Spatial Analysis cont. Density Estimation, Summary Spatial Statistics, Routing

1
Spatial Analysis cont.Density Estimation,
Summary Spatial Statistics, Routing
Longley et al., chs. 13,14
2
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)

3
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

4
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)

5
Density Estimation Using Kernels

• Mathematical function
• each point replaced by a pile of sand of
  constant shape
• add the piles to create a surface

6
Width of Kernel

• Determines smoothness of surface
• narrow kernels produce bumpy surfaces
• wide kernels produce smooth surfaces

7
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

8
Using Spatial Analyst
9
Kernal too small?(radius of 16 km)
10
Kernel radius of 150 km
11
Whats the Difference?
12
Summary Spatial Statistics

• Longley et al., chs. 5,14

13
Descriptive Summaries

• Ways of capturing the properties of data sets in
  simple summaries
• mean of attributes
• mean for spatial coordinates, e.g., centroid

14
Spatial Min, Max, Average
15
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.
16
Spatial Autocorrelation
Toblers 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.
17
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

18
Morans 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

19
Morans Indexsimilarity of attributes,
similarity of location
Dispersed, - SA
Extreme negative SA
Independent, 0 SA
Spatial Clustering, SA
Extreme positive SA
20
Crime Mapping

• Clustering - neighborhood scale

21
Gearys c Ratio

• Like Morans 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

22
Morans and Gearys
Lee and Marion, 1994, Analysis of spatial
autocorrelation of USGS 1250,000 DEMs. GIS/LIS
Proceedings.
23
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?

24
Note the increasing fragmentation of the natural
habitat as a result of settlement. Such
fragmentation can adversely affect the success of
wildlife populations.
25
Fragstats pattern analysis for landscape ecology
http//www.innovativegis.com/products/fragstatsarc
/
26
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

27
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

28
FRAGSTATS Fundamentals

• PATCH individual parcel (Polygon)
• CLASS sets of similar parcels
• All Mixed Wood polygons
• (stands)
• LANDSCAPE all parcels within an area

29
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)

30
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)

31
More Spatial Statistics Resources

• Spacestat (www.spacestat.com)
• S-Plus
• Alaska USGS freeware (www.absc.usgs.gov/glba/gisto
  ols/)
• Central Server for GIS Spatial Statistics on
  the Internet
• www.ai-geostats.org
• GEO 441/541 - Spatial Variation in Ecology
  Earth Science

32
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

33
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

34
Routing service technicians for Schindler
Elevator. Every day this companys service crews
must visit a different set of locations in Los
Angeles. GIS is used to partition the days
workload among the crews and trucks (color
coding) and to optimize the route to minimize
time and cost.
35
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

36
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.