Estimating direction fields in autonomous equation models, with an application to system identification from cross-sectional data

SUMMARY We consider a situation where 'cross-sectional' observations follow an underlying 'longitudinal' model with a population mean function. Measurements are available, made at unknown 'observation times', of the mean function values without noise and of derivatives with noise. The population mean function lies in a nonparametric smoothness class and, while not fully identifiable, is a trajectory in the direction field of an autonomous ordinary differential equation. An efficient reconstruction of that field is proposed which proceeds from nonparametric estimation of the function driving the differential equation. Rates of convergence are investigated. A simulation is included showing the recovery of the mean function. Signal and noise are chosen to be typical for T4-cell counts in a small cohort of undated HIV-seroconverters, and the reconstruction of the mean function is seen to be satisfactory.