Co-Kriging
Co-kriging fuses a primary variable with correlated secondary data or a lower-fidelity model to sharpen predictions and their uncertainty.
Borrowing strength across variables
Ordinary kriging predicts one variable from its own samples. Co-kriging extends this to predict a primary variable using not only its own data but also a correlated secondary variable that is cheaper or more abundant. When the two are correlated, the secondary data sharpens the prediction of the primary, especially where primary samples are sparse.
The multi-fidelity use
The most common engineering use casts fidelity levels as the two variables: a low-fidelity model provides the abundant secondary data, a high-fidelity model the scarce primary data. Autoregressive co-kriging models the high-fidelity output as a scaling of the low-fidelity output plus an independent Gaussian-process discrepancy, so many cheap runs and few expensive ones combine into one accurate surrogate.
The cross-covariance
Co-kriging requires modeling not just each variable's own covariance but the cross-covariance between them. This joint covariance structure must be valid - positive definite - which constrains the models. The linear model of coregionalization and the autoregressive scheme are standard ways to build admissible cross-covariances.
Benefits
- More accurate prediction of the primary variable from fewer expensive samples
- Uncertainty estimates that reflect information from both sources
- A principled way to fuse a hierarchy of model fidelities
- Natural extension to more than two levels of fidelity
Cautions
The benefit depends entirely on genuine correlation between the sources; a weakly correlated secondary variable helps little and can even mislead if the cross-covariance is misspecified. Fitting the joint model is harder than fitting a single kriging model, and it inherits kriging's cubic scaling with sample count.
In practice
Co-kriging is a core tool for multi-fidelity surrogate modeling. In Kronos work it fuses coarse and fine simulations of the machines so that a design surrogate reaches high-fidelity accuracy while spending most of its budget on cheap runs, with uncertainty that reflects both fidelity levels.