Areal data

Birth counts 1974-78, North Carolina counties

Elevation at raster grid cells covering Luxembourg

Cellular neighbourhood in colorectal cancer tissue

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We don’t always observe data at points, but summarised for areas.

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• $D$ is a fixed countable collection of areal units at which variables are observed
  • Fixed = set of regions decided a priori and not changed during analysis
• Information aggregated over fixed, non-overlapping geographic regions, rather than at specific points
• Examples
  • Number of individuals with a certain disease in provinces of a country
  • Number of trees in a mixed plantation
  • Average electricity consumption in districts of a city
• Regular lattice
  • Structured as a matrix in rows and columns
  • Areas are all of the same size equal to the cells of a raster grid
  • Spatially inhomogeneous distribution of the events

Mapping of cells

Spatial distribution of trees in a rain forest

• Irregular lattice

  • Real-life lattice data are seldom in a regular grid
  • Areas different in size and shape
  • Different number of neighbours

  

  Sudden Infant Death Syndrome (SIDS) in North Carolina, 1974-1978

Spatial neighbourhood

• Areal data → spatial autocorrelation
  • Do close areas have similar or dissimilar values?
  • Do neighbouring counties tend to have similar disease rates?
  • Do adjacent forest blocks have similar tree densities?
  • Do nearby districts have similar unemployment rates?
• Spatial neighbours have different definitions

Contiguity neighbour Queen: single shared boundary point

Contiguity neighbour Rook: more than one shared boundary point

k-nearest neighbour: example of k = 3

Distance-based neighbour: example of 0.4

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When is k-nearest or distance-based neighbour more useful than Queen or Rook?

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Contiguity Rook and Queen neighbours of k orders: example of k = 1 (dark grey) and k = 2 **(light grey)

Neighbourhood matrices