Topic 3 – Basic
Techniques in Spatial Analysis 
Map
Analysis book/CD 
Use a
Mapematical Framework for GIS Modeling — describes a conceptual
structure for map analysis operations and GIS modeling
Options
Seem Endless When Reclassifying Maps — discusses the basic
reclassifying map operations
Overlay
Operations Feature a Variety of Options — discusses the basic overlaying
map operations
Computers
Quickly Characterize Spatial Coincidence — discusses several human
considerations in implementing GIS
Further Reading
— two additional sections
<Click here>
for a printerfriendly version of this topic
(.pdf).
(Back to the Table of Contents)
______________________________
Use a
Mapematical Framework for GIS Modeling
(GeoWorld, March 2004)
As
While
map analysis tools might at first seem uncomfortable, they simply are
extensions of traditional analysis procedures brought on by the digital nature
of modern maps. Since maps are “number
first, pictures later,” a mapematical
framework can be can be used to organize the analytical operations. Like basic math, this approach uses
sequential processing of mathematical operations to perform a wide variety of
complex map analyses. By controlling the
order that the operations are executed, and using a common database to store
the intermediate results, a mathematicallike processing structure is
developed.
This
“map algebra” is similar to traditional algebra where basic operations, such as
addition, subtraction and exponentiation, are logically sequenced for specific
variables to form equations—however, in map algebra the variables represent
entire maps consisting of thousands of individual grid values. Most of traditional mathematical capabilities,
plus extensive set of advanced map processing operations, comprise the map
analysis toolbox.
In
gridbased map analysis, the spatial coincidence and juxtapositioning of
values among and within maps create new analytical operations, such as
coincidence, proximity, visual exposure and optimal routes. These operators are accessed through general
purpose map analysis software available in many
There
are two fundamental conditions required by any map analysis package—a consistent data structure and an iterative processing environment. Previous discussion (Beyond Mapping columns
July and September, 2002) described the characteristics of a gridbased data
structure by introducing the concepts of an analysis frame, map stack and data
types. This discussion extended the
traditional discrete set of map features (points, lines and polygons) to map
surfaces that characterize geographic space as a continuum of uniformlyspaced
grid cells.
The
second condition is the focus of this and the next couple of sections. It provides an iterative processing
environment by logically sequencing map analysis operations and involves: 1) retrieval of one or more map layers from
the database, 2) processing that data
as specified by the user, 3) creation
of a new map containing the processing results, and ) storage of the new map for subsequent processing.
Each
new map derived as processing continues aligns with the analysis frame so it is
automatically georegistered to the other maps in the database. The values comprising the derived maps are a
function of the processing specified for the “input map(s).”
This
cyclical processing provides an extremely flexible structure similar to
“evaluating nested parentheticals” in traditional math. Within this structure, one first defines the
values for each variable and then solves the equation by performing the
mathematical operations on those numbers in the order prescribed by the
equation. For example, the equation for
calculating percent change in your investment portfolio—
—identifies
that the variables B and C are first defined, then subtracted and the
difference stored as an intermediate solution.
The intermediate solution is divided by variable C to generate another
intermediate solution that, in turn is multiplied by 100 to calculate the
solution variable A, the percent change value.
The
same mathematical structure provides the framework for computerassisted map
analysis. The only difference is that
the variables are represented by mapped data composed of thousands of organized
numbers. Figure 1 shows a similar
solution for calculating the percent change in animal activity for an
area. Maps of activity in two periods
serve as input; a difference map is calculated then divided by the earlier
period and multiplied by 100. The procedure
uses the same equation, just derives a different form of output—a map of
percent change.
Figure 1. An iterative processing environment,
analogous to basic math, is used to derive new map variables.
The
processing steps shown in the figure are identical to the traditional solution
except the calculations are performed for each grid cell in the study area and
the result is a map that identifies the percent change at each location (a
decrease of 8.51% for the example location; red tones indicate decreased and
green tones indicate increased animal activity).
Map
analysis identifies what kind of change (termed the thematic attribute)
occurred where (termed the spatial attribute).
The characterization of what
and where provides information needed
for further
Within
this iterative processing structure, four fundamental classes of map analysis
operations can be identified. These
include:

Reclassifying Maps –
involving the reassignment of the values of an existing map as a function of
its initial value, position, size, shape or contiguity of the spatial
configuration associated with each map category.

Overlaying Maps –
resulting in the creation of a new map where the value assigned to every
location is computed as a function of the independent values associated with
that location on two or more maps.

Measuring Distance and Connectivity –
involving the creation of a new map expressing the distance and route between
locations as straightline length (simple proximity) or as a function of
absolute or relative barriers (effective proximity).

