Geography Reference
In-Depth Information
To obtain the TPS weights (
l
i
) and the values of
a
0
,
a
1
, and
a
2
, the inverse (see Section
3.3 and Appendices E and F for a discussion about matrix inversion) of the matrix
R
is
multiplied by the vector of data values,
z
:
-
=
Rz
l
Using the same data as for the IDW example in Section 9.5, following Equation 9.5
the TPS system is given as:
È
0
352.128
270.779
367.512
1
292.500
329.100
˘
È
l
l
l
l
˘
È
68
29
48
53
0
0
˘
1
Í
˙
Í
˙
Í
˙
352.128
0
101.483
451.925
1
305.700
339.700
Í
˙
Í
˙
Í
˙
2
Í
˙
Í
˙
Í
˙
270.779
101.483
0
902.999
1
307.629
329.826
3
Í
˙
Í
˙
Í
˙
367.512
451.925
902.999
0
1
287.854
345.702
¥
=
Í
˙
Í
˙
Í
˙
4
Í
˙
Í
˙
Í
˙
1
1
1
1
0
0
0
a
a
a
0
Í
˙
Í
˙
Í
˙
292.500
305.700
307.629
287.854
0
0
0
Í
˙
Í
˙
Í
˙
1
Í
˙
Í
˙
Í
˙
329.100
339.700
329.826
345.702
0
0
0
0
Î
˚
Î
˚
Î
˚
2
Note that the distances were divided by 1000 prior to calculating the values for
R
. h is
gives the same results but reduces the size of the values obtained for
R
(
x
-
x
i
) and
makes the process more manageable. h e diagonals in the matrix
R
are all 0 and
they indicate the distance between an observation and itself (obviously 0); where
a smoothing parameter is used (and the spline function is not forced to i t to the data),
the smoothing parameter value is added to the diagonals (for this example, the i rst
four 0 components, reading from the let ), as described by Lloyd (2006).
Solving the TPS system, the weights are as follows:
l
1
= -0.0319,
l
2
= -0.0493,
l
3
= 0.0520,
l
4
= 0.0292,
a
0
= 1078.474,
a
1
= -1.5906, and
a
2
= -1.6794.
We then put the distances between each observation and the prediction location
into the equation
d
2
log
d
. For each observation this gives the following values:
x
1
= 34.105,
x
2
= 111.261,
x
3
= 113.049, and
x
4
= 306.863.
h e predicted value is then given by multiplying the weights (
l
i
) by the basis
function values (
R
(
x
-
x
i
)), adding
a
0
(the constant, note that the intercept is also called
the constant), and multiplying the trend coei cients (
a
1
and
a
2
) by the coordinates
(
x
and
y
). In this case this leads to: (34.105 ¥ -0.0319) + (111.261 ¥ -0.0493) + (113.049 ¥
0.0520) + (306.863 ¥ 0.0292) + 1078.474 + (297.624 ¥ -1.5906) + (333.070 ¥ -1.6794) =
60.569.
h e 'true' value is 61 and the TPS prediction error is smaller than the IDW prediction
error (with an IDW prediction of 55.003; see Section 9.5).
Figure 9.8 shows a map of precipitation amount generated using TPS with the same
data used to illustrate the application of IDW (see Figure 9.7). Lloyd (2006) gives a
summary account of variants of the TPS approach, which may provide more robust
and more accurate predictions than standard TPS in some circumstances.
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