Spatial Analysis
Early Spatial Analysis
• John Snow, 1854
• Cholera via polluted
water, not air
• “John Snow’s pump”
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www.jsi.com
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Categories of Spatial Analysis
(Longley et al.)
• Hypothesis testing
• Queries and reasoning
– Map & database/catalog queries, buffer, polygon
overlay
• Measurements
– Aspects of geographic data, length, area, etc.
Categories of Spatial Analysis
(Longley et al.)
• Transformations
– New data, raster to vector, geometric rules
– Buffer, polygon overlay
– Interpolation, Density Estimation, Terrain Analysis
(Lab 6)
• Descriptive summaries
– Essence of data in 1 or 2 parameters
– Spatial statistics (including fragmentation statistics)
• Optimization - ideal locations, routes
– Network analysis (Lab 5), Routing
Interpolation
Nonlinear Interpolation
• When things aren't or shouldn’t be so simple
• Basic types:
1. Trend surface analysis / Polynomial
2. Minimum Curvature Spline
3. Inverse Distance Weighted
4. Kriging
Fitting Continuous
Surfaces to Data
(1) FLAT plane
(2) flat but TILTED to fit data better
(3) tilted but WARPED to fit data even better
1. Trend Surface/Polynomial
• point-based
• Fits a polynomial to input points
• When calculating function that will describe
surface, uses least-square regression fit
• approximate interpolater
– Resulting surface doesn’t pass through all data points
– global trend in data, varying slowly overlain by local but
rapid fluctuations
1. Trend Surface cont.
• flat but TILTED plane to fit data
– surface is approximated by linear
equation (polynomial degree 1)
– z = a + bx + cy
• tilted but WARPED plane to fit data
– surface is approximated by quadratic equation
(polynomial degree 2)
– z = a + bx + cy + dx2 + exy + fy2
Trend Surfaces
• Simplifies the surface
representation to allow
visualization of general trends.
• Polynomials of higher order
Windows (not Microsoft’s)
• generates estimates based on existing data in the
“region”
• “region” = “roving window”
–
–
–
–
–
moves about study area
summarizes data it encounters
reach (search radius)
number of samples
Direction
• WHERE might you find unusual responses?
• results extend non-spatial concept of central
tendency
2. Minimum Curvature Splines
• Fits a minimum-curvature surface through input
points
• Like bending a sheet of rubber to pass through points
– While minimizing curvature of that sheet
• repeatedly applies a smoothing equation (piecewise
polynomial) to the surface
– Resulting surface passes through all points
• best for gently varying surfaces, not for rugged ones
(can overshoot data values)
3. Distance Weighted Methods
3. Inverse Distance Weighted
• Each input point has local influence that diminishes
with distance
• estimates are averages of values at n known points
within window
z (x)   wi zi
w
i
i
i
– where w is some function of distance
wi  1 d
2
i
point i
known value zi
location xi
weight wi distance di
unknown value (to be
interpolated)
location x
z (x)   wi zi
i
wi  1 d
w
i
The estimate is a weighted
average
i
2
i
Weights decline with distance
IDW (cont.)
• an almost infinite variety of algorithms
may be used, variations include:
– the nature of the distance function (w)
– varying the number of points used
– the direction from which they are selected
IDW (cont.)
• IDW is popular, easy, but not panacea
• interpolated values limited by the range
of the data
• no interpolated value will be outside the
observed range of z values
• how many points should be included in
the averaging?
• what to do about irregularly spaced
points?
A potentially undesirable characteristic of IDW
interpolation
This set of six data
points clearly suggests
a hill profile. But in
areas where there is
little or no data the
interpolator will move
towards the overall
mean. Blue line shows
the profile interpolated
by IDW
IDW Example
• ozone concentrations at CA
measurement stations
1. estimate a complete field, make a map
2. estimate ozone concentrations at
specific locations (e.g., Los Angeles)
Data for IDW Example
measuring stations and concentrations
(point shapefile)
CA cities (point shapefile)
CA outline (polygon shapefile)
DEM (raster)
IDW Wizard in Geostatistical Analyst
define data source
Further define interpolation
Power of distance
4 sectors
wi  1 di2
Cross validation
• removing one of the n observation points and using the
remaining n-1 points to predict its value.
• Error = observed - predicted
Results
amount of detail where
there is no data
generally smooth
surface
highs in LA, S central
valley
4. Kriging
• Assumes distance or direction betw. sample points
shows a spatial correlation that help describe the
surface
• Fits function to
– Specified number of points OR
– All points within a window of specified radius
• based on an analysis of the data, then an
application of the results of this analysis to
interpolation
• Most appropriate when you already know about
spatially correlated distance or directional bias in
data
Kriging (cont.)