Characterizing and Summarizing
Neighborhoods – resulting in the creation of a new map based on
the consideration of values within the general vicinity of target locations.
Reclassification
operations merely repackage existing information on a single map. Overlay operations, on the other hand,
involve two or more maps and result in the delineation of new boundaries. Distance and connectivity operations are more
advanced techniques that generate entirely new information by characterizing
the relative positioning of map features.
Neighborhood operations summarize the conditions occurring in the
general vicinity of a location. See the
Author’s Notes for links to more detailed discussions of the types of map
analysis operations.
The
reclassifying and overlaying operations based on point processing are the
backbone of current
The
mathematical structure and classification scheme of Reclassify, Overlay, Distance and Neighbors form a conceptual framework that is easily adapted to modeling
spatial relationships in both physical and abstract systems. A major advantage is flexibility. For example, a model for siting a new highway
can be developed as a series of processing steps. The analysis might consider economic and
social concerns (e.g., proximity to high housing density, visual exposure to
houses), as well as purely engineering ones (e.g., steep slopes, water
bodies). The combined expression of both
physical and nonphysical concerns within a quantified spatial context is another
significant major benefit.
However,
the ability to simulate various scenarios (e.g., steepness is twice as
important as visual exposure and proximity to housing is four times more
important than all other considerations) provides an opportunity to fully
integrate spatial information into the decisionmaking process. By noting how often and where the proposed
route changes as successive runs are made under varying assumptions,
information on the unique sensitivity to siting a highway in a particular
locale is described.
In the
old environment, decisionmakers attempted to interpret results, bounded by
vague assumptions and system expressions of a specialist. Gridbased map analysis, on the other hand,
engages decisionmakers in the analytic process, as it both documents the
thought process and encourages interaction.
It’s sort of like a “spatial spreadsheet” containing mapmatical equations (or recipes) that
encapsulates the spatial reasoning of a problem and solves it using digital map
variables.
Options
Seem Endless When Reclassifying Maps
(GeoWorld, April 2004)
The
previous section described a mapematical
framework for
The
reassignment of existing values can be made as a function of the initial value, position, contiguity, size,
or shape of the spatial configuration of the individual map
categories. Each reclassification
operation involves the simple repackaging of information on a single map, and
results in no new boundary delineation.
Such operations can be thought of as the purposeful
"recoloring" of maps.
Figure
1 shows the result of simply reclassifying a map as a function of its initial
thematic values. For display, a unique
symbol is associated with each value. In
the figure, the cover type map has categories of Open Water, Meadow and
The
binary map on the right side of the figure isolates the Open Water locations by simply assigning zero to the areas of Meadow and Forest and displaying as the categories as grey. Although the operation seems trivial by
itself, it has map analysis implications far beyond simply recoloring the map
categories. And it graphically
demonstrate the basic characteristic of reclassify operations—values change but
the spatial pattern of the data doesn’t.
Figure 1. Areas of meadow and forest on a COVERTYPE map
can be reclassified to isolate large areas of open water.
A
similar reclassification operation might involve the ranking or weighing of
qualitative map categories to generate a new map with quantitative values. For example, a map of soil types could be
assigned values that indicate the relative suitability of each soil type for
residential development.
Quantitative
values also might be reclassified to yield new quantitative values. This might involve a specified reordering of
map categories (e.g., given a map of soil moisture content, generate a map of
suitability levels for plant growth).
Or, it could involve the application of a generalized reclassifying
function, such as "level slicing," which splits a continuous range of
map category values into discrete intervals (e.g., derivation of a contour map
of just 10 contour intervals from an elevation surface composed of thousands of
specific elevation values).
Other
quantitative reclassification functions include a variety of arithmetic
operations involving map category values and a specified or computed
constant. Among these operations are
addition, subtraction, multiplication, division, exponentiation, maximization,
minimization, normalization and other scalar mathematical and statistical
operators. For example, an elevation
surface expressed in feet could be converted to meters by multiplying each map
value by the appropriate conversion factor of 3.28083 feet per meter.
Reclassification
operations can also relate to location, as well as purely thematic. One such characteristic is position. An overlay category represented by a single
"point" location, for example, might be reclassified according to its
latitude and longitude. Similarly, a
line segment or area feature could be reassigned values indicating its center
or general orientation.
A
related operation, termed parceling, characterizes category contiguity. This procedure identifies individual
"clumps" of one or more points that have the same numerical value and
are spatially contiguous (e.g., generation of a map identifying each lake as a
unique value from a generalized map of water representing all lakes as a single
category).
