Affine control-variate bias correction across fidelities

Debiases a cheap-fidelity output ensemble against a sparse expensive sample, per observation dimension. For each observation j it fits the first-order autoregressive control variate (the linear term of the Kennedy-O'Hagan multi-fidelity model) high_j ~ a_j + b_j * low_j on the paired subset, with b_j = cov(high_j, low_j) / var(low_j) the variance-minimising coefficient, then predicts the corrected high-fidelity output for every realisation as a_j + b_j * low_all_j.

Usage

mf_control_variate(low_all, high_sub, low_sub)

Arguments

low_all: Numeric matrix, nreal x nobs. Cheap-fidelity output for every realisation.
high_sub: Numeric matrix, nsub x nobs. Expensive-fidelity output for the paired subset.
low_sub: Numeric matrix, nsub x nobs. Cheap-fidelity output for the same subset, row-aligned with high_sub.

Value

A nreal x nobs matrix of bias-corrected outputs, with attributes "intercept" (a_j), "slope" (b_j), and "subset_cor" (per-dimension subset correlation; NA for a degenerate dimension).

Details

The estimator degrades gracefully: where the cheap output has zero variance on the subset it falls back to the expensive subset mean (b_j = 0), and where the two fidelities are weakly correlated the correction shrinks toward that mean rather than amplifying noise.

This is the plug-in primitive for surrogate cascades: a cascade runs the cheap level over the full ensemble, the expensive level over a chosen subset, and calls this to lift the cheap ensemble toward the expensive one at a fraction of the cost.

References

Kennedy, M. C. & O'Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87(1), 1–13.

Examples

set.seed(1L)
low_all  <- matrix(rnorm(40L), ncol = 2L)
sub      <- 1:5
low_sub  <- low_all[sub, , drop = FALSE]
high_sub <- 0.3 + 1.2 * low_sub + matrix(rnorm(10L, sd = 0.01), ncol = 2L)
corrected <- mf_control_variate(low_all, high_sub, low_sub)
attr(corrected, "slope")
#> [1] 1.200315 1.200632