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University of California, Los Angeles
Department of Statistics
Statistics C173/C273
Instructor: Nicolas Christou
Cokriging
Suppose that at each spatial location
s
i
,i
= 1
,...,n
we observe
k
variables as follows:
Z
1
(
s
1
)
Z
1
(
s
2
)
...
Z
1
(
s
n
)
Z
2
(
s
1
)
Z
2
(
s
2
)
Z
2
(
s
n
)
.
.
.
.
.
.
.
.
.
.
.
.
Z
k
(
s
1
)
Z
k
(
s
2
)
Z
k
(
s
n
)
We want to predict
Z
1
(
s
0
), i.e. the value of variable
Z
1
at location
s
0
.
This situation that the variable under consideration (the target variable) occurs with other vari
ables (colocated variables) arises many times in practice and we want to explore the possibility
of improving the prediction of variable
Z
1
by taking into account the correlation of
Z
1
with these
other variables. For example, the prediction of lead can be improved if we also know the values of
zinc at each spatial location. The value of lead will be predicted by the observed values of lead but
also by the observed values of zinc.
The predictor assumption:
ˆ
Z
1
(
s
0
) =
k
X
j
=1
n
X
i
=1
w
ji
Z
j
(
s
i
)
=
w
11
z
1
(
s
1
) +
w
12
z
1
(
s
2
) +
+
w
1
n
z
1
(
s
n
)
+
w
21
z
2
(
s
1
) +
w
22
z
2
(
s
2
) +
+
w
2
n
z
2
(
s
n
)
+
.
.
.
+
.
.
.
+
+
.
.
.
+
w
k
1
z
k
(
s
1
) +
w
k
2
z
k
(
s
2
) +
+
w
kn
z
k
(
s
n
)
We see that there are weights associated with variable
Z
1
but also with each one of the other
variables. We will examine ordinary cokriging, which means that
E
(
Z
j
(
s
i
)) =
μ
j
for all
j
and
i
.
In vector form:
E
(
Z
(
s
)) =
E
(
Z
1
(
s
))
E
(
Z
2
(
s
))
.
.
.
.
.
.
E
(
Z
k
(
s
)
=
μ
1
μ
2
.
.
.
.
.
.
μ
k
=
μ
.
1
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View Full Document We want the predictor
ˆ
Z
1
(
s
0
) to be unbiased, that is
E
±
ˆ
Z
1
(
s
0
)
²
=
μ
1
:
E
±
ˆ
Z
1
(
s
0
)
²
=
k
X
j
=1
n
X
i
=1
w
ji
E
(
Z
j
(
s
i
))
=
w
11
E
(
z
1
(
s
1
)) +
w
12
E
(
z
1
(
s
2
)) +
...
+
w
1
n
E
(
z
1
(
s
n
))
+
w
21
E
(
z
2
(
s
1
)) +
w
22
E
(
z
2
(
s
2
)) +
+
w
2
n
E
(
z
2
(
s
n
))
+
.
.
.
+
.
.
.
+
+
.
.
.
+
w
k
1
E
(
z
k
(
s
1
)) +
w
k
2
E
(
z
k
(
s
2
)) +
+
w
kn
E
(
z
k
(
s
n
))
=
n
X
i
=1
w
1
i
μ
1
+
n
X
i
=1
w
2
i
μ
2
+
+
n
X
i
=1
w
ki
μ
k
=
μ
1
.
Therefore, we must have the following set of constraints:
n
X
i
=1
w
1
i
= 1
n
X
i
=1
w
2
i
= 0
.
.
.
.
.
.
.
.
.
n
X
i
=1
w
ki
= 0
As with the other forms of kriging, cokriging minimizes the mean squared error of prediction
(MSE):
min σ
2
e
=
E
[
Z
(
s
0
)

ˆ
Z
(
s
0
)]
2
or
min σ
2
e
=
E
Z
(
s
0
)

k
X
j
=1
n
X
i
=1
w
ji
Z
j
(
s
i
)
2
subject to the constraints:
n
X
i
=1
w
1
i
= 1
n
X
i
=1
w
2
i
= 0
.
.
.
.
.
.
.
.
.
n
X
i
=1
w
ki
= 0
2
For smplicity, lets assume
k
= 2, in other words, we observe variables
Z
1
and
Z
2
and we want to
predict
Z
1
. Therefore,
min σ
2
e
=
E
"
Z
(
s
0
)

n
X
i
=1
w
1
i
Z
1
(
s
i
)

n
X
i
=1
w
2
i
Z
2
(
s
i
)
#
2
Let’s add the following quantities:

μ
1
+
μ
1
+
∑
n
i
=1
w
2
i
μ
2
:
min σ
2
e
=
E
"
Z
(
s
0
)

n
X
i
=1
w
1
i
Z
1
(
s
i
)

n
X
i
=1
w
2
i
Z
2
(
s
i
)

μ
1
+
μ
1
+
n
X
i
=1
w
2
i
μ
2
#
2
or
min σ
2
e
=
E
"
(
Z
(
s
0
)

μ
1
)

n
X
i
=1
w
1
i
[
Z
1
(
s
i
)

μ
1
]

n
X
i
=1
w
2
i
[
Z
2
(
s
i
)

μ
2
]
#
2
We complete the square above to get:
[
Z
(
s
0
)

μ
1
]
2

2
n
X
i
=1
w
1
i
[
Z
1
(
s
0
)

μ
1
][
Z
1
(
s
i
)

μ
1
]

2
n
X
i
=1
w
2
i
[
Z
1
(
s
0
)

μ
1
][
Z
2
(
s
i
)

μ
2
]
+
n
X
i
=1
n
X
j
=1
w
1
i
w
1
j
[
Z
1
(
s
i
)

μ
1
][
Z
1
(
s
j
)

μ
1
]
+
n
X
i
=1
n
X
j
=1
w
2
i
w
2
j
[
Z
2
(
s
i
)

μ
2
][
Z
2
(
s
j
)

μ
2
]
+ 2
"
n
X
i
=1
w
1
i
[
Z
1
(
s
i
)

μ
1
]
#"
n
X
i
=1
w
2
i
[
Z
2
(
s
i
)

μ
2
]
#
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This note was uploaded on 02/11/2012 for the course STATS c173/c273 taught by Professor Nicolaschristou during the Spring '11 term at UCLA.
 Spring '11
 NicolasChristou
 Statistics

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