and we can spatially discover and
subsequently evaluate the spatial relationships inherent in sets of grid-based
mapped data as true map-ematical
expressions. All that is needed is a new
perspective of what a map is (and isn’t).
_____________________________
Author’s Notes: 1) in the Beyond Mapping Compilation Series posted
at www.innovativegis.com see book III , Topic 6, sections 9 through 12 on Analyzing Landscape
Patterns; 2) see book III, Topic 9 on Basic Techniques in Spatial Statistics;
3) refers to C. Dana Tomlin’s four data acquisition classes; 4) for more
discussion on data acquisition techniques, see book IV, Topic 5, Section 3
“Getting the Numbers Right.”
Laying the
Foundation for SpatialSTEM: Spatial Mathematics, Map Algebra
and Map Analysis
(GeoWorld, October 2013)
Mathematics in general
and geometry and trigonometry in particular have long been the keystone to mapping—from
Spatial Mathematics that enables the development of mapped data; to a
generalized Map Algebra for expressing math/stat relationships among map
variables; to a comprehensive Map Analysis toolbox that extends traditional
quantitative data analysis procedures by considering the spatial distribution
and interaction of mapped data layers.
Several years ago, Nigel
Waters wrote a short synopsis on “The Most Beautiful Formulae in GIS” where he
described the ten most useful Spatial Formulae and the ten most useful
Attribute-related Formulae chosen for their elegance, simplicity, and
generality, as well as their wide applicability and power (see author’s note
1). More recently, the book “Spatial
Mathematics: Theory and Practice through Mapping” by Arlinghaus
and Kerski further develops the wealth of enabling Spatial
Mathematics equations and techniques (see author’s note 2).
These and a host of
similar treatises provide a comfortable conceptual springboard for STEM
disciplines to extend traditional scalar mathematics into the spatial
realm. The digital map expressed as an
organized set of numbers fuels this transition— today “maps are numbers first,
pictures later.” The result is a
generalized Map Algebra (see author’s note 3) enabling a user to add,
subtract, divide, raise to a power, root, log and even differentiate and
integrate digital maps— all of the functionality of a pocket calculator (and
then some) operating on geo-registered stacks of digital maps.
This algebraic framework provides
a comprehensive toolbox of primitive mathematical operations transitioning
traditional quantitative data analysis into Map Analysis that infuses
the consideration of spatial patterns and relationships into the analysis. From this perspective, the spatial
distribution of data is as important as its numerical distribution in analyzing
map variables.
Figure 1. GIS can be viewed as both a “Technological Tool”
and an “Analytical Tool.”
Figure 1 provides a
40,000-foot overview of the evolving field of Geotechnology, one of the three
mega-technologies for the 21^{st} century as identified by the U.S.
Department of Labor (the other two are Biotechnology and Nanotechnology). The left side of the figure depicts the
“spatial triad” of technologies (GPS, GIS and RS) comprising Geotechnology that
collects, stores, retrieves, processes, and displays digital mapped data. The mapping and analysis capabilities of GIS
can be characterized as both a “Technological Tool” involving mapping, display
and geo-query and an “Analytical Tool” involving spatial mathematics, analysis
and statistics.
As a technological tool,
GIS greatly extends traditional mapping and inventory techniques involving
laborious, inefficient and generally ineffective manual procedures employed
just a few decades ago. Today it is
commonplace to get real-time routing directions, superimposed on an interactive
map with a satellite image backdrop and a street view of your destination; all
from a smartphone that rivals the computing power of a mainframe computer a few
decades ago. For the most part, static
paper maps have given way to dynamic digital mapped data that can be
interactively viewed and processed in radically new ways—a revolution that is
simply amazing for anyone over thirty, yet commonplace for those who are
younger.
The meteoric rise in the
technical expressions of Geotechnology is in large part due to its easily
envisioned extension of its manual mapping and inventory legacies. Database systems replaced the walls of file
cabinets (attribute data) and digital maps replaced paper maps (spatial data). Linking the two data set perspectives spawned
a radically new paradigm of what a map is and isn’t and catapulted mapping to
“mega-technology” status.
Is a similar canonic step
and radically changed paradigm in the future for traditional quantitative data
analysis concepts, procedures and applications?
What are the impediments holding back GIS as an analytical tool? What are the inducements needed for advancing
spatially-aware quantitative data analysis?
