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Title: Optimisation and selection strategies for parameter estimation in ODE models with generalised smoothing Authors:  Beatrice Laroche - INRA (France) [presenting]
Nicolas Brunel - ENSIIE (France)
Daniel Goujot - INRA (France)
Simon Labarthe - INRA (France)
Abstract: Let $(Y_{ik})_{i=1\ldots n;k=0\ldots K}$ be observations of the coordinates of a $n$-dimensional vector data collected at times $(t_{ik})\in [0,T]$, modelled by a parametric ODE $\frac{dX}{dt}=f(X,\theta)+u$ and an observation equation $Y_{ik}=X_i(t_{ik})+\eta_{ik}$, where $X$ is the $n$-dimensional vector state of the model, $u=\frac{dX}{dt}-f(X,\theta)$ is the model error, $\eta_{ik}=Y_{ik}-X_i(t_{ik})$ are measurement errors and $\theta$ is the parameter. Methods from functional data analysis consist in replacing $X$ by an approximation $\widehat X$ on a functional base (e.g. B-splines). Inspired by one of them, the Generalized Smoothing, the estimation problem can be formulated as the joint estimation of the parameters and the coefficients in the approximation basis (gathered for all coordinates of $\widehat X$ in the vector $C$) as $\min_{\theta,\, C}\left(\sum_{i,k}|Y_{ik}-\widehat X_i(t_{ik})|^2+\frac{\lambda}{T}\int_0^T\|\frac{d\widehat X}{dt}-f(\widehat X,\theta)\|^2dt\right)$ where $\lambda\geq 0$ allows the trade-off between the measurement and model errors. The formulation, properties and numerical resolution of the estimation problem for a fixed $\lambda$ will be discussed, as well as strategies for choosing $\lambda$. The influence of optimisation and hyperparameter selection will be tackled, both on synthetic and real data, on models of interactions networks in ecology or chemistry.