Another
location characteristic is size. In the
case of map categories associated with linear features or point locations,
overall length or number of points might be used as the basis for reclassifying
the categories. Similarly, an overlay
category associated with a planar area could be reclassified according to its
total acreage or the length of its perimeter.
A map
of water types, for example, could be reassigned values to indicate the area of
individual lakes or the length of stream channels. The same sort of technique also could be used
to deal with volume. Given a map of
depth to bottom for a group of lakes, each lake might be assigned a value
indicating total water volume based on the area of each depth category.
Figure 2. A sequence of reclassification operations
(renumber, clump, size and renumber) can be used to isolate large water bodies
from a cover type map.
Figure
2 identifies a similar processing sequence using the information derived in
figure 1. Although your eye sees two
distinct blobs of water on the OPEN WATER map, the computer only “sees”
distinctions by different map category values.
Because both water bodies are assigned the same value of 1, there isn’t
a mapematical distinction that the computer cannot see the distinction.
The Clump operation is used to identify the
contiguous features as separate values—clump #1 (Larry’s
In
addition to the initial value, position, contiguity, and size of features,
shape characteristics also can be used as the basis for reclassifying map
categories. Shape characteristics
associated with linear forms identify the patterns formed by multiple line
segments (e.g., dendritic stream pattern).
The primary shape characteristics associated with polygonal forms
include feature integrity, boundary convexity, and nature of edge.
Feature
integrity relates to an area’s “intactness”.
A category that is broken into numerous "fragments" and/or
contains several interior "holes" is said to have less spatial
integrity than categories without such violations. Feature integrity can be summarized as the
Euler Number that’s computed as the number of holes within a feature less one
short of the number of fragments. Euler
Numbers of zero indicates features that are spatially balanced, whereas larger
negative or positive numbers indicate less spatial integrity—either broken into
more pieces or poked with more holes.
Convexity
and edge are other shape indices that relate to the configuration of polygonal
features’ boundaries. The Convexity
Index for a feature is computed by the ratio of its perimeter to its area. The most regular configuration is that of a
circle which is totally convex and, therefore, not enclosed by the background
at any point along its boundary.
Comparison
of a feature's computed convexity to a circle of the same area, results in a
standard measure of boundary regularity.
The nature of the boundary at each point can be used for a detailed
description of boundary configuration.
At some locations the boundary might be an entirely concave intrusion,
whereas others might be at entirely convex protrusions. Depending on the "degree of
edginess," each point can be assigned a value indicating the actual
boundary convexity at that location.
This
explicit use of cartographic shape as an analytic parameter is unfamiliar to
most GIS users. However, a non‑quantitative
consideration of shape is implicit in any visual assessment of mapped
data. Particularly promising is the
potential for applying quantitative shape analysis techniques in the areas of
digital image classification and wildlife habitat modeling.
A map
of forest stands, for example, could be reclassified such that each stand is
characterized according to the relative amount of forest edge with respect to
total acreage and the frequency of interior forest canopy gaps. Stands with a large proportion of edge and a
high frequency of gaps will generally indicate better wildlife habitat for many
species. In any event, reclassify
operations simply assign new values to old category values—sometimes seeming
trivial and sometimes seemingly a bit conceptually complex.
Overlay Operations Feature a Variety of Options
(GeoWorld, May 2004)
The
general class of overlay operations can be characterized as "light‑table
gymnastics." These involve the
creation of a new map where the value assigned to every point, or set of
points, is a function of the independent values associated with that location
on two or more existing map layers.
In location‑specific
overlaying, the value assigned is a function of the point‑by‑point
coincidence of the existing maps. In category‑wide
composites, values are assigned to entire thematic regions as a function of the
values on other overlays that are associated with the categories. Whereas the first overlay approach
conceptually involves the vertical spearing of a set of map layers, the latter
approach uses one map to identify boundaries by which information is extracted
from other maps.
Figure
1 shows an example of location‑specific overlaying. Here, maps of COVERTYPE and topographic
SLOPE_CLASSES are combined to create a new map identifying the particular
cover/slope combination at each location.
A specific function used to compute new category values from those of
existing maps being overlaid can vary according to the nature of the data being
processed and the specific use of that data within a modeling context.
Environmental
analyses typically involve the manipulation of quantitative values to generate
new values that are likewise quantitative in nature. Among these are the basic arithmetic
operations such as addition, subtraction, multiplication, division, roots, and
exponentials. Functions that relate to
simple statistical parameters such as maximum, minimum, median, mode, majority,
standard deviation or weighted average also can be applied. The type of data being manipulated dictates
the appropriateness of the mathematical or statistical procedure used.
