Transcript No Slide Title

Spatial Analysis cont.
Density Estimation, Summary
Spatial Statistics, Routing
Longley et al., chs. 13,14
Density Estimation
• point interpolation to estimate a
continuous surface
vs.
• density estimation - surface is estimated
from counts within polygons
– (e.g., population density surface derived from total
population counts in each reporting zone)
Objects to Fields
• map of discrete objects and want to
calculate their density
– density of population
– density of cases of a disease
– density of roads in an area
• density would form a field
• one way of creating a field from a set of
discrete objects
Density Estimation and Potential
• Spatial interpolation is used to fill the gaps in a
field
• Density estimation creates a field from discrete
objects
– the field’s 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)
Density Estimation Using Kernels
• Mathematical function
• each point replaced by a “pile of sand” of
constant shape
• add the piles to create a surface
Width of Kernel
• Determines
smoothness of
surface
– narrow kernels
produce bumpy
surfaces
– wide kernels
produce smooth
surfaces
Example
• Density estimation and spatial
interpolation applied to the same data
• density of ozone measuring stations
vs.
• Interpolating surface based on locations
of ozone measuring stations
Using Spatial Analyst
Kernal too small?
(radius of 16 km)
Kernel radius of 150 km
What’s the Difference?
Summary Spatial Statistics
Longley et al., chs. 5,14
Descriptive Summaries
• Ways of capturing the properties of data
sets in simple summaries
• mean of attributes
• mean for spatial coordinates, e.g.,
centroid
Spatial Min, Max, Average
An example of the use of centroids to summarize the changes in point patterns through
time. The centroids of four land use classes are shown for London, Ontario, Canada
from 1850 to 1960. Circles show the associated dispersions of sites within each class.
Note how the industrial class has moved east, remaining concentrated, while the
commercial class has remained concentrated in the core, and the residential class has
dispersed but remained centered on the core. In contrast the institutional class moved
to a center in the northern part of the city.
Spatial Autocorrelation
Tobler’s 1st Law of Geography: everything is related to
everything else, but near things are more related than
distant things
S. autocorrelation: formal property that measures the
degree to which near and distant things are related.
Close in space
Dissimilar in attributes
Attributes
independent
of location
Close in space
Similar in attributes
Arrangements of dark and light colored cells exhibiting negative, zero, and positive spatial autocorrelation.
Why Spatial Dependence?
• evaluate the amount of clustering or
randomness in a pattern
– e.g., of disease rates, accident rates, wealth,
ethnicity
• random: causative factors operate at
scales finer than “reporting zones”
• clustered: causative factors operate at
scales coarser than “reporting zones”
Moran’s Index
• positive when attributes of nearby
objects are more similar than expected
• 0 when arrangements are random
• negative when attributes of nearby
objects are less similar than expected
I = nS
S wijcij / S S wij S(zi - zavg)2
n = number of objects in sample
i,j - any 2 of the objects
Z = value of attribute for I
cij = similarity of i and j attributes
wij= similarity of i and j locations
Moran’s Index
similarity of attributes, similarity of location
Dispersed, - SA
Extreme negative SA
Independent, 0 SA
Spatial Clustering, + SA
Extreme positive SA
Crime Mapping
• Clustering - neighborhood scale
Geary’s c Ratio
• Like Moran’s Index, use a single value to
describe spatial distribution
– e.g., of elevations in DEM cells
less than 1 (clustered)
1
greater than 1 (random)
• e.g., spatial autocorrelation indicator of
information loss during conversions
between DEMs and TINs
Moran’s and Geary’s
Lee and Marion, 1994, Analysis of spatial autocorrelation of USGS 1:250,000 DEMs. GIS/LIS Proceedings.
Fragmentation Statistics
• how fragmented is the pattern of areas
and attributes?
• are areas small or large?
• how contorted are their boundaries?
• what impact does this have on habitat,
species, conservation in general?
1975
1992
1986
Note the increasing fragmentation
of the natural habitat as a result of
settlement. Such fragmentation
can adversely affect the success
of wildlife populations.
Fragstats
pattern analysis for landscape ecology
http://www.innovativegis.com/products/fragstatsarc/
FRAGSTATS Overview
• derives a comprehensive set of useful
landscape metrics
• Public domain code developed by
Kevin McGarigal and Barbara Marks
under U.S.F.S. funding
• Exists as two separate programs
– AML version for ARC/INFO vector data
– C version for raster data
FRAGSTATS Fundamentals
• PATCH… individual parcel (Polygon)
A single homogeneous landscape unit
with consistent vegetation characteristics,
e.g. dominant species, avg. tree height,
horizontal density ,etc.
A single Mixed Wood polygon
(stand)
CLASS… sets of similar parcels
LANDSCAPE… all parcels within an area
FRAGSTATS Fundamentals
PATCH… individual parcel (Polygon)
• CLASS… sets of similar parcels
All Mixed Wood polygons
(stands)
LANDSCAPE… all parcels within an area
FRAGSTATS Fundamentals
PATCH… individual parcel (Polygon)
CLASS… sets of similar parcels
• LANDSCAPE… all parcels within an
area “of interacting ecosystems”
e.g., all polygons
within a given
geographic area
(landscape mosaic)
FRAGSTATS Output Metrics
•
•
•
•
•
•
•
•
Area Metrics (6),
Patch Density, Size and Variability Metrics (5),
Edge Metrics (8),
Shape Metrics (8),
Core Area Metrics (15),
Nearest Neighbor Metrics (6),
Diversity Metrics (9),
Contagion and Interspersion Metrics (2)
• …59 individual indices
(US Forest Service 1995 Report PNW-GTR-351)
More Spatial Statistics Resources
• Spacestat (www.spacestat.com)
• S-Plus
• Alaska USGS freeware
(www.absc.usgs.gov/glba/gistools/)
• Central Server for GIS & Spatial Statistics
on the Internet
– www.ai-geostats.org
• GEO 441/541 - Spatial Variation in
Ecology & Earth Science
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
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
Routing service technicians for Schindler Elevator. Every day this company’s
service crews must visit a different set of locations in Los Angeles. GIS is used to
partition the day’s workload among the crews and trucks (color coding) and to
optimize the route to minimize time and cost.
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
Solution of a least-cost path
problem. The white line
represents the optimum
solution, or path of least total
cost, across a friction surface
represented as a raster. The area
is dominated by a mountain
range, and cost is determined by
elevation and slope. The best
route uses a narrow pass
through the range. The blue line
results from solving the same
problem using a coarser raster.