Week 8: Advanced Spatial Analysis

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Week 8Advanced Spatial Analysis
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This week

• Guest speaker - Greg Robillard
• Geographic data bases
• Data download and manipulation demo
• Spatial interpolation
• Advanced spatial analysis

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Last weekGeographic query and analysis

• What is spatial analysis?
• A method of analysis is spatial if the results
  depend on the locations of the objects being
  analyzed
• Types of analysis
• Interrogation and reasoning
• Measurements
• Transformations

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Spatial Interpolation

• Values of a field have been measured at a number
  of sample points
• There is a need to estimate the complete field
• to estimate values at points where the field was
  not measured
• to create a contour map by drawing isolines
  between the data points
• Methods of spatial interpolation are designed to
  solve this problem

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Spatial Autocorrelation

• Spatial autocorrelation measures the extent to
  which similarities in position match similarities
  in attributes
• Sampling interval
• Self-similarity

Tobler
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Spatial autocorrelation

• Interpretation depends on how we conceptualize
  the phenomena
• Measure of smoothness for field data
• Measure of how attribute values are distributed
  among objects (object view)
• Clustered
• Random
• Locally contrasting

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Spatial autocorrelation measures
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Spatial autocorrelation measures
n number of objects in the sample i,j any two of
the objects z the value of the attribute
of interest for object i c the similarity
of is and js attributes w the similarity
of is and js locations
i
i,j
i,j
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Spatial Interpolation
ORIGINAL SAMPLE POINTS
Interpolated Values
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Types of interpolation

• Theissen polygons
• IDW
• Spline
• Kriging

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Inverse Distance Weighting (IDW)

• The unknown value of a field at a point is
  estimated by taking an average over the known
  values
• weighting each known value by its distance from
  the point, giving greatest weight to the nearest
  points

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point i known value zi location xi weight wi
distance di
unknown value (to be interpolated) location x
The estimate is a weighted average
Weights decline with distance
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Issues with IDW

• The range of interpolated values cannot exceed
  the range of observed values
• it is important to position sample points to
  include the extremes of the field
• this can be very difficult

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A Potentially Undesirable Characteristic of IDW
interpolation
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Spline

• Fits a mathmetical function to input points
  insuring reulting surface passes through all
  sample points
• Minimizes curvature
• Good for smooth surfaces

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Kriging

• A technique of spatial interpolation firmly
  grounded in geostatistical theory
• The semivariogram reflects Toblers Law
• differences within a small neighborhood are
  likely to be small
• differences rise with distance

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Stages of Kriging

• Analyze observed data to estimate a semivariogram
• Estimate values at unknown points as weighted
  averages
• obtaining weights based on the semivariogram
• the interpolated surface replicates statistical
  properties of the semivariogram

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Density Estimation and Potential

• Spatial interpolation is used to fill the gaps in
  a field
• Density estimation creates a field from discrete
  objects
• the fields value at any point is an estimate of
  the density of discrete objects at that point
• e.g., estimating a map of population density (a
  field) from a map of individual people (discrete
  objects)

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The Kernel Function

• Each discrete object is replaced by a
  mathematical function known as a kernel
• Kernels are summed to obtain a composite surface
  of density
• The smoothness of the resulting field depends on
  the width of the kernel
• narrow kernels produce bumpy surfaces
• wide kernels produce smooth surfaces

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typical kernel function
smooth statistical surface
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Street Intersections
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Street Intersections
2 mile kernel
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Street Intersections
½ mile kernel
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Spatial analysis

• Six categories, each having a distinct conceptual
  basis
• Queries and interrogation
• Measurements
• Transformations
• Descriptive summaries
• Optimization
• Hypothesis testing

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Spatial analysis (Openshaw)

• Exploratory
• Descriptive
• Model based
• Inferential
• User driven
• Visualization
• Machine driven
• Automated