The
addition of qualitative maps such as soils and land use, for example, would
result in mathematically meaningless sums, since their thematic values have no
numerical relationship. Other map
overlay techniques include several that might be used to process either
quantitative or qualitative data and generate values which can likewise take
either form. Among these are masking,
comparison, calculation of diversity, and permutations of map categories (as
depicted in figure 1).
Figure 1. Pointby point overlaying operations
summarize the coincidence of two or more maps, such as assigning a unique value
identifying the COVERTYPE and SLOPE_CLASS conditions at each location.
More complex
statistical techniques may also be applied in this manner, assuming that the
inherent interdependence among spatial observations can be taken into
account. This approach treats each map
as a variable, each point as a case, and each value as an observation. A predictive statistical model can then be
evaluated for each location, resulting in a spatially continuous surface of
predicted values. The mapped predictions
contain additional information over traditional non‑spatial procedures,
such as direct consideration of coincidence among regression variables and the
ability to spatially locate areas of a given level of prediction. Topic
12 investigates the considerations in spatial data mining derived by
statistically overlaying mapped data.
Figure 2. Categorywide
overlay operations summarize the spatial coincidence of map categories, such as
generating the average SLOPE for each COVERTYPE category.
An
entirely different approach to overlaying maps involves category‑wide
summarization of values. Rather than
combining information on a point‑by‑point basis, this group
summarizes the spatial coincidence of entire categories shown on one map with
the values contained on another map(s).
Figure 2 contains an example of a category‑wide overlay
operation. In this example, the
categories of the COVERTYPE map are used to define an area over which the coincidental
values of the SLOPE map are averaged.
The computed values of average slope within each category area are then
assigned to each of the cover type categories.
Summary
statistics which can be used in this way include the total, average, maximum,
minimum, median, mode, or minority value; the standard deviation, variance, or
diversity of values; and the correlation, deviation, or uniqueness of
particular value combinations. For
example, a map indicating the proportion of undeveloped land within each of
several counties could be generated by superimposing a map of county boundaries
on a map of land use and computing the ratio of undeveloped land to the total
land area for each county. Or a map of
zip code boundaries could be superimposed over maps of demographic data to
determine the average income, average age, and dominant ethnic group within
each zip code.
As with
location‑specific overlay techniques, data types must be consistent with
the summary procedure used. Also of
concern is the order of data processing.
Operations such as addition and multiplication are independent of the
order of processing. Other operations,
such as subtraction and division, however, yield different results depending on
the order in which a group of numbers is processed. This latter type of operations, termed non‑commutative,
cannot be used for category‑wide summaries.
Computers Quickly Characterize Spatial Coincidence
(GeoWorld, June 2004)
As previously
noted,
Everybody
knows the 'bread and butter' of a
Let's
compare how you and your computer might approach the task of identifying
coincidence. Your eye moves randomly
about the stack, pausing for a nanosecond at each location and mentally
establishing the conditions by interpreting the color. Your summary might conclude that the
northeastern portion of the area is unfavorable as it has "kind of a
magenta tone." This is the result
of visually combining steep slopes portrayed as bright red with unstable soils
portrayed as bright blue with minimal vegetation portrayed as dark green. If you want to express the result in map form,
you would tape a clear acetate sheet on top and delineate globs of color
differences and label each parcel with your interpretation. Whew!
No wonder you want a
The
A
raster system has things a bit easier.
As all locations are predefined as a consistent set of cells within a
matrix, the computer merely 'goes' to a location, retrieves the information
stored for each map layer and assigns a value indicating the combined map
conditions. The result is a new set of
values for the matrix identifying the coincidence of the maps.
The big
difference between ocular and computer approaches to map overlay is not so much
in technique, as it is in the treatment of the data. If you have several maps to overlay you
quickly run out of distinct colors and the whole stack of maps goes to an
indistinguishable dark, purplish hue.
One remedy is to classify each map layer into just two categories, such
as suitable and unsuitable. Keep one as
clear acetate (good) and shade the other as light grey (bad). The resulting stack avoids the ambiguities of
color combinations, and depicts the best areas as lighter tones. However, in making the technique operable you
have severely limited the content of the data—just good and bad.
The
computer can mimic this technique by using binary maps. A "0" is assigned to good
conditions and a "1" is assigned to bad conditions. The sum of the maps has the same information
as the brightness scale you observe—the smaller the value the better. The two basic forms of logical combination
can be computed. "Find those
locations which have good slopes .
But how
would you handle, "Find those locations which have good slopes .OR. good
soils .
In fact
any combination is easy to identify.
