Statistics C173/C273. Instructor: Nicolas Christou. Ordinary kriging in terms of the covariance function. The model: The
University of California, Los Angeles Department of Statistics Statistics C173/C273
Instructor: Nicolas Christou
Ordinary kriging in terms of the covariance function The model: The model assumption is: Z(s) = µ + δ(s) where δ(s) is a zero mean stochastic term with variogram 2γ(·). The Kriging System The predictor assumption is ˆ 0) = Z(s
n X
wi Z(si )
i=1
It is a weighted average of the sample values, and wi ’s are the weights that will be estimated.
Pn
i=1
wi = 1 to ensure unbiasedness. The
Kriging minimizes the mean squared error of prediction ˆ 0 )]2 min σe2 = E[(Z(s0 ) − Z(s or "
min
σe2
= E (Z(s0 ) −
n X
#2
wi Z(si )
i=1
For second order stationary process the last equation can be written as: σe2 = C(0) − 2
n X
wi C(s0 , si ) +
i=1
n n X X
wi wj C(si , sj )
i=1 j=1
See next page for the proof:
1
(1)
Let’s examine (Z(s0 ) −
Pn
i=1
wi Z(si ))2 : z(s0 ) −
n X
!2
wi z(si ) + µ − µ
=
i=1
(
[z(s0 ) − µ] −
n X
)2
wi [z(si ) − µ]
=
i=1
[z(s0 ) − µ]2 − 2
n X
wi [z(si ) − µ][z(s0 ) − µ] +
i=1
n X n X
wi wj [z(si ) − µ][z(sj ) − µ] .
i=1 j=1
If we take expectations on the last expression we have E [z(s0 ) − µ]2 − 2
n X
wi E [z(si ) − µ][z(s0 ) − µ] +
i=1
n X n X
wi wj E [z(si ) − µ][z(sj ) − µ]
i=1 j=1
The expectations above are the covariances: C(0) − 2
n X
wi C(s0 , si ) +
i=1
n X n X
wi wj C(si , sj )
i=1 j=1
Therefore kriging minimizes σe2 = E[(Z(s0 ) −
n X
wi Z(si )]2 =
i=1
C(0) − 2
n X
wi C(s0 , si ) +
i=1
n X n X
wi wj C(si , sj )
i=1 j=1
subject to n X
wi = 1
i=1
The minimization is carried out over (w1 , w2 , ..., wn ), subject to the constraint Therefore the minimization problem can be written as: min C(0) − 2
n X i=1
wi C(s0 , si ) +
n n X X
wi wj C(si , sj ) − 2λ(
n X
wi − 1)
Pn
i=1
wi = 1.
(2)
i=1
i=1 j=1
where λ is the Lagrange multiplier. After differentiating (2) with respect to w1 , w2 , ..., wn , and λ and set the derivatives equal to zero we find that 2
n X
wj C(si , sj ) − 2C(s0 , si ) − 2λ = 0, i = 1, ..., n
j=1 n X
wj C(si , sj ) − C(s0 , si ) − λ = 0, i = 1, ..., n
j=1
and n X
wi = 1
i=1
2
Using matrix notation the previous system of equations can be written as CW = c Therefore the weights w1 , w2 , ..., wn and the Lagrange multiplier λ can be obtained by W = C−1 c where W = (w1 , w2 , ..., wn , −λ) c = (C(s0 , s1 ), C(s0 , s2 ), ..., C(s0 , sn ), 1)0
C(si , sj ), 1, C= 1, 0,
i = 1, 2, ..., n, j = 1, 2, ..., n, i = n + 1, j = 1, ..., n, j = n + 1, i = 1, ..., n, i = n + 1, j = n + 1.
The variance of the estimator: ˆ 0 ) = Pn wi Z(si ). So far, we found the weights and therefore we can compute the estimator: Z(s i=1 How about the variance of the estimator, namely σe2 ? We multiply n X
wj C(si , sj ) − C(s0 , si ) − λ = 0, i = 1, ..., n
j=1
by wi and we sum over all i = 1, · · · , n to get: n X n X
wi wj C(si , sj ) −
i=1 j=1
n X
wi C(s0 , si ) −
i=1
n X
wi λ = 0
i=1
Therefore, n n X X
wi wj C(si , sj ) =
n X
wi C(s0 , si ) + λ
i=1
i=1 j=1
If we substitute this result into equation (1) we finally get: σe2 = C(0) −
n X
wi C(si , s0 ) + λ
(3)
i=1
3
4
C(s2 , s2 ) ··· .. . C(sn , s2 ) 1
··· .. . C(sn , s1 ) 1
C(s1 , s2 )
C(s2 , s1 )
C(s1 , s1 )
···
··· 1
C(sn , sn )
···
···
C(sn , s3 )
..
···
C(s2 , sn )
C(s1 , sn )
···
.
···
···
.
.. .
..
C(s2 , s3 )
C(s1 , s3 )
0
1
1
1
1
1
w2 .. . .. . wn −λ
w1
=
1
C(s0 , sn )
.. .
.. .
C(s0 , s2 )
C(s0 , s1 )
Again we observe that the matrix C must be positive definite and this ensured by a choice of a model covariance function.
The kriging system in terms of covariance
Short code for ordinary kriging in terms of variogram: a