Spatial Analysis cont.
Density Estimation, Summary
Spatial Statistics, Routing
Density Estimation
• 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)
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 Using Kernels
• Mathematical function
• each point replaced by a “pile of sand” of
constant shape
• add the piles to create a surface
A
B
(A) A collection of point objects (B) A kernel function for one of the points
The kernel’s shape depends on a distance parameter—
increasing the value of the parameter results in a broader
and lower kernel, and reducing it results in a narrower and
sharper kernel. When each point is replaced by a kernel
and the kernels are added, the result is a density surface
whose smoothness depends on the value of the distance
parameter.
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 measured
level of ozone at measuring stations
Using Spatial Analyst
Kernal too small?(radius of 16 km)
each kernal isolated from neighbors
Kernel radius of 150 km
What’s the Difference?
A dataset with two possible interpretations:
First, a continuous field of atmospheric temperature measured at eight
irregularly spaced sample points, and second, eight discrete objects representing
cities, with associated populations in thousands. Spatial interpolation makes
sense only for the former, and density estimation only for the latter
GEO 565: Best locations for a new Beanery
more factors: proximity to a highway, zoning concerns, income levels, population density, age, etc.
Origins of Computer Viruses
Directory Harvest Attacks
Origins of Email Spam
Density of “The Dance”
Density of “The Dance”
From Transformations to
Descriptive Summaries:
Summary Spatial Statistics
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
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
Moran’s Index
similarity of attributes, similarity of location
Maps from Luc Anselin, University of Illinois U-C, Spatial Analysis Lab
The Local Moran statistic, applied using the
GeoDa software (available via geodacenter.asu.edu)
to describe local aspects of the pattern of
housing value among U.S. states (darker =
more expensive)
Below avg surrounded by above-avg
Oregon, Arizona, Pennsylvania
Moran’s I = 0.4011, clustering
Hotspot Analysis, Getis-Ord
Video in 3 Parts
• http://bit.ly/cGzF7G
• http://bit.ly/a5342O
• http://bit.ly/9BFnHa
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
See GEO 580 site for direct links
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
• GeoDA (geodacenter.asu.edu)
• 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 541 – Spatio-Temporal Variation in
Ecology & Earth Science