Identification and curvature bias for shared inputs in multichannel dynamical models

Abstract

Pooling many channels with a shared unknown input can leave persistent bias when each channel retains a fixed number of observations and its own incidental parameters. For heterogeneous smooth channel manifolds under isotropic noise, the second-order bias is an information-weighted regression of mean curvature on normal input velocity. A feasible correction gives centred Gaussian inference as the noise level $σ$ decreases with $Rσ^6\to0$, compared with $Rσ^2\to0$ without correction. We extend the stochastic expansion and studentization to deterministic channel arrays with densely correlated Gaussian errors of bounded covariance operator norm. General marginal covariance adds a forcing term that can bias even flat channels, while cross-channel covariance changes uncertainty. For Gaussian errors with a separable frame-by-channel covariance whose channel factor is autoregressive in a known ordering, an independent zero-mean calibration field yields a covariance estimator accurate enough for corrected inference. Boundary accumulation and fixed working population models introduce distinct bias terms; rational-multiplier and zero-count arguments connect identification to frame-integrated compartment models. Simulations test both oracle and estimated-covariance inference, including independent PET replications. Moderate-noise undercoverage, optimization ambiguity and the need to validate reconstruction covariance limit practical inference. The real-data example supports sensitivity analysis, not clinically calibrated intervals.