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Data Mining

• Analysis of massive data sets in search for
  patterns, anomalies, and trends
• spatial analysis applied on a large scale
• must be semi-automated because of data volumes
• widely used in practice, e.g. to detect unusual
  patterns in credit card use

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Descriptive Summaries

• Attempt to summarize useful properties of data
  sets in one or two statistics
• The mean or average is widely used to summarize
  data
• centers are the spatial equivalent
• there are several ways of defining centers

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The Histogram

• A useful summary of the values of an attribute
• showing the relative frequencies of different
  values
• A histogram view can be linked to other views
• e.g., click on a bar in the histogram view and
  objects with attributes in that range are
  highlighted in a linked map view

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A histogram or bar graph, showing the relative
frequencies of values of a selected attribute.
The attribute is the length of street between
intersections. Lengths of around 100m are
commonest.
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The Centroid

• Found for a point set by taking the weighted
  average of coordinates
• The balance point

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Fragmentation Statistics

• Measure the patchiness of data sets
• e.g., of vegetation cover in an area
• Useful in landscape ecology, because of the
  importance of habitat fragmentation in
  determining the success of animal and bird
  populations
• populations are less likely to survive in highly
  fragmented landscapes

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1975
Rainforest Fragmentation Rondonia, Brazil
1986
1992
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Zonal statistics

• Summarize information to a pre-defined zone
• Aggregation of more micro-scaled observations
• Pixel
• Census data

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Implications of a zone

• Are all zones in a feature layer comparable?
• Eg census tracts
• Were zone boundaries identified with the feature
  in question in mind?
• Eg summarizing number of stores by census block

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MAUP

• Modifiable Arial Unit Problem
• Scale aggregation MAUP
• can be investigated through simulation of large
  numbers of alternative zoning schemes

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Example of MAUP overlay census blocks With
something then census tracks with the Same thing
and quantify results. (distance to Schools?)
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The Ecological Fallacy
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Optimization

• Spatial optimization is a methodology used to
  maximize or minimize a management objective,
  given the limited area, finite resources, and
  spatial relationships in an system
• Spatial analysis can be used to solve many
  problems of design
• A spatial decision support system (DST) is an
  adaptation of GIS aimed at solving a particular
  design problem

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Optimization Properties

• The centroid minimizes the sum of distances
  squared
• Not the sum of distances from each point
• the center with that property is called the point
  of minimum aggregate travel (MAT)
• the properties have frequently been confused,
  e.g. by the U.S. Bureau of the Census in
  calculating the center of U.S. population
• the MAT must be found by iteration rather than by
  calculation

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Applications of the MAT

• Because it minimizes distance the MAT is a useful
  point at which to locate any central service
• e.g., a school, hospital, store, fire station
• finding the MAT is a simple instance of using
  spatial analysis for optimization

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Optimizing Point Locations

• The MAT is a simple case one service location
  and the goal of minimizing total distance
  traveled
• The operator of a chain of convenience stores or
  fire stations might want to solve for many
  locations at once
• where are the best locations to add new services?
• which existing services should be dropped?

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Location-allocation Problems

• Design locations for services, and allocate
  demand to them, to achieve specified goals
• Goals might include
• minimizing total distance traveled
• minimizing the largest distance traveled by any
  customer
• maximizing profit
• minimizing a combination of travel distance and
  facility operating cost

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Routing Problems

• Search for optimum routes among several
  destinations
• The traveling salesman problem
• find the shortest tour from an origin, through a
  set of destinations, and back to the origin

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Optimum Paths

• Find the best path across a continuous cost
  surface
• between defined origin and destination
• to minimize total cost
• cost may combine construction, environmental
  impact, land acquisition, and operating cost
• used to locate highways, power lines, pipelines
• requires a raster representation

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Solution of a least-cost path problem.
Example of cost path
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Advanced Analysis

• Spatial statistics (hypothesis testing, spatial
  regression)
• Spatial Multi-criteria analysis
• Heuristic models
• Linear programming
• Simulated annealing