Residual Model Criticism for Fitted SDE Implementations

Abstract

Stochastic differential equation (SDE) models can reproduce trajectories while missing temporal dependence. We compare residual-based scores from a held-out path with scores from paths simulated by a fitted implementation, keeping fitted parameters, simulation rules and the scoring rule fixed. We give conditions for Monte Carlo calibration and bound conditional rejection error by the gap between conditional score CDFs. Residuals use zero-noise centres and covariance approximations that track noise propagation over observation intervals; report residuals also include reporting variation. Residual weights affect sensitivity and the influence of fitting error. For a quadratic energy score in a linear example with singular true covariance, whitening with nonsingular fitted covariance matrices leaves a nonvanishing score-CDF gap as nonzero coefficient estimates approach the truth. A fixed positive-definite weighting matrix makes the gap vanish. In a controlled SIR experiment using reference paths simulated at fitted parameters, adding susceptible-state endpoints to training or downweighting a low-variance residual reduces null rejection. A separate comparison using reference paths simulated at the true parameter shows reduced sensitivity to added susceptible-coordinate noise after downweighting. An exploratory weekly-report application incorporates reporting and reconstruction into reference simulations, revealing excess correlation between squared residuals in Czechia and persistently negative residuals in Finland.