Figure 2. Types of GIS data, users and
applications.
Figure 2 outlines the
data, users and application approaches that is fueling this
transformation. A major hurdle is the
historical perspective of maps as being comprised of discrete spatial objects
(point, line and areal patterns) as depicted in the 2D vector-based map in the
upper-left portion of the figure. While
this vector data format is comfortable and ideal for human visual
interpretation, it lacks the spatial specificity and consistency required by
advanced analysis procedures needed by most the STEM research and
applications.
Alternatively, raster
data depicted in the lower-left portion of the figure provides a continuous
and consistent data form that is preconditioned for quantitative data
analysis. A grid-based map surface
tracks subtle spatial variations of a map variable as an uninterrupted gradient
instead of aggregating the detailed data into discrete ranges (i.e., contour
intervals).
In addition, the matrix
structuring provides a consistent “analysis frame” for a geo-registered stack
of map layers for a project area. Within
this grid structure the row, column locators implicitly carry all of the
necessary spatial topology relating each grid location to the positioning of
all other locations within a single map layer and among multiple layers in a
geo-registered map stack.
The right side of figure
2 identifies several types of GIS users.
Currently, most of the GIS community is comprised of Data Providers, GIS
Specialists, and General Users who are primarily involved with the technical
aspects of GIS and their vector processing expressions— creating, maintaining
and accessing mapped data and then executing standardized processing routines. These users can be thought of as “of the
technology.”
The Power Users,
Developers and Modelers, on the other hand, are more “of the application.” Within this context, domain expertise
identifies the scope of a problem and the map variables involved and then map
analysis capabilities are used to uncover spatial relationships that then forms
a spatially-aware solution. It is in
this arena that a “newly developing niche for SpatialSTEM” is poised to
take-hold (see author’s note 4).
Einstein noted that “we
cannot solve our problems with the same level of thinking that created them”
and that “the formulation of the problem is often more essential than its
solution, which may be merely a matter of mathematical or experimental skill.” This thinking suggests that the STEM
disciplines need to be actively engaged and leading the search for
spatially-aware solutions to today’s complex spatial problems. Also, it recognizes that geospatial
technologists need to fully recognize the quantitative nature mapped data and
embrace its analytical potential, as well as its technical application.
However when it comes to
Map Analysis (grid-based Spatial Analysis and Spatial Statistics operations),
the old adage that “they who know not, know not they
know not” takes center stage and the status quo paradigms of science and
technology continue to dominate education, research and application
development. As long as a conceptual
chasm exists between the mapping and quantitative analysis communities, spatially-aware
solutions to complex problems will continue to be mostly side-lined.
_____________________________
Author’s Notes: 1) See “The Most Beautiful Formulae in GIS” by
Nigel Waters (1995) posted at www.innovativegis.com/basis/MapAnalysis/Topic30/Beautiful_Formulae.pdf. 2) See “Spatial Mathematics: Theory and Practice through Mapping” by
Sandra Arlinghaus and Joseph Kerski
(2013, www.crcpress.com/product/isbn/9781466505322). 3) The concepts and procedures behind Spatial
Mathematics was introduced by David Unwin with the
University of London (Introductory Spatial Analysis, 1981, Methuen New York)
and subsequently developed as a set-based Map Algebra for manipulating raster map
layers by Dana Tomlin as a doctoral
student at Yale University (Geographic Information Systems and Cartographic
Modeling, 1990, Prentice-Hall, Englewood, New Jersey). 4) See “SpatialSTEM
– Seminar, Workshop and Teaching Materials for Understanding Grid-based Map
Analysis” posted at www.innovativegis.com/Basis/Courses/SpatialSTEM/.
_____________________
Further Online Reading: (Chronological listing posted at www.innovativegis.com/basis/BeyondMappingSeries/)
Map-ematically
Messing with Mapped Data — discusses the nature of
grid-based mapped data and Spatial Analysis operations (February 2012)
Paint by Numbers Outside
the Traditional Statistics Box — discusses the nature of
Spatial Statistics operations (March 2012)
The Spatial Key to Seeing the Big Picture
— describes a five step process for generating grid map layers from
spatially tagged data (September 2013)
Recasting Map Analysis Operations for
General Consumption — reorganizes ArcGIS’s Spatial Analyst tools into
the SpatialSTEM framework that extends traditional math/stat procedures
(February 2013)
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