• Involves several steps
– Exploratory statistical analysis of data
– Variogram modeling
– Creating the surface based on variogram
Explore with Trend analysis
• You may wish to remove
a trend from the dataset
before using kriging. The
Trend Analysis tool can
help identify global
trends in the input
dataset.
SemiVariogram
in
Kriging
how avg. difference between values at points changes with distance between points
Range – no
more surprises
sill
nugget
A semivariogram. Each cross represents a pair of points. The solid circles are obtained by averaging within
the ranges or bins of the distance axis. The solid line represents the best fit to these five points, using one of a
small number of standard mathematical functions.
Kriging Results
• once the variogram has been developed,
it is used to estimate distance weights
for interpolation
• computationally very intensive w/ lots of
data points
• estimation of the variogram complex
– No one method is absolute best
– Results never absolute, assumptions about
distance, directional bias
Kriging Example
Surface has no constant mean
Maybe no underlying trend
surface has a constant mean,
no underlying trend
allows for a trend
binary data
Analysis of Variogram
Fitting a Model, Directional Effects
How Many Neighbors?
Cross Validation
Kriging Result
• similar pattern to IDW
• less detail in remote
areas
• smooth
Slightly Better Cross Validation
IDW vs. Kriging
• Kriging appears to
give a more
“natural” look to
the data
• Kriging avoids the
“bulls eye” effect
• Kriging gives us a
standard error
Which Method to Use?
• Trend - rarely goes through your original points
• Spline - best for surfaces that are already
smooth
– Elevations, water table heights, etc.
• IDW - assumes variable decreases in
influence w/distance from sampled location
– Interpolating a surface of consumer
purchasing power for a retail store
• Kriging - if you already know correlated
distances or directional bias in data
– Geology, soil science
Which to Use? cont.
• Kriging - Allows user greater flexibility in
defining the model to be used in the
interpolation
– Tracks changes in spatial dependence
across study area (may not be linear)
– Produces
• a smooth, interpolated surface
• variogram (how well pixel value fits overall model)
– Diagnostic tool to refine model
– Want to get variances close as possible to zero
Interpolation Software
• ArcGIS with Geostatistical Analyst
• Surfer (Golden Software)
• Surface II package (Kansas Geological
Survey)
• GEOEAS (EPA)
• Spherekit (NCGIA, UCSB)
• Matlab
ArcInfo Workstation
Interpolation Methods
• TREND (Grid function)
• SPLINE (Grid function, minimum curvature
spline)
• IDW (Grid function)
• KRIGING (Arc command)
Research Issues...
• "easy to use"
– choose correct technique w/o having a Ph.D. in
math or stats
• “effective"
– techniques should be informative,
– highlighting the essential nature of the data and/or surface
– meet needs of the study
• “natural language” interface
– series of questions about the intentions, goals and aims of the
user and about the nature of the data
• articles on prototypes in the literature
Gateway to the Literature
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•
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•
•
•
•
Lam, N.S.-N., Spatial interpolation methods: A review, Am. Cartogr.,
10 (2), 129-149, 1983.
Gold, C.M., Surface interpolation, spatial adjacency, and GIS, in
Three Dimensional Applications in Geographic Information
Systems, edited by J. Raper, pp. 21-35, Taylor and Francis, Ltd.,
London, 1989.
Robeson, S.M., Spherical methods for spatial interpolation: Review
and evaluation, Cartog. Geog. Inf. Sys., 24 (1), 3-20, 1997.
Mulugeta, G., The elusive nature of expertise in spatial
interpolation, Cart. Geog. Inf. Sys., 25 (1), 33-41, 1999.
Wang, F., Towards a natural language user interface: An approach
of fuzzy query, Int. J. Geog. Inf. Sys., 8 (2), 143-162, 1994.
Davies, C., and D. Medyckyj-Scott, GIS usability: Recommendations
based on the user's view, Int. J. Geographical Info. Sys., 8 (2), 175189, 1994.
Blaser, A.D., M. Sester, and M.J. Egenhofer, Visualization in an early
stage of the problem-solving process in GIS, Comp. Geosci, 26, 5766, 2000.
More Resources
... a link to a USDA geostatistical workshop
http://www.ars.usda.gov/News/docs.htm?docid=12555
... an EPA workshop with presentations on geostatistical applications for
stream networks:
http://oregonstate.edu/dept/statistics/epa_program/sac2005js.htm