Geography Reference
In-Depth Information
where
K
is the
n
+ 1 ¥
n
+ 1 (with
n
nearest neighbours used for prediction) matrix of
semivariances between each of the observations:
È
g
(
xx
-
)
g
(
xx
-
)
1
˘
1
1
1
n
Í
˙
Í
˙
K
=
Í
˙
g
(
xx
-
)
g
(
xx
-
)
1
n
1
n
n
Í
˙
1
1
0
Í
˙
Î
˚
l
are the OK weights and
k
are semivariances for the observations to the prediction
location (with one placed in the bottom position):
È˘
l
È
g
(
xx
-
)
˘
1
1
0
Í˙
Í
˙
Í˙
Í
˙
l
=
k
=
Í˙
Í
˙
g
(
xx
-
)
l
y
n
0
Í˙
n
Í
˙
1
Í˙
Í
˙
Î
˚
Î˚
To obtain the OK weights, the inverse of the data semivariance matrix is multiplied
by the vector of data to prediction semivariances:
l
=
Kk
-
(9.17)
h e OK variance is then obtained from:
T
s=
k l
2
OK
(9.18)
Using the same data as for the example in Sections 9.5 (IDW) and 9.6 (TPS), the OK
system is given as:
0
376.905
359.589
379.853
1
l
l
l
l
y
268.116
È
˘
È
˘
È
˘
1
Í
˙
Í
˙
Í
˙
376.905
0
307.108
394.601
1
311.250
Í
˙
Í
˙
Í
˙
2
Í
˙
Í
˙
Í
˙
359.589
307.108
0
448.401
1
311.983
¥
=
3
Í
˙
Í
˙
Í
˙
379.853
394.601
448.401
0
1
367.662
Í
˙
Í
˙
Í
˙
4
Í
˙
Í
˙
Í
˙
1
1
1
1
0
1
Î
˚
Î
˚
Î
˚
Note that the semivariance between a given location and itself is set to 0.
h is account does not show how the weights are obtained. To see how this is done
(i.e. to see how the OK system is solved) go to Appendix F, where the same example is
given (but exactly how the system is solved is shown).
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