Geography Reference
In-Depth Information
8.5.3
Geographically weighted regression
h is topic has outlined various geographically weighted statistics and argued that such
approaches are intuitively sensible since we expect places close together to be more
alike than places a greater distance apart. Geographically weighted regression (GWR)
extends the same principle to regression analysis. GWR has become a core tool in
many analyses of spatial data. h e approach is described in detail by Fotheringham
et al.
(2002) but an account of some key principles is given here. Essentially, the key
steps in a GWR analysis are as follows:
1.
Go to a location (zone or point).
2.
Conduct regression using all data (or some subset) but give greater weight
(inl uence) to locations that are close to the location of interest—a geographical
weighting scheme is used.
Move to the next location and go back to stage 2 until all locations have been
3.
visited.
h e output is a set of regression coei cients (e.g. for bivariate regression (with one
independent and one dependent variable), the intercept, and slope) at each location.
GWR coei cients are obtained using:
T
-
1
Y Wx z
T
(8.5)
b
()(
x
=
Y Wx Y
())
()
i
i
i
h is is the same as for ordinary unweighted regression (Equation 3.9), except that the
regression coei cients are computed for each location
x
i
and there are geographical
weights given by
W
(
x
i
). If all weights were equal to 1 then this would correspond to
standard unweighted regression. h e weights matrix is given by:
È
w
000
˘
i
1
Í
˙
0
w
0
0
Í
˙
i
2
Wx
()
=
Í
i
˙
00 0
000
Í
˙
w
Í
˙
Î
˚
in
w
i
1
is the weight given the distance between the location
i
and observation 1. h e
diagonal dots (
) indicate that the matrix can be expanded—that is, if
n
(the number
of observations) is 5 then the matrix will have 5 ¥ 5 entries, with non-zero values only
in the diagonal of the matrix.
One weighting scheme used widely in GWR contexts is the Gaussian weighting
scheme, detailed in Equation 8.1. Note that MWR is a special case of GWR where the
weights for the
n
nearest neighbours are set to 1 and all other weights are 0.
GWR is illustrated using the data listed in Table 8.3 and mapped in Figure 8.5. Table 8.3
gives the coordinates of the observations, variable 1 (independent) and variable 2 (depen-
dent) values, distance from the i rst observation, and geographical weights ('Geog. wt.';
using the Gaussian weighting function) for a bandwidth of 10 units. h e computation
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