History matching for functional data

Traditional history matching (HM) is widely used as a computationally tractable alternative to Bayesian calibration for ruling out implausible regions of the input space of expensive computer models.

Its standard formulation, however, is primarily tailored to scalar or finite-dimensional vector outputs, whereas observations in many physical systems are naturally functional, taking the form of time series or spatial fields.

We propose Functional History Matching (FHM), an extension of HM to functional data.

FHM defines implausibility through a function-level discrepancy measure together with a scalarized uncertainty term that combines observation error, model discrepancy, and emulator uncertainty.

To quantify functional mismatch, we introduce a Wiener-process-based random projection criterion that is sensitive to differences in overall shape and temporal alignment, and we allow derivative information to be incorporated when additional discriminatory power is required.

For functional emulation, we adopt a multi-output Gaussian process framework implemented through the Outer Product Emulator, which yields predictive means and uncertainty summaries for functional outputs at practical computational cost.

We further motivate threshold selection through a conservative, distribution-free argument based on Chebyshev's inequality.

In a synthetic tsunami forecasting case study, FHM progressively contracts the not-ruled-out-yet region and yields more concentrated downstream coastal predictions than landmark-based HM.