Geography Reference
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where
1
is an n-by-1 vector of ones, and T denotes the matrix transpose operator.
The decomposition generates n eigenvectors and their associated n eigenvalues.
In descending order, the n eigenvalues, usually are denoted by
œ D
(œ
1
, œ
2
,
œ
3
, :::, œ
n
), where the subscript represents descending order of value. These
eigenvalues quantify the value of the Moran Coefficient for their corresponding n-
by-1 eigenvectors, which usually are denoted by
E
D
(
E
1
,
E
2
,
E
3
, :::,
E
n
)-i.e.,
MC
j
D
n
1
T
C1
œ
j
,for
E
j
.
The eigenfunctions have some important properties: the eigenvectors are mutu-
ally orthogonal and uncorrelated (Griffith
2000
); the eigenvectors portray distinct
map patterns exhibiting a specified level of spatial autocorrelation when mapped
onto the n areal units associated with the corresponding spatial weight matrix
C
(Tiefelsdorf and Boots
1995
); and, given a spatial weight matrix
C
, the feasible
range of the Moran Coefficient values is determined by the largest and smallest
eigenvalues, œ
1
and œ
n
(de Jong et al.
1984
). Based upon these properties, the
eigenvectors can be interpreted as (Griffith
2003
) follows:
The first eigenvector,
E
1
, is the set of real numbers that has the largest MC value achievable
by any set of real numbers for the spatial arrangement defined by the spatial weight matrix
C
; the second eigenvector,
E
2,
is the set of real numbers that has the largest achievable MC
value by any set that is uncorrelated with
E
1
; the third eigenvector,
E
3,
is the set of real
numbers that has the largest achievable MC value by any set that is uncorrelated with both
E
1
and
E
2
; the fourth eigenvector is the fourth such set of values; and so on through
E
n
,
the set of real numbers that has the largest negative MC value achievable by any set that is
uncorrelated with the preceding (n
1) eigenvectors.
Frequently the eigenfunctions of interest relate to positive spatial autocorrelation,
with a threshold value of MC
j
/MC
max
> 0.25 indicating noticeable positive spatial
autocorrelation.
Griffith (
2004
) formulates an eigenfunction spatial filter binomial regression
model.
9.2.2
Random Effects
A random effects term accounts for within-areal-unit heterogeneity, and is time
invariant for space-time data. Spatially structured random effects models are
increasing in popularity (see, for example, Griffith and Paelinck
2011
; Hughes and
Haran
2012
), partially because they can be conceptualized in terms of eigenvector
spatial filtering, and partially because they have become estimable. One common
specification is for the intercept term to be written as a random effects term, resulting
in it representing variability about the conventional single-value, constant mean.
One role of a random effects term when analyzing space-time data is to account
for correlation between repeated measures in time for each areal unit. With n very
short time series, each constituting a non-random sample, a random effects term
represents a common constant across time that introduces correlation among values.
In other words, it captures temporal correlation effects.
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