[SOLVED] CS AREAL DATA

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)

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

Examples:
Number of cancer cases in counties (USA)
Number of road accidents in provinces
Proportion of people living in poverty in census tracts

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 js closer to i than those farther away from I

SPATIAL NEIGHBORHOOD MATRIX

Simplest neighborhood definition:
wij = 1 if regions i and j share common boundary
wij = 0 otherwise Customarily, wii = 0

ELEMENTS OF A SPATIAL MODEL

BESAG-YORK-MOLLIE 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

Model includes:

Spatial random effect that smoothes the data according to a neighborhood structure
Unstructured exchangeable component that models uncorrelated noise

BESAG-YORK-MOLLIE MODEL: EXAMPLE
Assume we are interested in observed counts Yi in area i for spatial small area disease risk estimation
Yi Po(Eii), i = 1,, n
log(i) = + ui + vi
v N(0,2) iv
Ei expected counts
i relative risk in area i

BESAG-YORK-MOLLIE MODEL: CAR
ui|uiN u i, (n)
i u = n 1 u j
i iji

i = set of neighbours
ni = number of neighbours over area i
u
2

NEXT TIME
Well put everything weve 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!