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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
2
This week
• Guest speaker - Greg Robillard
• Geographic data bases
• Spatial interpolation

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

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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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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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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)

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?

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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

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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