Spatial Statistics
PM599, Fall 2013
Lecture 5: October 7, 2013
1 / 45
Areal Data
Link between geostatistical/point referenced and areal data
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For geostatistical/point referenced data, we use functions of
distance to estimate the variogram/covariance that defines
spatial relationships
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Geostatistical prediction involves using fitted covariance
functions (kriging), spatial interpolation, or basis spline
smoothing
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For areal data (lattices), we use neighbour information to
define spatial relationships
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Areal Data
Link between geostatistical/point referenced and areal data
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In general, areal units are irregular (e.g. zip code, county) but
methods may also apply to regular grids
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We care about how areal units connect to each other
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We will see some analogies between geostatistical data and
areal data. Sometimes geostatistical methods are used for
areal data prediction, but autoregressive models employing
neighbourhood information are more commonly used
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We will use the R package spdep() for areal data analysis
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Areal Data
Inferential issues with areal data
Is there a spatial pattern?
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Spatial pattern suggest that areal observations close to each
other have more similar values than those far from each other.
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You might think that there is a pattern through visualization,
but this is often subjective.
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Independent measurements will have no pattern, and would
look completely random, but there may actually be an
underlying pattern.
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If there is a spatial pattern, how strong is it?
4 / 45
Areal Data
Misrepresentation with maps
Crude birth rates by state based on equal-interval cut points
Figure: Monomier, N. Lying with Maps. Statistical Science 2005, 20(3)
215222.
5 / 45
Areal Data
Misrepresentation with maps
Crude birth rates by state based on quantile cut points
Figure: Monomier, N. Lying with Maps. Statistical Science 2005, 20(3)
215222.
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Areal Data
Is there a spatial pattern?
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Response of interest Yi measured in block or areal unit Bi
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The Bi are supplemented with neighbourhood information
(distance between Bi and Bj , area of Bi , boundary/edge
connections)
Areal data analysis involves:
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Representation of spatial proximity in areal data using
weighted graphs
Testing for spatial pattern: Global testing using Morans I or
Gearys C statistic
Testing for spatial pattern: Local testing using local Moran’s I
or Getis-Ord G∗ statistic
Modeling spatial pattern for prediction and inference:
autoregressive models (Simultaneous Autoregressive (SAR)
models and Conditional Autoregressive (CAR) models
7 / 45
Areal Data
Areal Data Example: SIDS
37
38
Sudden Infant Deaths in North Carolina
Ashe
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Watauga
Madison
Swain
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Graham
Cherokee
Clay
Macon
Avery
Mitchell
Yancey
Alleghany
Wilkes
Caldwell Alexander
Surry
Stokes
Yadkin
Forsyth
Rockingham Caswell Person
Northampton
Vance Warren
Granville
Bertie
Nash
Edgecombe
Iredell
Davidson
Wake
Burke
Randolph
Wilson
Chatham
McDowell
Catawba
Rowan
Buncombe
Haywood
Johnston
Greene
Lincoln
Lee
Rutherford
Cabarrus
Harnett
Wayne
Henderson
Cleveland Gaston
StanlyMontgomery Moore
Jackson
Polk
Mecklenburg
Lenoir
Transylvania
Union
Anson Richmond
Hoke
Cumberland
Sampson
Duplin
Martin
Pitt
34
Columbus
Camden
Currituck
Pasquotank
Perquimans
Chowan
Washington Tyrrell
Beaufort
Dare
Hyde
Pamlico
Jones
Onslow
Bladen
Gates
Craven
Scotland
Robeson
Hertford
Halifax
Franklin
Guilford Alamance
OrangeDurham
Davie
Carteret
Pender
New Hanover
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Brunswick
-84
-82
-80
-78
-76
8 / 45
Areal Data
Areal Data Example: SIDS
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Data for 100 counties in North Carolina
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Includes counts of live births and sudden infant deaths for two
periods: July 1974-June 1978 and July 1979 to June 1984.
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SIDS is defined as sudden death of infant up to 12 months old.
