Ordinary Differential Equations (ODE) are routinely calibrated on real data for estimating unknown parameters or for reverse-engineering of biological systems. Nevertheless, standard statistical technics can give disappointing results because of the complex relationship between parameters and states, that makes the corresponding estimation problem ill-posed. Moreover, ODE are mechanistic models that are prone to modelling errors, whose influences on inference are often neglected during statistical analysis. We propose a regularised estimation framework that consists in adding a perturbation to the original ODE. This perturbation facilitates data fitting and represents also possible model misspecifications, so that parameter estimation is done by solving a trade-off between data fidelity and model fidelity. We show that the underlying optimisation problem is an optimal control problem, that can be solved by the Pontryagin Maximum Principle for general nonlinear and partially observed ODE. The same methodology can be used for the joint estimation of finite and time-varying parameters. We show, in the case of a well-specified parametric model, that our estimator is consistent and reaches the root-$n$ rate. Numerical experiments considering various sources of model misspecifications shows that Tracking still furnish accurate estimates.
In a second part of the talk, we present a novel algorithm that deals directly with the log-likelihood of the observations and avoid the use of a nonparametric proxy. The inference still uses a perturbed model that is estimated based on the discretisation in time and of the perturbation function (piecewise constant). We focus on linear ODEs and show that we can compute efficiently the parameter estimator by dynamic programming.
The computational speed enables to address the estimation of (relatively) high-dimensional systems, and to implement standard computationally intensive procedure such as cross-validation and bootstrap. Finally, we show that we can use our approach to estimate nonlinear ODEs used for modelling bacteria interactions in microbiome.