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Week 8: Advanced Spatial Analysis

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Interrogation and reasoning. Measurements. Transformations ... Methods of spatial interpolation are designed to solve this problem. Spatial Autocorrelation ... – PowerPoint PPT presentation

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Title: Week 8: Advanced Spatial Analysis


1
Week 8 Advanced Spatial Analysis
2
This week
  • Guest speaker - Greg Robillard
  • Geographic data bases
  • Data download and manipulation demo
  • Spatial interpolation
  • Advanced spatial analysis

3
Last week Geographic 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

4
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

5
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6
Spatial Autocorrelation
  • Spatial autocorrelation measures the extent to
    which similarities in position match similarities
    in attributes
  • Sampling interval
  • Self-similarity

Tobler
7
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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9
Spatial autocorrelation measures
10
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
11
Spatial Interpolation
ORIGINAL SAMPLE POINTS
Interpolated Values
12
Types of interpolation
  • Theissen polygons
  • IDW
  • Spline
  • Kriging

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14
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

15
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
16
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

17
A Potentially Undesirable Characteristic of IDW
interpolation
18
Spline
  • Fits a mathmetical function to input points
    insuring reulting surface passes through all
    sample points
  • Minimizes curvature
  • Good for smooth surfaces

19
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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21
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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23
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)

24
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

25
typical kernel function
smooth statistical surface
26
Street Intersections
27
Street Intersections
2 mile kernel
28
Street Intersections
½ mile kernel
29
Spatial analysis
  • Six categories, each having a distinct conceptual
    basis
  • Queries and interrogation
  • Measurements
  • Transformations
  • Descriptive summaries
  • Optimization
  • Hypothesis testing

30
Spatial analysis (Openshaw)
  • Exploratory
  • Descriptive
  • Model based
  • Inferential
  • User driven
  • Visualization
  • Machine driven
  • Automated

31
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

32
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

33
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

34
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.
35
The Centroid
  • Found for a point set by taking the weighted
    average of coordinates
  • The balance point

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37
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

38
1975
Rainforest Fragmentation Rondonia, Brazil
1986
1992
39
Zonal statistics
  • Summarize information to a pre-defined zone
  • Aggregation of more micro-scaled observations
  • Pixel
  • Census data

40
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

41
MAUP
  • Modifiable Arial Unit Problem
  • Scale aggregation MAUP
  • can be investigated through simulation of large
    numbers of alternative zoning schemes

42
Example of MAUP overlay census blocks With
something then census tracks with the Same thing
and quantify results. (distance to Schools?)
43
The Ecological Fallacy
44
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

45
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

46
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

47
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?

48
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49
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

50
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52
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

53
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

54
Solution of a least-cost path problem.
Example of cost path
55
Advanced Analysis
  • Spatial statistics (hypothesis testing, spatial
    regression)
  • Spatial Multi-criteria analysis
  • Heuristic models
  • Linear programming
  • Simulated annealing
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