Geography Reference
In-Depth Information
Table 8.4
Calculations for the geographically we
ig
hted coeffi cient of
determination (data as for Table 8.3), with mean (
z
) of 33
ˆ
j
z
ˆ
w
(
zz
2
-
)
w
(
zz
2
-
)
Obs. (j)
ij
j
ij
j
j
1
729.000
15.649
93.103
2
359.195
46.581
29.219
3
56.480
43.769
6.766
4
46.470
15.649
63.491
5
33.638
14.243
16.728
6
51.206
18.461
7.957
7
3.400
77.513
40.499
8
6.936
104.227
8.315
9
0.000
59.235
0.148
Sum
1286.326
266.227
Obs., observation.
interest is benei cial in this case. Again, it should be stressed that the number of obser-
vations is small for this example and this should be considered in any analysis.
h e GWR sot ware of ers two approaches to assessing the signii cance of the GWR
model. In essence, these approaches allow users to determine if any of the local param-
eter estimates are 'signii cantly non-stationary' (Fotheringham
et al.
, 2002, p. 213),
where a non-stationary model is one which has parameters that vary geographically.
In other words, the test enables assessment of the degree to which the GWR model
is an improvement over a standard global model.
To illustrate GWR, data on elevation and precipitation amount in Northern Ireland
for July 2006 (with data at 149 locations) are analysed. h e data locations are shown in
Figure 8.7 and an interpolated (see Section 9.2) map of precipitation amounts is given
in Figure 9.7. Figure 8.8 is a scatter plot for all observations (it is a global regression)
with a i tted regression line. h is indicates that the expected precipitation amount at a
location with an altitude of 0 m is 52.203 mm (i.e. the intercept) and that this increases
on average by 0.1162 mm (the slope) with an increase in altitude of 1 m. While the
coei cient of determination (
r
2
) indicates that the model explains some 42% of the
variation, there is clearly much variation around the regression line and GWR allows
for exploration of local variations in this relationship.
A GWR bandwidth can be selected using cross-validation. Cross-validation entails
removal of an observation, regression conducted using the remaining observations,
and prediction of the removed value—that is, the regression equation is used to pre-
dict the value of
z
(the dependent) given a value of
y
(the independent), as described
in Section 3.3. h e removed value is then added back and the observation removed at
the next location (in whatever order the locations are visited) at er which the proce-
dure is repeated for all remaining observations. h e dif erence between the observed
and predicted values is then computed. h e bandwidth that results in the smallest
cross-validation error is retained. An additional method for bandwidth selection,
which is employed by the GWR sot ware of Fotheringham
et al.
(2002), is called the
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