Geography Reference
In-Depth Information
Any standard statistic can be geographically weighted (see Fotheringham
et al.
(2002) for more information). As an example of a geographical weighting scheme in
practice, obtaining the locally weighted mean using inverse distance is illustrated
below. h e locally weighted mean is given by:
=
Â
Â
n
zw
j
ij
j
=
1
z
(4.4)
i
n
w
ij
j
=
1
Recall from Equation 3.3 that
z
indicates the mean of
z
, so
z
indicates the mean
at location
i
;
w
ij
indicates, as before, the weight for the distance between observations
i
and
j
. In the case where all the weights are one, Equation 4.4 corresponds to the stan-
dard mean average (it is then the sum of the values divided by the number of
observations).
In Table 4.2, a set of observations (which can be treated as measurements of pre-
cipitation in millimetres for illustrative purposes) and their distance from a i xed loca-
tion are given. In this case, the value at the i xed location is unknown and it will be
Table 4.2
Observations (
j
), distance from observation 1 (
d
ij
), weights (
w
ij
),
and weights multiplied by values (
z
j
w
ij
)
k
=
1
k
=
2
k
=
3
j
d
ij
z
j
w
ij
z
j
w
ij
w
ij
z
j
w
ij
w
ij
z
j
w
ij
1
4.404
14
0.227
3.179
0.052
0.722
0.012
0.164
2
9.699
43
0.103
4.434
0.011
0.457
0.001
0.047
3
10.408
12
0.096
1.153
0.009
0.111
0.001
0.011
4
10.871
34
0.092
3.127
0.008
0.288
0.001
0.026
5
12.958
26
0.077
2.007
0.006
0.155
0.000
0.012
6
13.959
24
0.072
1.719
0.005
0.123
0.000
0.009
7
14.066
33
0.071
2.346
0.005
0.167
0.000
0.012
8
15.506
34
0.064
2.193
0.004
0.141
0.000
0.009
9
17.256
10
0.058
0.579
0.003
0.034
0.000
0.002
10
17.606
8
0.057
0.454
0.003
0.026
0.000
0.001
11
18.018
13
0.055
0.721
0.003
0.040
0.000
0.002
12
18.025
11
0.055
0.610
0.003
0.034
0.000
0.002
13
18.285
24
0.055
1.313
0.003
0.072
0.000
0.004
14
19.253
9
0.052
0.467
0.003
0.024
0.000
0.001
15
21.335
15
0.047
0.703
0.002
0.033
0.000
0.002
16
23.845
14
0.042
0.587
0.002
0.025
0.000
0.001
17
23.988
34
0.042
1.417
0.002
0.059
0.000
0.002
18
24.464
3
0.041
0.123
0.002
0.005
0.000
0.000
19
24.522
11
0.041
0.449
0.002
0.018
0.000
0.001
Sum
372
1.347
27.582
0.128
2.533
0.017
0.309
Mean
19.579
20.474
19.844
17.804
Weights are obtained using the inverse distance weighting scheme with exponents of 1, 2, and 3.
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