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Risk factors include race, SES, physiologic (respiratory, sleep
rate, cardiac function)
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The primary analysis here is not only to see how often SIDS
occurs, but where and if there are clusters or spatial patterns.
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Areal Data
Example Areal Data: SIDS
SIDS74 Counts
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40
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30
25
20
15
10
5
0
10 / 45
Areal Data
Flowchart
How we analyze areal data
11 / 45
Areal Data
Proximity
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We represent proximity between areal units (blocks, Bi ) using
connected graphs
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Adjacency matrix (proximity matrix) is denoted W
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The entries of W are wij and are called weights
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The wij connect different values of the process Y1 , ..., Yn ,
i = 1, ..., n in some fashion
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Generally wii is set to zero
12 / 45
Areal Data
Proximity
Examples of weights
1. Border based (edge connections): areal units are neighbours if
they share a border
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wij = 1 if i and j share common boundary
2. Distance based: areal units are neighbours if they are within a
distance of of each other
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wij = 1 if the centroid of i is distance (ex. 25km) of the
centroid of j
wij = 1 if j is the nearest neighbour (smallest ) of i
wij = 1 if j is one of the k nearest neighbours of i, e.g. the two
and three closest areal units j to i are the k=2 and k=3 nearest
neighbors of i. This will result in multiple neighbours for each i
13 / 45
Areal Data
Proximity
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Distance can be defined several ways:
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Euclidean distance (or driving distance or driving time, etc)
between centroids (straight line path)
Mean driving distance, mean driving time, walking distance,
etc. (transit path, not necessarily a straight line)
The connections between blocks under proximity by k nearest
neighbour or distance neighbour can be examined using a
connected graph
14 / 45
Areal Data
Distance Neighbours
Counties connected if 30km or less apart
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Areal Data
Nearest Neighbours (1)
Counties each connected to its nearest neighbour
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Areal Data
Nearest Neighbours (2)
Counties each connected to its 2 nearest neighbours
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17 / 45
Areal Data
Proximity
Border/Edge based, binary connectivity. Two areal
units are neighbours if they share a border
wij =
1
0
if i and j share a boundary
otherwise
Where wij = wji (symmetric)
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Areal Data
Border/Edge Connectivity
Queen a single shared boundary point means they are neighbours.
Rook requires more than a single shared point to constitute
neighbours.
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Areal Data
Border/Edge Connectivity: Queen
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Areal Data
Border/Edge Connectivity: Rook
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Areal Data
Border/Edge Connectivity: Difference Queen-Rook
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22 / 45
Areal Data
Proximity
Fractional borders
(
wij =
lij
li
0
if regions i and j share a border
otherwise
Where l is the length of the common border
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Areal Data
Proximity
Neighbour Based
wij =
1
0
if centroid of j is a k nearest neighbour of i
otherwise
Where wij and wji not necessarily symmetric
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Areal Data
Proximity
k Nearest Neighbours (kNN)
k=2
k=3
k=1
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Areal Data
Proximity
Neighbour Based: 1NN
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Areal Data
Proximity
Neighbour Based: 2NN
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Areal Data
Proximity
Neighbour Based: Difference Between 1NN and 2NN
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28 / 45
Areal Data
Proximity
Distance Based
wij =
1
0
if dij < otherwise
For some specified distance threshold Alternatively,
−ρ
dij
if ρ > 0
wij =
0
otherwise
For some power, ρ (recall idw)
29 / 45
Areal Data
Proximity
Distance based neighbours, dist=1*max_k1
dist=1.5*max_k1
30 / 45
Areal Data
Proximity
Distance based neighbours, between 1 and 1.5 times
maximum kNN distance
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Areal Data
Proximity
Distance based neighbours, between 10 and 30km
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32 / 45
Areal Data
Adjacency Creating the adjacency matrix from connectivity
graphs
33 / 45
Areal Data
Neighbours
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2
3
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5
6
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2
