AREAL DATA
STA465: Theory and Methods for Complex Spatial Data
Instructor: Dr. Vianey Leos Barajas
ANNOUNCEMENTS
Hwk 1 Solution is online
Hwk 2 is posted and due on Feb 22nd by 11:59 pm EST Reading week: February 15-19 (no class)
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WHAT IS AREAL DATA?
Areal or lattice data — when a fixed domain is partitioned into a finite number of subregions.
Defined on a finite or countable subset in space — e.g. grid nodes, pixels, polygons, small areas
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Examples:
Number of cancer cases in counties (USA)
Number of road accidents in provinces
Proportion of people living in poverty in census tracts
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EXAMPLE DATA
SPATIAL NEIGHBOURS — ADJACENCY
SPATIAL NEIGHBOURS — DISTANCE
SPATIAL NEIGHBORHOOD MATRIX
Spatial neighborhood matrix: W
(i,j)th element of spatial neighborhood denoted by wi,j
spatially connects areas i and j
Elements can be viewed as ‘weights’
More weight is associated with j’s closer to i than those farther away from I
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SPATIAL NEIGHBORHOOD MATRIX
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Simplest neighborhood definition:
wij = 1 if regions i and j share common boundary
wij = 0 otherwise Customarily, wii = 0
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ELEMENTS OF A SPATIAL MODEL
BESAG-YORK-MOLLIÉ MODEL
Takes into account that data may be spatially correlated
Observations in neighbouring areas may be more similar than observations in areas that are farther away
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Model includes:
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Spatial random effect that smoothes the data according to a neighborhood structure
Unstructured exchangeable component that models uncorrelated noise
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BESAG-YORK-MOLLIÉ MODEL: EXAMPLE
Assume we are interested in observed counts Yi in area i for spatial small area disease risk estimation
Yi ∼ Po(Eiθi), i = 1,…, n
log(θi) = α + ui + vi
v ∼ N(0,σ2) iv
Ei — expected counts
θi — relative risk in area i
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BESAG-YORK-MOLLIÉ MODEL: CAR
➤ui|u−i∼N u ̄δi, (n)
δi u ̄ δ = n − 1 ∑ u j
➤i δij∈δi
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δi = set of neighbours
nδi = number of neighbours over area i
σ u
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NEXT TIME
We’ll put everything we’ve learned together:
Simulate from model with fixed values for parameters
Simulate from prior predictive
Fit the model
Simulation from posterior predictive + Maps along the way!
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