Geography Reference
In-Depth Information
scheme is used, observations at smaller distances have larger weights than obser-
vations at larger distances from the location of interest. Some other weighting func-
tions are described later in this topic, but a general summary of distance weighting is
provided below.
h e inverse distance weighting scheme is illustrated now. h e weight for location
i
can be given by
w
ij
, indicating the weight of sample
i
with respect to location
j
.
h e inverse distance weight is given by:
-
k
w
=
(4.3)
ij
ij
which indicates that the weight for location
i
with respect to location
j
is obtained by
raising the distance
d
between locations
i
and
j
(i.e.
d
ij
) to the power -
k
. As noted above
in the case of
k
= 2, this is obtained with
1
/
d
k
. h e inverse distance weighting scheme is
illustrated in Figure 4.6. h e value of the exponent determines the degree of weighting
by distance. With larger exponent values, the weights decline more sharply with dis-
tance, whereas with smaller exponent values distant observations receive, relatively,
larger weights. Note that, with an exponent of zero, all of the weights are equal to one.
An application of the inverse distance weighting scheme is outlined below.
Dif erent forms of weighting scheme have found favour in particular contexts. For
example, the Gaussian weighting scheme described in Section 8.4 has been used for
weighting observations as a part of a method called geographically weighted regres-
sion, which is detailed in Section 8.5.3, while the quartic weighting scheme (see
Section 7.3.2) has been used for point pattern analysis. Inverse distance weighting is
the basis of a spatial interpolation method (a method for predicting values at unsam-
pled locations), which is discussed in Section 9.5 and is illustrated briel y here.
1.0
k
=
1
k
=
2
0.9
k
=
4
k
=
3
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1
2
3
4
5
6
7
8
9
10
Distance
Figure 4.6
Inverse distance weighting scheme for exponents (
k
) of 1, 2, 3, and 4.
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