Let's say we expand our informational scale and redefine each map from
just good and bad to not suitable (0), poor (1), marginal (2), good (3) and
excellent (4). We could ask the computer
to INTERSECT SLOPES WITH SOILS WITH COVER COMPLETELY FOR
Another
way of combining these maps is by asking to COMPUTE SLOPES MINIMIZE SOILS
MINIMIZE COVER FOR WEAK
What
would happen if, for each location (be it a polygon or a cell), we computed the
sum of the three maps, then divided by the number of maps? That would yield the average rating for each
location. Those with the higher averages
are better. Right? You might want to take it a few steps
further. First, in a particular
application, some maps may be more important than others in determining the
best areas. Ask the computer to AVERAGE
SLOPES TIMES 5 WITH SOILS TIMES 3 WITH COVER TIMES 1 FOR WEIGHTEDAVERAGE. The result is a map whose average ratings are
more heavily influenced by slope and soil conditions.
Just to
get a handle on the variability of ratings at each location, you can determine
the standard deviation—either simple or weighted. Or for even more information, determine the
coefficient of variation, which is the ratio of the standard deviation to the
average, expressed as a percent. What will
that tell you? It hints at the degree of
confidence you should put into the average rating. A high COFFVAR indicates wildly fluctuating
ratings among the maps and you might want to look at the actual combinations
before making a decision.
A
statistical way to summarized coincidence between maps is a crosstab
table. If you CROSSTAB FORESTS WITH
SOILS a table results identifying how often each forest type jointly occurs
with each soil type. In a vector system,
this is the total area in each forest/soil combinations. In a raster system, this is simply a count of
all the cell locations for each forest/soil combination.
For
example, reading across the first row of table in figure 1 notes that Forest
category 1 (Deciduous) contains 303 cells distributed throughout the map. The total count for Soils category 1
(Lowland) is 427 cells. The next section
of the table notes that the joint condition of Deciduous/Lowland occurs 299
times for 47.84 percent of the total map area.
Contrast this result with that of Deciduous/Upland occurrence on the row
below indicating only four “crosses” for less than one percent of the map. The coincidence statistics for the Conifer
category is more balanced with 128 cells (20.48%) occurring with the Lowland
soil and 194 cells (31.04%) occurring with the Upland soil.
Figure 1. A crosstab table statistically summarizes
the coincidence among the categories on two maps.
These
data may cause you to jump to some conclusions, but you had better consider the
last section of the Table before you do.
This section normalizes the coincidence count to the total number of
cells in each category. For example, the
299 Deciduous/Lowland coincidence accounts for 98.68 percent of all occurrences
of Deciduous trees ((299/303)*100).
That's a very strong relationship.
However, from Lowland soil occurrence the 299 Deciduous/Lowland
coincidence is a bit weaker as it accounts for only 70.02 percent of all
occurrences of Lowland soils ((299/427)*100).
In a similar vein, the Conifer/Upland coincidence is very strong as it
accounts for 97.98 percent of the occurrence of all Upland soil
occurrences. Both columns of coincidence
percentages must be considered as a single high percent might be merely the result
of the other category occurring just about everywhere.
There
are still a couple of loose ends before we can wrapup pointbypoint overlay
summaries. One is direct map comparison,
or change detection. For example,
if you encode a series of land use maps for an area, then subtract each
successive pair of maps; the locations that underwent change will appear as
nonzero values for each time step. In
If you
are real tricky and think mapematically you will assign a binary progression
to the land use categories (1,2,4,8,16, etc.), as the differences will
automatically identify the nature of the change. The only way you can get a 1 is 21; a 2 is
42; a 3 is 41; a 6 is 82; etc. A
negative sign indicates the opposite change, and now all bases are
covered. .
The
last pointbypoint operation is a weird one—covering. This operation is truly spatial and has no
traditional math counterpart. Imagine
you prepared two acetate sheets by coloring all of the forested areas opaque
green on one sheet and all of the roads an opaque red on the other sheet. Now overlay them on a lighttable. If you place the forest sheet down first the
red roads will “cover” the green forests and you will see the roads passing
through the forests. If the roads map
goes down first, the red lines will stop abruptly at the green forest globs.
In a
_____________________
Further Online
Reading: (Chronological
listing posted at www.innovativegis.com/basis/BeyondMappingSeries/)
(Spatial Coincidence)
Key Concepts Characterize Unique
Conditions — describes a technique for handling unique combinations
of map layers (April 2006)
Use “Shadow Maps” to Understand
Overlay Errors — describes how shadow maps of certainty can be used
to estimate error and its propagation (September 2004)
(Back
to the Table of Contents)