Geography Reference
In-Depth Information
IDW is an exact interpolator, so the predicted values at locations where there are
observations are the same as at the observed values. h e IDW predictor can be given as:
=
Â
n
-
k
z
()
x
◊
d
i
i
0
ˆ
()
z
x
i
=
1
(9.1)
0
n
Â
-
k
d
i
0
i
=
1
where the prediction is made at the location
x
0
as a function of the
n
neighbouring
observations,
z
(
x
i
),
i
= 1,…,
n
(i.e. we feed only the
n
nearest neighbours into Equation 9.1),
k
is an exponent that determines the weight assigned to each of the observations, and
d
i
0
is the distance by which the prediction location
x
0
and the observation location
x
i
are separated. As noted in Section 4.7, as the exponent becomes larger, the weight
assigned to observations at large distances from the prediction location becomes
smaller. Conversely, as was shown in Figure 4.6, for smaller values of the exponent, the
weights are proportionately larger for more distant observations. h e exponent is usually
set to 2 (i.e.
d
i
-2
)
and the inverse squared distance (where
k
= 2) is obtained with
1
/
d
2
.
h e inverse square of a distance of 6481.996
m, is therefore
1
/
6481.996
2
= 0.00000002380,
as shown in Table 9.1.
In this section, a worked example of IDW is given using four observations, with the
objective of predicting at another location (Figure 9.6 shows the data coni guration).
Since an observation is available at the prediction location
x
0
, but it has been removed
for the present purpose, it is possible to assess the accuracy of the predictions. h e data
are given in Table 9.1. h e same data set is used to illustrate other interpolation
methods in this chapter.
Following the IDW equation we i rst calculate the inverse square of the distances
and then multiply these values by the value of the observations. Table 9.1 shows the
inverse square distances and the observation values multiplied by the inverse square
distances. h e IDW prediction is given by (using the i gures from Table 9.1)
0.000002526
/
0.00000004592
= 55.003. h e 'true' value at the prediction location is 61, so there is a pre-
diction error of 5.997. In practice, assessment of prediction can be conducted using
jackknii ng or cross-validation. Jackknii ng entails splitting the sample into two and
using one set of data to make predictions at the locations represented by the second
data set. h e accuracy of these predictions can obviously be assessed directly. h e
basic idea of cross-validation was described in another context in Section 8.5.3. As
detailed in that section, cross-validation entails removal of an observation, using the
Table 9.1
Precipitation (mm): IDW prediction using observations
x
1
to
x
4
z
(
x
i
)
.
d
i
-
2
i
x
i
y
i
z
(
x
i
)
d
i
0
d
i
-
2
1
292500
329100
68
6481.996
0.00000002380
0.000001618
2
305700
339700
29
10448.860
0.00000000916
0.000000266
3
307629
329826
48
10517.774
0.00000000904
0.000000434
4
287854
345702
53
15969.356
0.00000000392
0.000000208
Sum
0.00000004592
0.000002526
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