# Biased Kriging: A Theoretical Development

• Carol Spease
• James R. Carr
Part of the Computer Applications in the Earth Sciences book series (CAES)

## Abstract

Kriging was developed to be a best linear unbiased estimator using a theoretical development to assure a minimum variance of estimation error. The Lagrangian function which assures this minimization constrained such that the weights (λ) sum to one (unbiasedness) is
$$L({\lambda _i},\mu ) = {\sigma ^2} - 2\sum\limits_i {{\lambda _i}\sigma \left( {{x_O}{x_i}} \right)} + \sum\limits_i {\sum\limits_j {{\lambda _i}{\lambda _j}} \sigma \left( {{x_O}{x_j}} \right) - 2} \mu \left( {\sum {{\lambda _i} - 1} } \right)$$
(A)
In (A), biasedness can be introduced by changing (∑λ-l) to (∑λ-N), where N is the new sum of weights. Yet, differentiating either equation with respect to λ and ∑μ results in formula
$$\sum\limits_i {\sum\limits_j {{\lambda _i}} \sigma \left( {{x_i}{x_j}} \right) - } \mu = \sum\limits_i {\sigma \left( {{x_O}{x_i}} \right)}$$
(B)
Hence, the same kriging system is used except N is introduced in the right-hand vector instead of 1. This allows each covariance value, σ, in (B) to be computed using a variogram, as with unbiased kriging. Biased kriging is useful for favoring a particular portion of a histogram. By allowing the sum of weights to be greater than one, as an example, the high end of the histogram can be favored.

## Keywords

Lagrangian Function Peak Acceleration Foundation Design Earthquake Resistant Design Reserve Estimation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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