Spatial Analysis

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Spatial Analysis

• Longley et al., chs. 14 and 15

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What is spatial analysis?

• Methods for working with spatial data
• to detect patterns, anomalies
• to find answers to questions
• to test or confirm theories
• deductive reasoning
• to generate new theories and generalizations
• inductive reasoning
• "a set of methods whose results change when the
  locations of the objects being analyzed change"

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What is spatial analysis (cont.)

• Methods for adding value to data
• in doing scientific research
• in trying to convince others
• Turning raw data into useful information
• A collaboration between human and machine
• Human directs, makes interpretations and
  inferences
• Machine does tedious, complex stuff

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Early Spatial Analysis

• John Snow, 1854
• Cholera via polluted water, not air
• John Snows pump
• www.jsi.com

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Video
http//vid01.esri.com/winmmedia/avflu.wmv
http//www.esri.com/apps/esriclips/clip.cfm?ClipID
165
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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.

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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

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Interpolation
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Nonlinear Interpolation

• When things aren't or shouldnt be so simple
• Basic types1. Trend surface analysis /
  Polynomial
• 2. Minimum Curvature Spline
• 3. Inverse Distance Weighted 4. Kriging

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Fitting ContinuousSurfaces to Data

• (1) FLAT plane
• (2) flat but TILTED to fit data better
• (3) tilted but WARPED to fit data even better

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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 doesnt pass through all data
  points
• global trend in data, varying slowly overlain by
  local but rapid fluctuations

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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

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Trend Surfaces

• Simplifies the surface representation to allow
  visualization of general trends.
• Polynomials of higher order

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Windows (not Microsofts)

• 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

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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)

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3. Distance Weighted Methods
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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

• where w is some function of distance

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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

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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?

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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
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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)

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Data for IDW Example
measuring stations and concentrations (point
shapefile) CA cities (point shapefile) CA
outline (polygon shapefile) DEM (raster)
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IDW Wizard in Geostatistical Analystdefine data
source
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Further define interpolation
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Cross validation

• removing one of the n observation points and
  using the remaining n-1 points to predict its
  value.
• Error observed - predicted

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Results
amount of detail where there is no
data generally smooth surface highs in LA, S
central valley
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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

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Kriging (cont.)

• Involves several steps
• Exploratory statistical analysis of data
• Variogram modeling
• Creating the surface based on variogram

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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.

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SemiVariogram in Kriging
how avg. difference between values at points
changes with distance between points
Range no more surprises
sill
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.
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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

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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
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Analysis of Variogram
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Fitting a Model, Directional Effects
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How Many Neighbors?
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Cross Validation
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Kriging Result

• similar pattern to IDW
• less detail in remote areas
• smooth

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Slightly Better Cross Validation
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IDW vs. Kriging
Kriging

• Kriging appears to give a more natural look to
  the data
• Kriging avoids the bulls eye effect
• Kriging gives us a standard error

IDW
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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

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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

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Interpolation Software

• ArcGIS with Geostatistical Analyst
• ArcView 3.2
• Surfer (Golden Software)
• Surface II package (Kansas Geological Survey)
• GEOEAS (EPA)
• Spherekit (NCGIA, UCSB)
• Matlab

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ArcInfo Workstation Interpolation Methods

• TREND (Grid function)
• SPLINE (Grid function, minimum curvature spline)
• IDW (Grid function)
• KRIGING (Arc command)

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Interpolation in ArcView 3.2

• MakeTrend (Avenue request)
• Interpolate Surface (menu choice) or MakeSpline
  (Avenue request)
• Interpolate Surface (menu choice) or MakeIDW
  (Avenue request)
• MakeSemivariogram and MakeKriging(Avenue
  requests)

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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

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Gateway to the Literature

• 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), 175-189, 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,
  57-66, 2000.

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More Resources

• ... a link to a USDA geostatistical workshop
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• http//www.ars.usda.gov/News/docs.htm?docid12555
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• ... an EPA workshop with presentations on
  geostatistical applications for stream networks
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• http//oregonstate.edu/dept/statistics/epa_program
  /sac2005js.htm