Area Objects and Spatial
Autocorrelation
Chapter 7
Geographic Information Analysis
O’Sullivan and Unwin
Types of Area Objects:
Natural Areas
• Boundaries defined by natural phenomena
– Lake, forest, rock outcrop
• Self-defining
• Subjective mapping by surveyor
– Open to uncertainty
• Fussiness of boundaries
• Small unmapped inclusions
• E.g. soil maps
Types of Area Objects:
Fiat or Command Regions
• Boundaries imposed by humans
– Countries, states, census tracts
• Can be misleading sample of underlying social
reality
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–
–
–
Boundaries don’t relate to underlying patterns
Boundaries arbitrary or modifiable
Analyses often artifacts of chosen boundaries (MAUP)
Relationships on macrolevel not always same as
microlevel
Types of Area Objects:
Raster Areas
• Space divided into raster grid
• Area objects are uniform and identical and
tessellate the region
• Data structures on squares, hexagons, or
triangular mesh
Relationships of Areas
• Isolated
• Overlapping
• Completely contained
within each other
• Planar enforced
– Mesh together neatly
and completely cover
study region
– Fundamental
assumption of many
GIS data models
Storing Area Objects
• Complete polygons
– Doesn’t work for planar enforced areas
• Store boundary segments
– Link boundary segments to build areas
– Difficult to transfer data between systems
Geometric Properties of Areas:
Area
• Superficially obvious,
but difficult in practice
• Uses coordinates of
vertices to find areas
of multiple trapezoids
• Raster coded data
– Count pixels and
multiply
Geometric Properties of Areas:
Skeleton
• Internal network of lines
– Each point is equidistant nearest 2 edges of boundary
• Single central point is farthest from boundary
– Representative point object location f area object
Geometric Properties of Areas:
Shape
• Set of relationships of relative position between
point on their perimeters, unaffected by change in
scale
• Difficult to quantify, can relate to known shape
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Compactness ratio = a/a2
Elongation Ratio = L1/L2
Form Ratio = a/L12
Radial Line Index
Geometric Properties of Areas:
Spatial Pattern & Fragmentation
Spatial Pattern
• Patterns of multiple areas
• Evaluated by contact numbers
– No. of areas that share a common boundary with each area
Fragmentation
• Extent to which the spatila pattern is broken up.
– Used commonly in ecology
Spatial Autocorrelation:
Review
• Data from near locations more likely to be
similar than data from distant locations
• Any set of spatial data likely to have
characteristic distance at which it is
correlated with itself
• Samples from spatial data are not truly
random.
Runs on Serial Data
One-Dimensional Autocorrelation
• Is a series likely to have occurred randomly?
• Counts runs of same data and compares Z-scores
using calculated expected values
• Nonfree sampling
– Probabilities change based on previous trials (e.g.
dealing cards)
– Most common in GIS data
• Free sampling
– Probability constant (e.g. flipping coin)
– Math much easier, so used to estimate nonfree sampling
Joins Count
Two-Dimensional Autocorrelation
• Is a spatial pattern likely to have
occurred randomly?
• Count number of possible joins
between neighbors
– Rook’s Case = N-S-E-W neighbors
– Queen’s Case = Adds diagonal neighbors
• Compares Z-scores using expected
values from free sampling
probabilities
• Only works for binary data
Joins Count Statistic
Real World Uses?
• Was the spatial pattern of
2000 Bush-Gore electoral
outcomes random?
• Build an adjacency matrix
(49 x 49)
Join Type
Z-Score
Bush-Bush
3.7930
Gore-Gore
-0.7325
Bush-Gore
-5.0763
Other Measures of Spatial
Autocorrelation
Moran’s I
• Translates nonspatial correlation measures to
spatial context
• Applied to numerical ratio or interval data
• Evaluates summed covariances corrected for
sample size
• I < 0, Negative Autocorrelation
• I > 0, Positive Autocorrelation
n
I=
-2
Σ(yi-y)
- -y)
ΣΣwij(yi-y)(y
j
ΣΣwij
Other Measures of Spatial
Autocorrelation
Geary’s Contiguity Ratio C
• Similar to Moran’s I
• C = 1, No auto correlation
• 0 < C < 1, Positive autocorrelation
• C > 1 Negative autocorrelation
n-1
C=
-2
Σ(yi-y)
ΣΣwij(yi-yj)2
2ΣΣwij
Other Measures of Spatial
Autocorrelation
Weighted Matrices
• Weights can be added to calculations of Moran’s I
or Geary’s C
– e.g. weight state boundaries based on length of borders
Lagged autocorrelation
• weights in the matrix in which nonadjacent spatial
autocorrelation is tested for.
– e.g. CA and UT are neighbors at a lag of 2
Local Indicators of Spatial
Association (LISA)
• Where are the data patterns within the study
region?
• Disaggregate measures of autocorrelation
• Describe extent to which particular areal units are
similar to their neighbors
• Nonstationarity of data
– When clusters of similar values found in specific subregions of study
• Tests: G, I, &C