Autoregressive Co-Kriging
Autoregressive co-kriging couples Gaussian processes across fidelity levels so a few accurate runs correct many cheap ones.
Model structure
Kennedy and O'Hagan's autoregressive scheme writes the high-fidelity response as f_hi(x) = rho * f_lo(x) + delta(x), where f_lo is a Gaussian process fit to the low-fidelity data, rho is a scaling coefficient, and delta is an independent Gaussian process capturing the discrepancy between levels. The construction generalizes to any number of levels by chaining: each level is a scaled version of the one below plus its own correction.
The recursion
For levels t = 1..s, f_t(x) = rho_{t-1} f_{t-1}(x) + delta_t(x). Under the Markov assumption that, given f_{t-1} at x, nothing more about f_t can be learned from cheaper levels, the posterior factorizes level by level. Le Gratiet showed this recursion lets each Gaussian process be trained independently, avoiding a single large covariance matrix and greatly reducing computational burden.
Estimating parameters
- Fit the lowest level GP to its data by maximum likelihood
- Regress the next level's data on the predicted lower-level mean to estimate rho
- Fit the discrepancy GP delta to the residuals
- Propagate predictive means and variances upward
Design of experiments
Co-kriging is most efficient with nested designs, where high-fidelity sample locations are a subset of low-fidelity ones. Nesting lets the discrepancy be evaluated exactly at shared points. Non-nested designs are usable but complicate the covariance bookkeeping.
Predictive variance
The posterior variance accumulates contributions from each level, so it correctly widens where high-fidelity data are sparse even if low-fidelity coverage is dense. This is the property that makes co-kriging honest: cheap data alone never collapses uncertainty about the expensive truth. Reported predictions should always carry this variance, and analysts should check that the estimated rho is stable, since a poorly identified rho signals weak cross-fidelity correlation.