Geography Reference
In-Depth Information
h e distance
d
is that between the location prediction
x
and the location
x
i
, so
R
(
x
-
x
i
)
is, in this case, the distance between those two locations fed into Equation 9.4. As an
example, for a distance of 16.9292646 units:
dd
=
2
log
16.9292646 log16.9292646
2
=
352.128
i
i
In short, the TPS function comprises the local trend and weights (
l
i
) by which the
basis function values are multiplied (the process is illustrated below). Using matrix
notation, the coei cients
a
k
and
l
i
are the solution of:
l
Rz
(9.5)
Appendix F shows how to solve such equations (i.e. how to i nd the unknown values,
which in this case are the coei cients
a
k
and
l
i
).
R
is a matrix obtained by feeding the distances between local observations into the
equation
d
2
log
d
. As above, a given distance is squared and multiplied by the log of the
distances. h e matrix
R
is given by:
È
R
(
xx
-
)
R
(
xx
-
)
1
x
y
˘
1
1
1
n
1
1
Í
˙
Í
˙
Í
˙
R
(
xx
-
)
R
(
xx
-
)
1
x
y
n
1
n
n
n
n
R
= Í
˙
1
1
0 0 0
00 0
00 0
Í
˙
Í
˙
x
x
1
n
Í
˙
y
y
Í
˙
Î
˚
1
n
l
are the TPS weights and
z
are the observations:
È˘
l
z
()
x
È
˘
1
1
Í˙
Í
˙
Í˙
Í
˙
Í˙
Í
z
()
0
0
0
x
˙
l
n
Í˙
n
l
=
z
= Í
˙
a
a
a
Í˙
Í
˙
0
Í˙
Í
˙
Í˙
1
Í
˙
Í˙
Í
˙
Î
˚
Î˚
2
h e matrix
R
has the weights and three rows and columns corresponding to a con-
stant trend component (the 1s) and the
x
and
y
coordinates of each location, the
vector (a matrix with only one row or column)
l
has three extra rows, which are
values of
a
k
for the constant (
a
0
) and for the
x
and
y
coordinates of each location (i.e.
a
1
and
a
2
), and the vector
z
includes three zeros, corresponding to the three trend
components.
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