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Geography / Grid maps and scale / Graduate / gge02

Ordinary Kriging as Constrained Prediction

Derive two-site kriging weights from a stated covariance model.

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5 questions0m 0s

Prerequisites

Covariance matrices, quadratic minimization, spatial prediction, and unbiasedness.

Learn the skill

With an unknown constant mean, an unbiased linear predictor w^Tz requires weights summing to one. Minimize prediction-error variance C00-2w^Tc+w^TCw subject to that constraint. The weights depend on covariance, not only distance or map appearance.

Worked example

If two sites have symmetric covariance with the target and each other, the constrained optimum can give equal weights. Asymmetric target covariance can shift the optimum even if observed values happen to be equal.

Model and Assumptions

Invented covariance model: observation covariance C=[[4,1],[1,4]], target covariances c=(2,1), target variance C00=4. Both observations and target share an unknown constant mean. Observed values are z=(10,16). Assume this covariance is known and valid.

1. Which constraint makes the predictor unbiased for every constant mean?
2. What is the minimizing first weight w1?
3. What predicted value follows from those weights?
4. What is the model-based prediction-error variance?
5. Does this variance automatically include uncertainty from fitting C to data?

Further inquiry

Derive the constrained solution using both substitution and a Lagrange multiplier. Compare its variance with equal weighting, then describe how trend misspecification or uncertain covariance parameters would change your assessment.

Review criteria

  • Recover weights (2/3,1/3) and prediction 12.
  • Compare variance 10/3 with equal-weight variance 3.5.
  • Distinguish model-based error variance from realized error and omitted uncertainty.