mixle.doe.calibrate module

Kennedy-O’Hagan calibration: infer a simulator’s parameters with an explicit model-discrepancy term.

Field data rarely equals the simulator even at the true parameters – there is model-form error. Fitting parameters by plain least squares absorbs that bias and gives wrong (over-confident) parameters. The Kennedy-O’Hagan model writes y(x) = eta(x, theta) + delta(x) + noise with delta a GP discrepancy, and infers theta and delta jointly, so the parameters are not contaminated by the bias.

calibrate(simulator, x, y, theta0, *, discrepancy=True, discrepancy_lengthscale=None, seed=0, max_iter=300)[source]

Calibrate simulator(x, theta) to field data (x, y) with a GP discrepancy term.

Maximizes the marginal likelihood of the residual r(theta) = y - eta(x, theta) under a GP + noise model, over theta and the discrepancy amplitude + noise. discrepancy=False drops the GP (plain nonlinear least squares) – useful to show the bias the discrepancy removes.

The discrepancy correlation length is fixed (discrepancy_lengthscale, default 10% of the input domain) rather than fitted: this is the standard resolution of the Kennedy-O’Hagan theta/delta identifiability problem – a short discrepancy length forces the GP to model only local model-form error, leaving the smooth global trend to the parametric simulator so theta stays identifiable. Set it to the scale of model error you expect.

Parameters:

  • simulator (Callable[[ndarray, ndarray], ndarray]) – eta(x, theta) -> predictions (vectorized over the rows of x).

  • x (ndarray) – field inputs and observations.

  • y (ndarray) – field inputs and observations.

  • theta0 (Sequence[float]) – initial calibration parameters (its length sets the parameter count).

  • discrepancy (bool) – include the GP discrepancy term (the Kennedy-O’Hagan model).

  • discrepancy_lengthscale (float | None) – fixed GP correlation length (default: 10% of the input domain).

  • seed (int)

  • max_iter (int)

Return type:

KOCalibration

class KOCalibration(theta, ls, amp, noise, simulator, x, y)[source]

Bases: object

Result of calibrate(): the fitted parameters, discrepancy GP, and a calibrated predictor.

predict(x_new, *, with_discrepancy=True)[source]

Calibrated prediction at x_new: simulator at the fitted theta, plus the GP discrepancy (the bias-corrected estimate of reality) unless with_discrepancy=False (the pure simulator).

Parameters:
Return type:

ndarray