2014/10/28 by Nicolas Brunel, Brunel, Nicolas, Quentin Clairon +1 · 1 citation
Engineering · Decision Sciences · #Advanced Control Systems Optimization #Fault Detection and Control Systems #Advanced Statistical Process Monitoring
paper · pdf · doi:10.48550/arxiv.1410.7554
Ordinary Differential Equations are widespread tools to model chemical,\nphysical, biological process but they usually rely on parameters which are of\ncritical importance in terms of dynamic and need to be estimated directly from\nthe data. Classical statistical approaches (nonlinear least squares, maximum\nlikelihood estimator) can give unsatisfactory results because of computational\ndifficulties and ill-posedness of the statistical problem. New estimation\nmethods that use some nonparametric devices have been proposed to circumvent\nthese issues. We present a new estimator that shares properties with Two-Step\nestimator and Generalized Smoothing (introduced by Ramsay et al, 2007). We\nintroduce a perturbed model and we use optimal control theory for constructing\na criterion that aims at minimizing the discrepancy with data and the model.\nHere, we focus on the case of linear Ordinary Differential Equations as our\ncriterion has a closed-form expression that permits a detailed analysis. Our\napproach avoids the use of a nonparametric estimator of the derivative, which\nis one of the main cause of inaccuracy in Two-Step estimators. Moreover, we\ntake into account model discrepancy and our estimator is more robust to model\nmisspecification than classical methods. The discrepancy with the parametric\nODE model correspond to the minimum perturbation (or control) to apply to the\ninitial model. Its qualitative analysis can be informative for misspecification\ndiagnosis. In the case of well-specified model, we show the consistency of our\nestimator and that we reach the parametric root-n rate when regression splines\nare used in the first step.\n