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186 7 Spatial Data
kriging uses a weighted average of the neighboring points to estimate the
value of an unobserved point:
where λ are the weights which have to be estimated. The sum of the weights
ι
should be one to guarantee that the estimates are unbiased:
The expected (average) error of the estimation has to be zero. That is:
where z is the true, but unknown value. After some algebra, using the pre-
x0
ceding equations, we can compute the mean-squared error in terms of the
variogram:
where E is the estimation or kriging variance, which has to be minimized,
γ(x x ) is the variogram (semivariance) between the data point and the un-
0
i,
observed, γ(x x ) is the variogram between the data points x and x , and λ
i, j i j i
and λ are the weights of the ith and jth data point.
j
For kriging we have to minimize this equation (quadratic objective func-
tion) satisfying the condition that the sum of weights should be one (linear
constraint). This optimization problem can be solved using a Lagrange mul-
tiplier ν resulting in the linear kriging system of N+1 equations and N+1
unknowns:
After obtaining the weights λ , the kriging variance is given by
i
The kriging system can be presented in a matrix notation:
