Geography Reference
In-Depth Information
bandwidth therefore corresponds to a wide 'hump' and a small bandwidth corresponds
to a narrow 'hump. Another widely used weighting scheme is the Gaussian weighting
scheme. As in previous chapters, the weight for location i can be given by w ij , indicating
the weight of sample i with respect to location j . h e Gaussian weighting scheme is
given by (Fotheringham et al. , 2002):
È
2
˘
d
w
=
exp
ij
(8.1)
Í
-
0.5
˙
t
ij
Î
˚
which indicates that the weight for location i with respect to location j is obtained by
multiplying -0.5 by the square of the distance d between locations i and j (i.e. d ij ) divided
by the bandwidth t and then obtaining the exponential value of the product (this can
easily be computed in a spreadsheet package, where 'exp' is the standard abbreviation
for the exponential function). As an example, for a bandwidth of 10 and a distance of 15:
2
È
˘
15
È ˘
2
exp
-
0.5
=
exp[
-
0.5(1.5) ]
=
exp[
-
0.5
¥
2.25
]=
exp[
-
1.125]
=
0.3247
Í
˙
Í Î˚
10
Í
˙
Î
˚
where exp[-1.125] can be obtained with 2.718281828 -1.125 (= 1 / 2.718281828 1.125 ) and
2.718281828 is the approximate base of the natural logarithm (see Wilson and Kirkby
(1980) for more details). Appendix B shows one way to compute the exponential
function.
h e Gaussian weighting scheme, for bandwidths of 5, 10, and 15 units, is illustrated
in Figure 8.1. Note that, unlike the case for the quartic kernel, the bandwidth for the
Gaussian weighting scheme does not extend to the outer edge of the kernel, but of
course the bandwidth still determines the kernel size.
Any standard statistic can be geographically weighted (see Fotheringham et al.
(2002) for more information). As an example of this geographical weighting scheme
in practice, obtaining the locally weighted mean using the Gaussian weighting scheme
is illustrated below. Following Section 4.7, the locally weighted mean is given by:
n
= Â
Â
zw
j
ij
j
=
1
z
(8.2)
i
n
w
ij
j
=
1
Table 8.1 and Figure 8.2 detail the locations of a set of observations which are the
same as those used in Section 4.7. However, in this case, the i rst observation is treated
as known. h e weights, obtained using the Gaussian weighting scheme detailed above
(with bandwidths of 5, 10, and 15 units), are given for each distance. h e weights for
each location are then multiplied by the value at that location. As an example, follow-
ing Equation 8.1 for a bandwidth of 10 the products of the multiplications are summed,
giving a value of 136.733. h e weight values are also summed, giving a value of 6.643.
h e weighted mean is then obtained by 136.733 / 6.643 = 20.582. Given the 20 observations,
the unweighted mean is 19.050.
 
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