Geography Reference
In-Depth Information
n
Â
Iz wzj
=
,
π
i
(8.3)
i
i
ij
j
j
=
1
where
z
j
are dif erences of variable
y
from its global mean (
i
y
- . In cases where
zones are used (as opposed to points) the weights,
w
ij
, are ot en set to 1 for immediate
neighbours of a zone and 0 for all other zones. Local
I
ot en appears in modii ed
form:
)
È
˘
z
n
Í
Î˚
Â
I
=
i
w z
,
j
π
i
(8.4)
s
i
ij
j
2
j
=
1
where
s
2
is the sample variance (the square of Equation 3.3). Note that local
I
values
sum up to global Moran's
I
. Anselin (1995) describes an approach to testing for signii -
cant local autocorrelation based on random relocation of the data values, the objective
being to assess if the observed coni guration of values is signii cant. h e GeoDa sot -
ware of ers the capacity to test the signii cance of local
I
using randomization
1
and to
map signii cant clusters. Clusters are identii ed using the Moran scatter plot (Anselin,
1996).
Local
I
is demonstrated following Equation 8.4 using the following grid:
45
44
44
43
42
39
38
32
34
Local
I
is computed for the central cell and rook's case weighting is used—only the
cells in the same row or column are used. h e mean of values in the entire data set is
needed i rst. Here there are only nine values and their mean is 40.111 and the sample
variance is 21.861. Table 8.2 shows the original values (
y
j
), their deviations from
the mean (
z
j
), the weights (
w
ij
), and weights multiplied by the deviations from the
mean (
w
ij
z
j
).
In this case the weights are row standardized, i.e. they sum to 1 (there are four values
of 0.25 and these are for the four cells which share an edge with the central cell, which
has a value of 42).
z
i
is 1.889, the sum of the weights multiplied by the deviations from
the mean (
w
ij
z
j
) is -0.611, as shown in Table 8.2.
I
i
is then given by (
1.889
/
21.861
) ¥ -0.611 =
-0.053. In this case,
I
i
has a negative value, indicating negative spatial autocorrelation,
i.e. neighbouring values tend to be dissimilar.
Figure 8.3 gives a map of the log of the number of persons per hectare in Northern
Ireland in 2001; the values were logged as the raw population densities had a positively
skewed distribution and the transformed values have almost zero skew. Figure 8.4
gives an example of the application of
I
i
for measuring spatial autocorrelation (using
queen's case contiguity; the use of contiguity schemes with non-gridded data was
outlined in Section 4.8) in logged population density given the data shown in
1 see https://www.geoda.uiuc.edu/support/help/glossary.html
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