Compiled with assistance from Nigel M. Waters, University of
Calgary
NOTES
UNIT 41 - SPATIAL INTERPOLATION II
Compiled with assistance from Nigel M. Waters, University of
Calgary
A. INTRODUCTION
- this unit continues the examination of spatial
interpolation by looking at areal interpolation
techniques and some applications
- areal interpolation is the problem of transferring data
from one set of areas (source reporting zones) to another
(target reporting zones)
- this is easy if the target set is an aggregation of
the source set, but more difficult if the boundaries
of the target set are independent of the source set
- later we look at applications that do not fall easily
into either point or areal interpolation categories
B. AREAL INTERPOLATION - NON-VOLUME PRESERVING
- e.g. interpolating population counts from census tracts
to school districts
Procedure
overhead - Non-volume preserving areal interpolation
- calculate the population density for each source census
tract by dividing population by area
- identify a centroid for each region
- assign to the point located at each centroid, the
population density value determined for its
enclosing area
- using this set of points, interpolate a gridded
population density surface using any of the methods
described previously
- convert each grid cell's value to a population by
multiplying the estimated density by the cell's area
- overlay the interpolated grid on the target map and
assign each grid value to each its target region (school
district)
- calculate the total population in each target region
- this method is criticized because:
- choosing the center point is ill-defined
- inadequacy of point based interpolation methods
- most importantly, the total value of each zone is
not conserved
- e.g. if a source zone is divided into two
target zones, the total population of the
target zones after interpolation need not equal
the population of the source zone
C. AREAL INTERPOLATION - VOLUME-PRESERVING
1. Overlay
- discussed by MacDougall (1976) and Goodchild and Lam
(1980)
- procedure involves:
- overlay of target and source zones
- determining the proportion of each source zone that
is assigned to each target zone
- apportioning the total value of the attribute for
each source zone to target zones according to the
areal proportions
- assumes uniform density of the attribute within each zone
- e.g. uniform population density if the attribute is
total zone population
2. Pycnophylactic
Boundary conditions
- at the boundary of the reporting zones, pixels will have
neighbors outside the study area and therefore without
values
- some decision must be made about the behavior of the
surface outside the study area
- e.g. population density equals zero (a lake or
rural area)
- e.g. population density unknown, assumed equal
to the values of the outermost pixels of the
study area
D. SPECIAL CASES OF SPATIAL INTERPOLATION
1. Mapping populated areas
2. Estimating trade areas
- in marketing, it is often desirable to plot the boundary
of a trade area for e.g. a store, given information on
the home locations of customers
- simplest case is when the location of all customers and
non-customers is known
- simply draw a boundary contour between them
- if the location of non-customers is not known:
1. calculate the average distance to all customers
and draw a circle
or
2. divide the area into sectors, average the
distance to customers within the sectors and draw a
distance arc for each sector (see Huff and Batsell,
1977)
E. A GIS PERSPECTIVE ON INTERPOLATION
- both point and areal interpolation try to estimate a
continuous surface
- in the point case, the surface has been measured at
sample points
- in the areal case, the surface of population density
is estimated from total population counts in each
reporting zone
- in other cases it is impossible to conceive of a
continuous surface
- e.g. if each point is a city and the attribute is
city population
- if city A has population 1 million and city B
100 km away has population 2 million, there is
no reason to believe in the existence of a city
half way between A and B with population 1.5
million
- in this case, the variable population exists only at
the points, not as a continuous surface
- in other cases the variable might exist only along
lines
e.g. traffic density on a street network
- we must distinguish here between layer and object views
of the world
- a continuous surface of elevations is a layer view
of the world - there is one value of elevation at an
infinite number of possible places in the space
- the point map of cities is an object view of the
world - the space in between points is empty, and
has no value of the population variable
- the street map is an object view of the world - the
world is empty except where there are streets - only
along streets is traffic density defined
- spatial interpolation implies a layer view of the world,
and it requires special techniques (e.g. density
estimation) to apply it to objects such as store
customers
Expert systems for spatial interpolation algorithms
- a good GIS should include a range of spatial
interpolation routines so that the user can choose the
most appropriate method for the data and the task
- ideally, these routines should provide a natural language
interface which would lead the user through an
appropriate series of questions about the intentions,
goals and aims of the user and about the nature of the
data
- a number of prototype expert systems for guiding the
choice of a spatial interpolation algorithm have been
developed
- these may be written in the form of:
- an expert system shell (Waters, 1988)
- in one of the artificial intelligence languages such
as Prolog or LISP (see Dutton-Marion, 1988)
- or in a high level language such as Pascal (Maslyn,
1987)
Conclusion
- if computer contouring and surface generation techniques
are to be incorporated successfully into GIS, they must
be easy to use and effective
- "easy to use" implies that those without a detailed
knowledge of the mathematical and statistical
characteristics of the procedure should be able to
choose the correct technique for displaying a
particular data set for a particular purpose
- note: statisticians argue that this is not an
ideal goal as people may use techniques without
a proper understanding of the underlying
assumptions
- "effective" means that these techniques should be
informative, highlighting the essential nature of
the data and/or surface and serving the purpose of
the researcher/analyst
- the researcher's measure of success will be
largely subjective and visual - does the result
look right?
- this purpose may vary from an attempt to model all the
"real" intricacies of the surface to simply trying to
highlight the general, spatial trend of the data in order
to aid in the decision-making process
REFERENCES
Bracken, I. and D. Martin, 1989. "The generation of spatial
population distributions from census centroid data,"
Environment and Planning A 21:537-44.
Dutton-Marion, K.E., 1988. Principles of Interpolation
Procedures in the Display and Analysis of Spatial Data:
A Comparative Analysis of Conceptual and Computer
Contouring, unpublished Ph.D. Thesis, Department of
Geography, University of Calgary, Calgary, Alberta.
Goodchild, M.F., and Lam, N., 1980. "Areal Interpolation: A
Variant of the Traditional Spatial Problem," Geo-
Processing 1: 297-312.
Huff, D.L. and R.R. Batsell, 1977. "Delimiting the areal
extent of a market area," Journal of Marketing Research
14:581-5.
MacDougall, E.B., 1976. Computer Programming for Spatial
Problems, Arnold, London.
Maslyn, R.M., 1987. "Gridding Advisor: An Expert System for
Selecting Gridding Algorithms," Geobyte 2(4):42-43.
Silverman, B.W., 1986. Density Estimation for Statistics and
Data Analysis, Chapman and Hall, London.
Tobler, W.R., 1979. "Smooth pycnophylactic interpolation for
geographical regions," Journal of the American
Statistical Association 74:519-30.
Waters, N.M., 1988. "Expert Systems and Systems of Experts,"
Chapter 12 in W.J. Coffey, ed., Geographical Systems and
Systems of Geography: Essays in Honour of William
Warntz, Department of Geography, University of Western
Ontario, London, Ontario.
DISCUSSION AND EXAM QUESTIONS
1. What are the main considerations to be aware of in
computer contouring? What are the key aspects for the
design of an expert system to aid in choosing a computer
contouring algorithm within a GIS? How long do you think it
will be before such expert systems become widely available?
2. Describe how Tobler's pycnophylactic method differs from
volume-preserving overlay. What model of the underlying
spatial distribution is assumed by each? Give examples of
phenomena and application which fit each method's
assumptions.
3. Describe the application of areal interpolation in
political districting.
4. One test of a spatial interpolation method is that its
results would be judged as equal or better to hand
contouring by a specialist, e.g. a field geologist, with
detailed knowledge of the phenomenon being mapped. How well
do the methods discussed in these two units do against this
criterion?
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