Area Objects and Spatial Autocorrelation

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Area Objects and Spatial Autocorrelation

• Chapter 7
• Geographic Information Analysis OSullivan and
  Unwin

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Types of Area ObjectsNatural 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

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Types of Area Objects Fiat or Command Regions

• Boundaries imposed by humans
• Countries, states, census tracts
• Can be misleading sample of underlying social
  reality
• Boundaries dont relate to underlying patterns
• Boundaries arbitrary or modifiable
• Analyses often artifacts of chosen boundaries
  (MAUP)
• Relationships on macrolevel not always same as
  microlevel

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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

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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

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Storing Area Objects

• Complete polygons
• Doesnt work for planar enforced areas
• Store boundary segments
• Link boundary segments to build areas
• Difficult to transfer data between systems

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Geometric Properties of AreasArea

• Superficially obvious, but difficult in practice
• Uses coordinates of vertices to find areas of
  multiple trapezoids
• Raster coded data
• Count pixels and multiply

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Geometric Properties of AreasSkeleton

• 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

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Geometric Properties of AreasShape

• Set of relationships of relative position between
  point on their perimeters, unaffected by change
  in scale
• Difficult to quantify, can relate to known shape
• Compactness ratio ?a/a2
• Elongation Ratio L1/L2
• Form Ratio a/L12
• Radial Line Index

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Geometric Properties of AreasSpatial 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

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Spatial AutocorrelationReview

• 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.

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Runs on Serial DataOne-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

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Joins CountTwo-Dimensional Autocorrelation

• Is a spatial pattern likely to have occurred
  randomly?
• Count number of possible joins between neighbors
• Rooks Case N-S-E-W neighbors
• Queens Case Adds diagonal neighbors
• Compares Z-scores using expected values from free
  sampling probabilities
• Only works for binary data

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Joins Count Statistic Real World Uses?

• Was the spatial pattern of 2000 Bush-Gore
  electoral outcomes random?
• Build an adjacency matrix (49 x 49)

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Other Measures of Spatial Autocorrelation

• Morans I
• Translates nonspatial correlation measures to
  spatial context
• Applied to numerical ratio or interval data
• Evaluates summed covariances corrected for sample
  size
• I lt 0, Negative Autocorrelation
• I gt 0, Positive Autocorrelation

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Other Measures of Spatial Autocorrelation

• Gearys Contiguity Ratio C
• Similar to Morans I
• C 1, No auto correlation
• 0 lt C lt 1, Positive autocorrelation
• C gt 1 Negative autocorrelation

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Other Measures of Spatial Autocorrelation

• Weighted Matrices
• Weights can be added to calculations of Morans I
  or Gearys 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

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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
  sub-regions of study
• Tests G, I, C