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– 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

– 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 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

Joins CountTwo-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

Σ(yi-y)2-nI =

ΣΣwij

ΣΣwij(yi-y)(yj-y)--

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

Σ(yi-y)2-n-1C =

2ΣΣwij

ΣΣwij(yi-yj)2

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 sub-regions of study

• Tests: G, I, &C