1
2
3
1
5
6
5
3
4
5
4
7
5
6
7
34 / 45
Areal Data
Weights: Adjacency Matrix
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The adjacency matrix, W is a matrix of neighbours where
elements are weights wij
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Once our list of neighbours (fixed distance or kNN) has been
created, we assign spatial weights to each relationship
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Can be binary or variable
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Even when the values are binary 0/1, there is the issue of
what to do with no-neighbour observations arises
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Binary weighting will assign a value of 1 to neighboring
features and 0 to all other features
35 / 45
Areal Data
Weights: Adjacency Matrix
Binary weights
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Binary weights vary the influence of observations
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Those with many neighbours are up-weighted compared to
those with few
0
0
1
0
1
0
1
1
0
1
0
1
0
1
0
1
36 / 45
Areal Data
Weights: Adjacency Matrix
Row standardization is used to create proportional weights in cases
where features have an unequal number of neighbors
I Row-standardized weights increase the influence of links from
observations with few neighbours
I Divide each neighbour weight for a feature by the sum of all
neighbour weights
I Obs i has 3 neighbours, each has a weight of 1/3
I Obs j has 2 neighbours, each has a weight of 1/2
I Use is you want comparable spatial parameters across
different data sets with different connectivity structures
0
0
0.5
0
1
0
0.5
0.33
0
0.5
0
0.33
0
0.5
0
0.33
37 / 45
Areal Data
Binary weight matrix
0
1
0
0
1
0
0
1
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
0
0
1
0
0
1
0
1
1
0
0
0
0
1
0
1
0
0
0
0
1
1
0
38 / 45
Areal Data
Row standardized weight matrix
0
0.5
0
0
0.25
0
0
0.5
0
0.5
0
0
0
0
0
0.5
0
0.5
0
0
0
0
0
0.5
0
0.25
0
0
0.5
0
0
0.5
0
0.5
0.5
0
0
0
0
0.25
0
0.5
0
0
0
0
0.25
0.5
0
39 / 45
Areal Data
Spatial Smoothers
We can use the block values and weight matrices to obtain a
smooth value for each region by taking locally weighted averages
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If we have a measure of Yi , such as the SIDS rate in county i,
we can get a rough estimate of what it could be predicted as
from it’s j neighbours
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Essentially, we replace Yi with Ŷi where
Ŷi = P
1 X
wij Yj
j wij
j
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The ”new” Ŷi is a function of it’s spatial neighbours j
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This smooths things out because the areal units look more
like their neighbours
40 / 45
Areal Data
Spatial Similarity
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We want to summarize similarity between nearby areal units
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Spatial autocorrelation is the the correlation of the same
measurement taken at different areal units
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The similarity of values at locations Bi and Bj are weighted by
the proximity of i and j
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The weight wij defines proximity
41 / 45
Areal Data
Spatial Association
Measuring strength of association
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We want to measure how strong observations from nearby
areal units are more or less alike than those that are farther
apart
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We also want to decide whether the similarity (or
dissimilarity) is strong enough that it is not due to chance
For example:
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I
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Let Yi be the response at the ith areal unit, Bi and Yj be the
response at the jth areal unit, Bj
Let simij be a measure of how similar (or dissimilar) the
responses are at areal units Bi and Bj
Let wij be a measure of the spatial proximity between areal
units Bi and Bj
We can define a general statistic by the cross product of the
simij matrix and wij matrix
42 / 45
Areal Data
Spatial Association
Example con’t:
Y=
0
0
1
1
0
1
0
1
0
define simij = (Yi − Yj )2 and
1
if i and j share a boundary
wij =
0
otherwise
43 / 45
Areal Data
Spatial Association
Example con’t:
W=
0
1
0
1
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
1
0
0
0
1
0
1
0
1
0
1
0
0
0
1
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
1
0
1
0
FindPpairwise
similarity from Y and take the cross product to get
P
C= i j wij simij
44 / 45
Areal Data
Spatial Association
Example con’t
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If C is small that means similarity between neighbours is high
and we have positive spatial autocorrelation
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If C is large that means there is little similarity between
neighbours
Measuring strength of association
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In general the extent of similarity is represented by the
weighted average of similarity between areal units:
N P
N
P
wij simij
i=1 j=1
N P
N
P
wij
i=1 j=1
45 / 45